# Uniform Treatment of Jensen’s Inequality by Montgomery Identity

Authors: Tahir Rasheed; Saad Ihsan Butt; Đilda Pečarić; Josip Pečarić; Ahmet Ocak Akdemir
Journal: Journal of Mathematics
 (2021)

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## Abstract

We generalize Jensen’s integral inequality for real Stieltjes measure by using Montgomery identity under the effect ofn−convex functions; also, we give different versions of Jensen’s discrete inequality along with its converses for real weights. As an application, we give generalized variants of Hermite–Hadamard inequality. Montgomery identity has a great importance as many inequalities can be obtained from Montgomery identity in q−calculus and fractional integrals. Also, we give applications in information theory for our obtained results, especially for Zipf and Hybrid Zipf–Mandelbrot entropies.


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## Body

## 1. Introduction

Convex functions have a great importance in mathematical inequalities, and the well-known Jensen’s inequality is the characterization of convex functions. Jensen’s inequality for differentiable convex functions plays a significant role in the field of inequalities as several other inequalities can be seen as special cases of it. One can find the application of Jensen’s discrete inequality in discrete-time delay systems in [1].Taking into consideration the tremendous applications of Jensen’s inequality in various fields of mathematics and other applied sciences, the generalizations and improvements of Jensen’s inequality have been a topic of supreme interest for the researchers during the last few decades as evident from a large number of publications on the topic (see [2–4] and the references therein).The well-known Jensen’s inequality asserts that for the functionΓ it holds that(1)Ψ1Pm∑J=1mpJxJ≤1Pm∑J=1mpJΨxJ,if Ψ is a convex function on interval I⊂ℝ, where pJ are positive real numbers and xJ∈IJ=1,…,m, while Pm=∑J=1mpJ.However, the well-known integral analogue of Jensen’s inequality is as follows.Theorem 1.
Letℏ:a,b⟶α,β be a continuous function and λ:a,b⟶ℝ be an increasing and bounded function with λa≠λb. Then, for every continuous convex function Ψ:α,β⟶ℝ, the following inequality holds:(2)Ψℏ˜≤∫abΨℏζdλζ∫abdλζ,where(3)ℏ˜=∫abℏζdλζ∫abdλζ∈α,β.There are several inequalities coming from Jensen’s inequality both in integral and discrete cases which can be obtained by varying conditions on the functionℏ and measure λ defined in Theorem 1.Montgomery identity is used in quantum calculus orq−calculus. There are different identities of Montgomery, and several inequalities of Ostrowski type were formulated by using these identities. Budak and Sarikaya established the generalized Montgomery-type identities for differential mappings in [5]. Applications of Montgomery identity can be found in fractional integrals as well as in quantum integral operators. Here we utilize Montgomery’s identity for the generalization of Jensen’s inequality. In [6], Cerone and Dragomir developed a systematic study which produced some novel inequalities. Several interesting results related to inequalities and different types of convexity can be found in [7–21]. The class of convex functions is a very useful concept that has become a focus of interest for researchers in statistics, convex programming, and many other applied disciplines, as well as in inequality theory. The readers can find some motivated findings related to convex functions and some new integral inequalities in [22–27].In [28], Khan et al. have mentioned about n-convex functions as follows.Definition 1.
A functionf:I⟶ℝ is called convex of order n or n-convex if for all choices of n+1 distinct points xi,…,xi+n we have Δnfxi≥0.
Ifn-th order derivative fn exists, then f is n−convex if and only if fn≥0. For 1≤k≤n−2, a function f is n−convex if and only if fk exists and is n−k−convex.
In the present paper, we will use Montgomery identity that is presented as following.Theorem 2.
Letn∈ℕ, Ψ:I⟶ℝ be such that Ψn−1 is absolutely continuous, I⊂ℝ is an open interval, and α,β∈I, α<β. Then, the following identity holds:(4)Ψx=1β−α∫αβΨtdt+∑ℓ=0n−2Ψℓ+1αℓ!ℓ+2x−αℓ+2β−α−∑ℓ=0n−2Ψℓ+1βℓ!ℓ+2x−βℓ+2β−α+1n−1!∫αβRnx,sΨnsds,where(5)Rnx,s=−x−snnβ−α+x−αβ−αx−sn−1,α≤s≤x,−x−snnβ−α+x−ββ−αx−sn−1,x<s≤β.


## 2. Generalization of Jensen’s Integral Inequality by Using Montgomery Identity

Before giving our main results, we consider the following assumptions that we use throughout our paper:A1 Let ℏ:a,b⟶ℝ be continuous function.A2 Let λ:a,b⟶ℝ be a continuous function or the functions of bounded variation such that λa≠λb.


### 2.1. New Generalization of Jensen’s Integral Inequality

In our first main result, we employ Montgomery identity to obtain the following real Stieltjes measure’s theoretical representations of Jensen’s inequality.Theorem 3.
Letg,λ be as defined in A1,A2 such that ℏa,b⊂α,β. Also, let Ψ:α,β⟶ℝ be such that for n≥1, Ψn−1 is absolutely continuous. If Ψ is n−convex such that(6)Rnℏ˜,s≤∫abRnℏζ,sdλζ∫abdλζ,s∈α,β,with ℏ˜ and Rnx,s as defined in (3) and (5), respectively, then we have(7)Ψℏ˜−∫abΨℏζdλζ∫abdλζ≤∑ℓ=0n−21ℓ!ℓ+2β−α×Ψℓ+1αℏ˜−αℓ+2−∫abℏζ−αℓ+2dλζ∫abdλζ−Ψℓ+1βℏ˜−βℓ+2+∫abℏζ−βℓ+2dλζ∫abdλζ.Proof.
AsΨn−1 is absolutely continuous for n≥1, we can use the representation of Ψ using Montgomery identity (4) and can calculate(8)Ψℏ˜=1β−α∫αβΨζdζ+∑ℓ=0n−2Ψℓ+1αℓ!ℓ+2ℏ˜−αℓ+2β−α−∑ℓ=0n−2Ψℓ+1βℓ!ℓ+2ℏ˜−βℓ+2β−α+1n−1!∫αβRnℏ˜,sΨnsds.
The integration of the composition of functionsΨ∘ℏ for the real measure λ on a,b gives(9)∫abΨℏζdλζ∫abdλζ=1β−α∫αβΨζdζ+∑ℓ=0n−2Ψℓ+1αℓ!ℓ+2∫abℏζ−αℓ+2dλζβ−α∫abdλζ−∑ℓ=0n−2Ψℓ+1βℓ!ℓ+2∫abℏζ−βℓ+2dλζβ−α∫abdλζ+1n−1!∫αβRnℏζ,sΨnsds.
Now computing the differenceΨℏ˜−∫abΨℏζdλζ/∫abdλζ, we get the following generalized identity involving real Stieltjes measure:(10)Ψℏ˜−∫abΨℏζdλζ∫abdλζ=∑ℓ=0n−21ℓ!ℓ+2β−α×Ψℓ+1αℏ˜−αℓ+2−∫abℏζ−αℓ+2dλζ∫abdλζ−Ψℓ+1βℏ˜−βℓ+2+∫abℏζ−βℓ+2dλζ∫abdλζ+1n−1!∫αβRnℏ˜,s−∫abRnℏζ,sdλζ∫abdλζΨnsds.GI.1.
Finally, by our assumption,Ψn−1 is absolutely continuous on α,β; as a result, Ψn exists almost everywhere. Moreover, Ψ is supposed to be n−convex, so we have Ψnx≥0 almost everywhere on α,β. Therefore, by taking into account the last term in generalized identity GI.1 and integral analogue of Jensen’s inequality that is given in (6), we get (7).In the later part of this section, we will vary our conditions on functionsg and Stieltjes measure dλ to obtain generalized variants of Jensen–Steffensen, Jensen–Boas, Jensen–Brunk, and Jensen-type inequalities. We start with the following generalization of Jensen–Steffensen inequality for n−convex functions.Theorem 4.
LetΨ defined in Theorem 3 be n−convex and ℏ defined in M1 be monotonic. Then, the following results hold.(i)
Ifλ defined in M2 satisfies(11)λa≤λx≤λb,∀x∈a,b,λb>λa,then for evenn≥3, (6) is valid.(ii)
Moreover, if (6) is valid and the function(12)Hx≔∑ℓ=0n−21ℓ!ℓ+2β−αΨℓ+1αx−αℓ+2−Ψℓ+1βx−βℓ+2is convex, then we get inequality (2) which is called generalized Jensen–Steffensen inequality for n−convex function.Proof.
(i)
By applying second derivative test, we can show that the functionRnx,s is convex for even n>3. Now using the assumed conditions, one can employ Jensen–Steffensen inequality given by Boas (see [29] or [30], p. 59) for convex function Rnx,s to obtain (6).(ii)
Since we can rewrite the R.H.S. of (7) in the difference(13)Hℏ˜−∫abHℏζdλζ∫abdλζ,for convex function H and by our assumed conditions on functions ℏ and λ, this difference is non-positive by using Jensen–Steffensen inequality difference [29]. As a result, the R.H.S. of inequality (7) is non-positive and we get generalized Jensen–Steffensen inequality (2) for n−convex function.Now, we give similar results related to Jensen–Boas inequality [30], p. 59], which is a generalization of Jensen–Steffensen inequality.Corollary 1.
LetΨ defined in Theorem 3 be n−convex function. Also, let ℏ be as defined in M1 with a=y0<y1<…<yk<…<ym−1<ym=b and ℏ be monotonic in each of the m intervals yk−1,yk. Then, the following results hold.(i)
Ifλ as defined in M2 satisfies(14)λa≤λx1≤λy1≤λx2≤λy2≤…≤λym−1≤λxm≤λb,∀xk∈yk−1,yk and λb>λa, then for even n≥3, (6) is valid.(ii)
Moreover, if (6) is valid and the function H⋅ defined in (18) is convex, then again inequality (2) holds and is called Jensen–Boas inequality for n−convex function.Proof.
We follow the similar argument as in the proof of Theorem4, but under the conditions of this corollary, we utilize Jensen–Boas inequality (see [29] or [24], p. 59) instead of Jensen–Steffensen inequality.
Next, we give results for Jensen–Brunk inequality.Corollary 2.
LetΨ defined in Theorem 3 be n−convex and ℏ defined in M1 be an increasing function. Then, the following results hold.(i)
Ifλ defined in M2 with λb>λa and(15)∫axℏx−ℏζdλζ≥0,and(16)∫xbℏx−ℏζdλζ≤0,∀x∈a,b holds, then for even n≥3, (6) is valid.(ii)
Moreover, if (6) is valid and the function H⋅ defined in (18) is convex, then again inequality (2) holds and is called Jensen–Brunk inequality for n−convex function.Proof.
We proceed with the similar idea as in the proof of Theorem4, but under the conditions of this corollary, we employ Jensen–Brunk inequality (see [31] or [30], p. 59]) instead of Jensen–Steffensen inequality.Remark 1.
The similar result in Corollary2 is also valid provided that the function ℏ is decreasing. Also, assuming that the function ℏ is monotonic, one can replace the conditions in Corollary 2i by(17)0≤∫axℏx−ℏζdλζ≤∫xbgx−ℏζdλζ.Remark 2.
It is interesting to see that by employing similar method as in Theorem4, we can also get the generalization of classical Jensen’s inequality (2) for n−convex functions by assuming the functions ℏ and λ along with the respective conditions in Theorem 1.
Another important consequence of Theorem3 can be given by setting the function ℏ as ℏζ=ζ. This form is the generalized version of L.H.S. inequality of the Hermite-Hadamard inequality.Corollary 3.
Letλ:a,b⟶ℝ be a function of bounded variation such that λa≠λb with a,b⊂α,β and ζ˜=∫abζdλζ/∫abdλζ∈α,β. Under the assumptions of Theorem 3, if Ψ is n−convex such that(18)Rnℏ˜,s≤∫abRnℏζ,sdλζ∫abdλζ,s∈α,β,then we have(19)Ψζ˜≤∫abΨζdλζ∫abdλζ+∑ℓ=0n−21ℓ!ℓ+2β−α×Ψℓ+1αζ˜−αℓ+2−∫abζ−αℓ+2dλζ∫abdλζ−Ψℓ+1βζ˜−βℓ+2+∫abζ−βℓ+2dλζ∫abdλζ.
If the inequality (18) holds in reverse direction, then (19) also holds reversely.The special case of above corollary can be given in the form of following remark.Remark 3.
It is interesting to see that substitutingλζ=ζ gives ∫abdλζ=b−a and ζ˜=a+b/2. Using these substitutions in (2) and by following remark (20), we get the L.H.S. inequality of renowned Hermite–Hadamard inequality for n−convex functions.


### 2.2. New Generalization of Converse of Jensen’s Integral Inequality

In this section, we give the results for the converse of Jensen’s inequality to hold, giving the conditions on the real Stieltjes measuredλ, such that λa≠λb, allowing that the measure can also be negative, but employing Montgomery identity.To start with we need the following assumption for the results of this section:A3 Let m,M∈α,βm≠M be such that m≤ℏζ≤M for all ζ∈a,b where ℏ is defined in A1.For a given functionΨ:α,β⟶ℝ, we consider the difference(20)CJΨ,ℏm,M;λ=∫abΨℏζdλζ∫abdλζ−M−ℏ˜M−mΨm−ℏ˜−mM−mΨM,where ℏ˜ is defined in (3).Using Montgomery identity, we obtain the following representation of the converse of Jensen’s inequality.Theorem 5.
Letℏ,λ be as defined in A1,A2 and let Ψ:α,β⟶ℝ be such that for n≥1, Ψn−1 is absolutely continuous. If Ψ is n−convex such that(21)CJRnx,s,ℏm,M;λ≤0,s∈α,β,or(22)∫abRnℏζ,sdλζ∫abdλζ≤M−ℏ˜M−mRnm,s+ℏ˜−mM−mRnM,s,s∈α,β,then we get the following extension of the converse of Jensen’s difference:(23)∫abΨℏζdλζ∫abdλζ≤M−ℏ˜M−mΨm+ℏ˜−mM−mΨM+∑ℓ=0n−21ℓ!ℓ+2β−α×Ψℓ+1αCJx−αℓ+2,ℏm,M;λ−Ψℓ+1βCJx−βℓ+2,ℏm,M;λ,where Rn⋅,s is defined in (5).Proof.
AsΨn−1 is absolutely continuous for n≥1, we can use the representation of Ψ using Montgomery identity (4) in the difference CJΨ,ℏm,M;λ:(24)CJΨ,ℏm,M;λ=CJ1β−α∫αβΨζdζ,ℏm,M;λ+∑ℓ=0n−21ℓ!ℓ+2β−αΨℓ+1αCJx−αℓ+2,ℏm,M;λ−∑ℓ=0n−21ℓ!ℓ+2β−αΨℓ+1βCJx−βℓ+2,ℏm,M;λ+1n−1!∫αβCJRnx,s,ℏm,M;λΨnsds.
After simplification and following the fact thatCJΨ,ℏm,M;λ is zero for Ψ to be constant or linear, we get the following generalized identity:(25)CJΨ,ℏm,M;λj=∑ℓ=0n−2.ℓ!ℓ+2β−α×Ψℓ+1αCJx−αℓ+2,ℏm,M;λ−Ψℓ+1βCJx−βℓ+2,ℏm,M;λ+1n−1!∫αβCJRnx,s,ℏm,M;λΨnsds.CGI.1.
Now using characterizations ofn−convex functions like in the proof of Theorem 3, we get (23).The next result gives converse of Jensen’s inequality for higher-order convex functions.Theorem 6.
LetΨ defined in Theorem 5 be n−convex and ℏ be as defined in A3. Then, the following results hold.(i)
Ifλ is non-negative measure on a,b, then for even n≥3, (22) is valid.(ii)
Moreover, if (22) is valid and the function H⋅ defined in (12) is convex, then we get the following inequality for n−convex function to be valid:(26)∫abΨℏζdλζ∫abdλζ≤M−ℏ˜M−mΨm−ℏ˜−mM−mΨM.Proof.
The idea of the proof is similar to that of (6), but we use converse of Jensen’s inequality (see [32] or [30], p. 98).


### 2.3. Applications of Jensen’s Integral Inequality

In this section, we give applications of Jensen’s integral inequality.Another important consequence of Theorem3 is by setting the function ℏ as ℏζ=ζ gives generalized version of L. H. S. inequality of the Hermite–Hadamard inequality.Corollary 4.
Letλ:a,b⟶ℝ be a function of bounded variation such that λa≠λb with a,b⊂α,β and ζ˜=∫abζdλζ/∫abdλζ∈α,β. Under the assumptions of Theorem 5, if Ψ is n−convex such that(27)∫abRnζ,sdλζ∫abdλζ≤b−ζ˜b−aRna,s+ζ˜−ab−aRnb,s,s∈α,β,then we have(28)∫abΨζdλζ∫abdλζ≤b−ζ˜b−aΨa+ζ˜−ab−aΨb+∑ℓ=0n−2.ℓ!ℓ+2β−α×Ψℓ+1αCJx−αℓ+2,ida,b;λ−Ψℓ+1βCJx−βℓ+2,ida,b;λ.
If the inequality (27) holds in reverse direction, then (28) also holds reversely.The special case of above corollary can be given in the form of following remark.Remark 4.
It is interesting to see that substitutingλζ=ζ and by following Theorem 6, we get the R.H.S. inequality of renowned Hermite–Hadamard inequality for n−convex functions.


## 2.1. New Generalization of Jensen’s Integral Inequality

In our first main result, we employ Montgomery identity to obtain the following real Stieltjes measure’s theoretical representations of Jensen’s inequality.Theorem 3.
Letg,λ be as defined in A1,A2 such that ℏa,b⊂α,β. Also, let Ψ:α,β⟶ℝ be such that for n≥1, Ψn−1 is absolutely continuous. If Ψ is n−convex such that(6)Rnℏ˜,s≤∫abRnℏζ,sdλζ∫abdλζ,s∈α,β,with ℏ˜ and Rnx,s as defined in (3) and (5), respectively, then we have(7)Ψℏ˜−∫abΨℏζdλζ∫abdλζ≤∑ℓ=0n−21ℓ!ℓ+2β−α×Ψℓ+1αℏ˜−αℓ+2−∫abℏζ−αℓ+2dλζ∫abdλζ−Ψℓ+1βℏ˜−βℓ+2+∫abℏζ−βℓ+2dλζ∫abdλζ.Proof.
AsΨn−1 is absolutely continuous for n≥1, we can use the representation of Ψ using Montgomery identity (4) and can calculate(8)Ψℏ˜=1β−α∫αβΨζdζ+∑ℓ=0n−2Ψℓ+1αℓ!ℓ+2ℏ˜−αℓ+2β−α−∑ℓ=0n−2Ψℓ+1βℓ!ℓ+2ℏ˜−βℓ+2β−α+1n−1!∫αβRnℏ˜,sΨnsds.
The integration of the composition of functionsΨ∘ℏ for the real measure λ on a,b gives(9)∫abΨℏζdλζ∫abdλζ=1β−α∫αβΨζdζ+∑ℓ=0n−2Ψℓ+1αℓ!ℓ+2∫abℏζ−αℓ+2dλζβ−α∫abdλζ−∑ℓ=0n−2Ψℓ+1βℓ!ℓ+2∫abℏζ−βℓ+2dλζβ−α∫abdλζ+1n−1!∫αβRnℏζ,sΨnsds.
Now computing the differenceΨℏ˜−∫abΨℏζdλζ/∫abdλζ, we get the following generalized identity involving real Stieltjes measure:(10)Ψℏ˜−∫abΨℏζdλζ∫abdλζ=∑ℓ=0n−21ℓ!ℓ+2β−α×Ψℓ+1αℏ˜−αℓ+2−∫abℏζ−αℓ+2dλζ∫abdλζ−Ψℓ+1βℏ˜−βℓ+2+∫abℏζ−βℓ+2dλζ∫abdλζ+1n−1!∫αβRnℏ˜,s−∫abRnℏζ,sdλζ∫abdλζΨnsds.GI.1.
Finally, by our assumption,Ψn−1 is absolutely continuous on α,β; as a result, Ψn exists almost everywhere. Moreover, Ψ is supposed to be n−convex, so we have Ψnx≥0 almost everywhere on α,β. Therefore, by taking into account the last term in generalized identity GI.1 and integral analogue of Jensen’s inequality that is given in (6), we get (7).In the later part of this section, we will vary our conditions on functionsg and Stieltjes measure dλ to obtain generalized variants of Jensen–Steffensen, Jensen–Boas, Jensen–Brunk, and Jensen-type inequalities. We start with the following generalization of Jensen–Steffensen inequality for n−convex functions.Theorem 4.
LetΨ defined in Theorem 3 be n−convex and ℏ defined in M1 be monotonic. Then, the following results hold.(i)
Ifλ defined in M2 satisfies(11)λa≤λx≤λb,∀x∈a,b,λb>λa,then for evenn≥3, (6) is valid.(ii)
Moreover, if (6) is valid and the function(12)Hx≔∑ℓ=0n−21ℓ!ℓ+2β−αΨℓ+1αx−αℓ+2−Ψℓ+1βx−βℓ+2is convex, then we get inequality (2) which is called generalized Jensen–Steffensen inequality for n−convex function.Proof.
(i)
By applying second derivative test, we can show that the functionRnx,s is convex for even n>3. Now using the assumed conditions, one can employ Jensen–Steffensen inequality given by Boas (see [29] or [30], p. 59) for convex function Rnx,s to obtain (6).(ii)
Since we can rewrite the R.H.S. of (7) in the difference(13)Hℏ˜−∫abHℏζdλζ∫abdλζ,for convex function H and by our assumed conditions on functions ℏ and λ, this difference is non-positive by using Jensen–Steffensen inequality difference [29]. As a result, the R.H.S. of inequality (7) is non-positive and we get generalized Jensen–Steffensen inequality (2) for n−convex function.Now, we give similar results related to Jensen–Boas inequality [30], p. 59], which is a generalization of Jensen–Steffensen inequality.Corollary 1.
LetΨ defined in Theorem 3 be n−convex function. Also, let ℏ be as defined in M1 with a=y0<y1<…<yk<…<ym−1<ym=b and ℏ be monotonic in each of the m intervals yk−1,yk. Then, the following results hold.(i)
Ifλ as defined in M2 satisfies(14)λa≤λx1≤λy1≤λx2≤λy2≤…≤λym−1≤λxm≤λb,∀xk∈yk−1,yk and λb>λa, then for even n≥3, (6) is valid.(ii)
Moreover, if (6) is valid and the function H⋅ defined in (18) is convex, then again inequality (2) holds and is called Jensen–Boas inequality for n−convex function.Proof.
We follow the similar argument as in the proof of Theorem4, but under the conditions of this corollary, we utilize Jensen–Boas inequality (see [29] or [24], p. 59) instead of Jensen–Steffensen inequality.
Next, we give results for Jensen–Brunk inequality.Corollary 2.
LetΨ defined in Theorem 3 be n−convex and ℏ defined in M1 be an increasing function. Then, the following results hold.(i)
Ifλ defined in M2 with λb>λa and(15)∫axℏx−ℏζdλζ≥0,and(16)∫xbℏx−ℏζdλζ≤0,∀x∈a,b holds, then for even n≥3, (6) is valid.(ii)
Moreover, if (6) is valid and the function H⋅ defined in (18) is convex, then again inequality (2) holds and is called Jensen–Brunk inequality for n−convex function.Proof.
We proceed with the similar idea as in the proof of Theorem4, but under the conditions of this corollary, we employ Jensen–Brunk inequality (see [31] or [30], p. 59]) instead of Jensen–Steffensen inequality.Remark 1.
The similar result in Corollary2 is also valid provided that the function ℏ is decreasing. Also, assuming that the function ℏ is monotonic, one can replace the conditions in Corollary 2i by(17)0≤∫axℏx−ℏζdλζ≤∫xbgx−ℏζdλζ.Remark 2.
It is interesting to see that by employing similar method as in Theorem4, we can also get the generalization of classical Jensen’s inequality (2) for n−convex functions by assuming the functions ℏ and λ along with the respective conditions in Theorem 1.
Another important consequence of Theorem3 can be given by setting the function ℏ as ℏζ=ζ. This form is the generalized version of L.H.S. inequality of the Hermite-Hadamard inequality.Corollary 3.
Letλ:a,b⟶ℝ be a function of bounded variation such that λa≠λb with a,b⊂α,β and ζ˜=∫abζdλζ/∫abdλζ∈α,β. Under the assumptions of Theorem 3, if Ψ is n−convex such that(18)Rnℏ˜,s≤∫abRnℏζ,sdλζ∫abdλζ,s∈α,β,then we have(19)Ψζ˜≤∫abΨζdλζ∫abdλζ+∑ℓ=0n−21ℓ!ℓ+2β−α×Ψℓ+1αζ˜−αℓ+2−∫abζ−αℓ+2dλζ∫abdλζ−Ψℓ+1βζ˜−βℓ+2+∫abζ−βℓ+2dλζ∫abdλζ.
If the inequality (18) holds in reverse direction, then (19) also holds reversely.The special case of above corollary can be given in the form of following remark.Remark 3.
It is interesting to see that substitutingλζ=ζ gives ∫abdλζ=b−a and ζ˜=a+b/2. Using these substitutions in (2) and by following remark (20), we get the L.H.S. inequality of renowned Hermite–Hadamard inequality for n−convex functions.


## 2.2. New Generalization of Converse of Jensen’s Integral Inequality

In this section, we give the results for the converse of Jensen’s inequality to hold, giving the conditions on the real Stieltjes measuredλ, such that λa≠λb, allowing that the measure can also be negative, but employing Montgomery identity.To start with we need the following assumption for the results of this section:A3 Let m,M∈α,βm≠M be such that m≤ℏζ≤M for all ζ∈a,b where ℏ is defined in A1.For a given functionΨ:α,β⟶ℝ, we consider the difference(20)CJΨ,ℏm,M;λ=∫abΨℏζdλζ∫abdλζ−M−ℏ˜M−mΨm−ℏ˜−mM−mΨM,where ℏ˜ is defined in (3).Using Montgomery identity, we obtain the following representation of the converse of Jensen’s inequality.Theorem 5.
Letℏ,λ be as defined in A1,A2 and let Ψ:α,β⟶ℝ be such that for n≥1, Ψn−1 is absolutely continuous. If Ψ is n−convex such that(21)CJRnx,s,ℏm,M;λ≤0,s∈α,β,or(22)∫abRnℏζ,sdλζ∫abdλζ≤M−ℏ˜M−mRnm,s+ℏ˜−mM−mRnM,s,s∈α,β,then we get the following extension of the converse of Jensen’s difference:(23)∫abΨℏζdλζ∫abdλζ≤M−ℏ˜M−mΨm+ℏ˜−mM−mΨM+∑ℓ=0n−21ℓ!ℓ+2β−α×Ψℓ+1αCJx−αℓ+2,ℏm,M;λ−Ψℓ+1βCJx−βℓ+2,ℏm,M;λ,where Rn⋅,s is defined in (5).Proof.
AsΨn−1 is absolutely continuous for n≥1, we can use the representation of Ψ using Montgomery identity (4) in the difference CJΨ,ℏm,M;λ:(24)CJΨ,ℏm,M;λ=CJ1β−α∫αβΨζdζ,ℏm,M;λ+∑ℓ=0n−21ℓ!ℓ+2β−αΨℓ+1αCJx−αℓ+2,ℏm,M;λ−∑ℓ=0n−21ℓ!ℓ+2β−αΨℓ+1βCJx−βℓ+2,ℏm,M;λ+1n−1!∫αβCJRnx,s,ℏm,M;λΨnsds.
After simplification and following the fact thatCJΨ,ℏm,M;λ is zero for Ψ to be constant or linear, we get the following generalized identity:(25)CJΨ,ℏm,M;λj=∑ℓ=0n−2.ℓ!ℓ+2β−α×Ψℓ+1αCJx−αℓ+2,ℏm,M;λ−Ψℓ+1βCJx−βℓ+2,ℏm,M;λ+1n−1!∫αβCJRnx,s,ℏm,M;λΨnsds.CGI.1.
Now using characterizations ofn−convex functions like in the proof of Theorem 3, we get (23).The next result gives converse of Jensen’s inequality for higher-order convex functions.Theorem 6.
LetΨ defined in Theorem 5 be n−convex and ℏ be as defined in A3. Then, the following results hold.(i)
Ifλ is non-negative measure on a,b, then for even n≥3, (22) is valid.(ii)
Moreover, if (22) is valid and the function H⋅ defined in (12) is convex, then we get the following inequality for n−convex function to be valid:(26)∫abΨℏζdλζ∫abdλζ≤M−ℏ˜M−mΨm−ℏ˜−mM−mΨM.Proof.
The idea of the proof is similar to that of (6), but we use converse of Jensen’s inequality (see [32] or [30], p. 98).


## 2.3. Applications of Jensen’s Integral Inequality

In this section, we give applications of Jensen’s integral inequality.Another important consequence of Theorem3 is by setting the function ℏ as ℏζ=ζ gives generalized version of L. H. S. inequality of the Hermite–Hadamard inequality.Corollary 4.
Letλ:a,b⟶ℝ be a function of bounded variation such that λa≠λb with a,b⊂α,β and ζ˜=∫abζdλζ/∫abdλζ∈α,β. Under the assumptions of Theorem 5, if Ψ is n−convex such that(27)∫abRnζ,sdλζ∫abdλζ≤b−ζ˜b−aRna,s+ζ˜−ab−aRnb,s,s∈α,β,then we have(28)∫abΨζdλζ∫abdλζ≤b−ζ˜b−aΨa+ζ˜−ab−aΨb+∑ℓ=0n−2.ℓ!ℓ+2β−α×Ψℓ+1αCJx−αℓ+2,ida,b;λ−Ψℓ+1βCJx−βℓ+2,ida,b;λ.
If the inequality (27) holds in reverse direction, then (28) also holds reversely.The special case of above corollary can be given in the form of following remark.Remark 4.
It is interesting to see that substitutingλζ=ζ and by following Theorem 6, we get the R.H.S. inequality of renowned Hermite–Hadamard inequality for n−convex functions.


## 3. Generalization of Jensen’s Discrete Inequality by Using Montgomery Identity

In this section, we give generalizations for Jensen’s discrete inequality by using Montgomery identity. The proofs are similar to those of continuous case as given in previous section; therefore, we give results directly.


### 3.1. Generalization of Jensen’s Discrete Inequality for Real Weights

In discrete case, we have thatpJ>0 for all J=1,2,…,m. Here we give generalizations of results allowing pJ to be negative real numbers. Also, with usual notations for pJxJJ=1,2,…,n, we notate(29)x=x1,x2,…,xmandp=p1,p2,…,pmto be m−tuples.(30)Pv=∑J=1vpJ,P¯v=Pm−Pv−1v=1,2,…,m,and(31)x¯=1Pm∑J=1mpJxJ.Using Montgomery identity (4), we obtain the following representations of Jensen’s discrete inequality.Theorem 7.
LetΨ:α,β⟶ℝ be such that for n≥1, Ψn−1 is absolutely continuous. Also, let xJ∈a,b⊆α,β, pJ∈ℝJ=1,…,m be such that Pm≠0 and x¯∈α,β.(i)
Then, the following generalized identity holds:(32)Ψx¯−1Pm∑J=1mpJΨxJ=∑ℓ=2n−21ℓ!ℓ+2β−α×Ψℓ+1αx¯−αℓ+2−1Pm∑J=1mpJxJ−αℓ+2−Ψℓ+1βx¯−βℓ+2−1Pm∑J=1mpJxJ−βℓ+2+1n−1!∫αβRnx¯,s−1Pm∑J=1mpJRnxJ,sΨnsds,DGI.1,whereRn⋅,s is defined in (5).(ii)
Moreover, ifΨ is n−convex and the inequality(33)Rnx¯,s≤1Pm∑J=1mpJRnxJ,sholds, then we have the following generalized inequality:(34)Ψx¯−1Pm∑J=1mpJΨxJ≤∑ℓ=2n−21ℓ!ℓ+2β−α×Ψℓ+1αx¯−αℓ+2−1Pm∑J=1mpJxJ−αℓ+2−Ψℓ+1βx¯−βℓ+2−1Pm∑J=1mpJxJ−βℓ+2.
If inequality (33) holds in reverse direction, then (34) also holds reversely.Proof.
Similar to that of Theorem3.
In the later part of this section, we will vary our conditions onpJxJJ=1,2,…,n to obtain generalized discrete variants of Jensen–Steffensen, Jensen’s, and Jensen–Petrovic type inequalities. We start with the following generalization of Jensen–Steffensen discrete inequality for n−convex functions.Theorem 8.
LetΨ be as defined in Theorem 7. Also, let x be monotonic n−tuple, xJ∈a,b⊆α,β, and p be a real n−tuple such that(35)0≤Pv≤Pm,v=1,2,…,m−1,Pm>0is satisfied.(i)
IfΨ is n−convex, then for even n≥3, (33) is valid.(ii)
Moreover, if (33) is valid and the function H⋅ defined in (12) is convex, then we get the following generalized Jensen–Steffensen discrete inequality:(36)Ψx¯≤1Pm∑J=1mpJΨxJ.Proof.
It is interesting to see that under the assumed conditions on tuplesx and p, we have that x¯∈a,b. For x1≥x2≥…≥xm,(37)Pmx1−x¯=∑J=2mpJx1−xJ=∑v=2mxv−1−xvPm−Pv−1≥0.
This shows thatx1≥x¯. Also, x¯≥xn, since we have(38)Pmx¯−xm=∑J=1m−1pJxJ−xm=∑v=1m−1xv−xv−1Pv≥0.
For further details, see the proof of Jensen–Steffensen discrete inequality ([24], p. 57). The idea of the rest of the proof is similar to that of Theorem 3, but here we employ Theorem 7 and Jensen–Steffensen discrete inequality.Corollary 5.
LetΨ be as defined in Theorem 7 and let xJ∈a,b⊆α,β with p being a positive n−tuple.(i)
IfΨ is n−convex, then for even n≥3, (34) is valid.(ii)
Moreover, if (33) is valid and the function H⋅ defined in (12) is convex, then again we get (36) which is called Jensen’s inequality for n−convex functions.Proof.
ForpJ>0,xJ∈a,bJ=1,2,3,…,m ensures that x¯∈a,b. So, by applying classical Jensen’s discrete inequality (1) and idea of Theorem 8, we will get the required results.Remark 5.
Under the assumptions of Corollary5, if we choose Pm=1, then Corollary 5ii gives the following inequality for n−convex functions:(39)Ψ∑J=1mpJxJ≤∑J=1mpJΨxJ.
Now we give following reverses of Jensen–Steffensen and Jensen-type inequalities.Corollary 6.
LetΨ be as defined in Theorem 7. Also, let x be monotonic m−tuple, xJ∈a,b⊆α,β, and p be a real m−tuple such that there exist m∈1,2,…,m such that(40)0≥Pv,forv<mand0≥P¯v,forv>m,where Pm>0 and x¯∈α,β.(i)
If isn−convex, then for even n≥3, then reverse of inequality (33) holds.(ii)
Moreover if (33) holds reversely and the function H⋅ defined in (12) is convex, then we get reverse of generalized Jensen–Steffensen inequality (36) for n−convex functions.Proof.
We follow the idea of Theorem8, but according to our assumed conditions, we employ reverse of Jensen–Steffensen inequality to obtain results.In the next corollary, we give explicit conditions on real tuplep such that we get reverse of classical Jensen inequality.Corollary 7.
LetΨ be as defined in Theorem 7 and let xJ∈a,b⊆α,β such that x¯∈α,β. Let p be a real n−tuple such that(41)0<p1,0≥p2,p3,…,pm,0<Pmis satisfied.(i)
IfΨ is n−convex, then for even n≥3, the reverse of inequality (33) is valid.(ii)
Also, if reverse of (33) is valid and the function H⋅ defined in (12) is convex, then we get reverse of (36).Proof.
We follow the idea of Theorem8, but according to our assumed conditions, we employ reverse of Jensen inequality to obtain results.In [33] (see also [30]), one can find the result which is equivalent to the Jensen–Steffensen and the reverse Jensen–Steffensen inequality together. It is the so-called Jensen–Petrović inequality. Here, without the proof, we give the adequate corollary which uses that result. The proof goes the same way as in the previous corollaries.Corollary 8.
LetΨ be as defined in Theorem 7 and let xi∈a,b⊆α,β be such that xm≥xm−1,…,x2≤x1. Let p be a real m−tuple with Pm=1 such that(42)0≤Pv,for1≤v<m−1and0≤P¯v,for2≤v<m,is satisfied. Then, we get the equivalent results given in Theorem 8i and ii, respectively.Remark 6.
Under the assumptions of Corollary8, if there exist m∈1,2,…,n such that(43)0≥Pv,forv<mand0≥P¯v,forv>m,and x¯∈α,β, then we get the equivalent results for reverse Jensen–Steffensen inequality given in Corollary 6i and ii, respectively.Remark 7.
It is interesting to see that the conditions onpJ,J=1,2,…,m given in Corollary 8 and Remark 6 are coming from Jensen–Petrović inequality which become equivalent to conditions for pJ,J=1,2,…,m for Jensen–Steffensen results given in Theorem 8 and Corollary 6, respectively, when Pm=1.
Now we give results for Jensen and its reverses forn−tuples x and p when n is an odd number.Corollary 9.
LetΨ be as defined in Theorem 7 and let xJ∈a,b⊆α,β for J=1,2,…,m be such that x, p be real m−tuples, m=2m+1,m∈ℕ and x^=1/∑J=12k+1pJ∑J=12k+1pJxJ∈α,β for all k=1,2,…,m. If for every k=1,2,…,m, we havei∗p1>0,p2k≤0,p2k+p2k+1≤0,∑J=12kpJ≥0,∑J=12k+1pJ>0ii∗x2k≤x2k+1,∑J=12k+1pJxJ−x2k+1≥0,then we have the following statements to be valid.(i)
IfΨ is n−convex, then for even n≥3, the inequality(44)Rnx^,s≥1P2m+1∑J=12m+1pJRnxJ,s.(ii)
Also if (44) is valid and the function H⋅ defined in (12) is convex, then we get the following generalized inequality:(45)Ψx^≥1P2m+1∑J=12m+1pJΨxJ.Proof.
We employ the idea of the proofs of Theorems7 and 8 for n=odd along with inequality of Vasić and Janic [34].Remark 8.
We can also discuss the following important cases by considering the explicit conditions given in [34].
We conclude this section by giving the following important cases:(Case 1)Let the conditioni∗ hold and the reverse inequalities in condition ii∗ hold. Then, again we can give inequalities (44) and (45), respectively, given in Corollary 9.(Case 2)If in case of conditionsi∗ and ii∗, the following are valid:iii∗p1>0,p2k+1≥0,p2k+p2k+1≥0,∑J=12kpJ≥0,∑J=12k+1pJ>0iv∗x2k≤x2k+1,∑J=12k−1pJxJ−x2k≤0,then we can give reverses of inequalities (44) and (45), respectively, given in Corollary 9.(Case 3)Finally, we can also give reverses of inequalities (44) and (45), respectively, given in Corollary 9 provided that the condition iii∗ holds and the reverse inequalities in condition iv∗ hold.
The result given inCase3 is type of generalization of inequality by Szegö [35].


### 3.2. Generalization of Converse Jensen’s Discrete Inequality for Real Weights

In this section, we give the results for converse of Jensen’s inequality in discrete case by using the Montgomery identity.LetxJ∈a,b⊆α,β, a≠b, pJ∈ℝJ=1,…,n be such that Pm≠0. Then, we have the following difference of converse of Jensen’s inequality for Ψ:α,β⟶ℝ:(46)CJdisΨ=1Pm∑J=1mpJΨxJ−b−x¯b−aΨa−x¯−ab−aΨb.Similarly, we assume the Giaccardi difference [36] given as(47)GcardiΨ=∑J=1mpJΨxJ−AΨ∑J=1mpJxJ−B∑J=1mpJ−1Ψx0,where(48)A=∑J=1mpJxJ−x0∑J=1mpJxJ−x0,B=∑J=1mpJxJ∑J=1mpJxJ−x0and∑J=1mpJxJ≠x0.Theorem 9.
LetΨ:α,β⟶ℝ be such that for n≥1, Ψn−1 is absolutely continuous. Also, let x0,xJ∈a,b⊆α,β, pJ∈ℝJ=1,…,m, be such that ∑J=1mpJxJ≠x0.(i)
Then, the following generalized identity holds:(49)CJdisΨ=∑ℓ=0n−21ℓ!ℓ+2β−αΨℓ+1αCJdisxJ−αℓ+2−Ψℓ+1βCJdisxJ−βℓ+2+1n−1!∫αβCJdisRnxJ,sΨnsds,DC.GI,whereRn⋅,s is defined in (5).(ii)
Moreover, ifΨ is n−convex and the inequality(50)CJdisRnxJ,s≤0holds, then we have the following generalized inequality:(51)CJdisΨ≤∑ℓ=0n−21ℓ!ℓ+2β−αΨℓ+1αCJdisxJ−αℓ+2−Ψℓ+1βCJdisxJ−βℓ+2.
If inequality (50) holds in reverse direction, then (51) also holds reversely.Theorem 10.
LetΨ:α,β⟶ℝ be such that for n≥1, Ψn−1 is absolutely continuous. Also, let xJ∈a,b⊆α,β, pJ∈ℝJ=1,…,m, be such that Pm≠0 and x¯∈α,β.(i)
Then, the following generalized Giaccardi identity holds:(52)GcardiΨ=∑ℓ=0n−21ℓ!ℓ+2β−αΨℓ+1αGcardixJ−αℓ+2−Ψℓ+1βGcardixJ−βℓ+2+1n−1!∫αβGcardiRnxJ,sΨnsds,GIA.GI,whereRn⋅,s is defined in (5).(ii)
Moreover, ifΨ is n−convex and the inequality(53)GcardiRnxJ,s≤0holds, then we have the following generalized Giaccardi inequality:(54)GcardiΨ≤∑ℓ=0n−21ℓ!ℓ+2β−αΨℓ+1αGcardixJ−αℓ+2−Ψℓ+1βGcardixJ−βℓ+2.
If inequality (53) holds in reverse direction, then (54) also holds reversely.In the later part of this section, we will vary our conditions onpJxJJ=1,2,…,m to obtain generalized converse discrete variants of Jensen’s inequality and Giaccardi inequality for n−convex functions.Theorem 11.
LetΨ be as defined in Theorem 9. Also, let xJ∈a,b⊆α,β and p be a positive m−tuple.(i)
IfΨ is n−convex, then for even n≥3, (50) is valid.(ii)
Moreover, if (50) is valid and the function H⋅ defined in (12) is convex, then we get the following generalized converse of Jensen’s inequality:(55)1Pm∑J=1mpJΨxJ≤b−x¯b−aΨa+x¯−ab−aΨb.Proof.
We follow the idea of Theorem8, but according to our assumed conditions, we employ converse of Jensen’s inequality (see [32] or [30], p. 98) to obtain results.Finally, in this section, we give Giaccardi inequality for higher-order convex functions.Theorem 12.
LetΨ be as defined in Theorem 9. Also, let x0,xJ∈a,b⊆α,β and p be a positive m−tuple such that(56)∑J=1mpJxJ≠x0andxv−x0∑J=1mpJxJ−xv≥0,v=1,…,m.(i)
IfΨ is n−convex, then for even n≥3, (53) is valid.(ii)
Moreover, if (53) is valid and the function H⋅ defined in (12) is convex, then we get the following generalized Giaccardi inequality:(57)∑J=1mpJΨxJ≤AΨ∑J=1mpJxJ+B∑J=1mpJ−1Ψx0,where A and B are defined in (47).Proof.
We follow the idea of Theorem8, but according to our assumed conditions, we employ Giaccardi inequality (see [36] or [37], p. 11) to obtain results.


### 3.3. Applications in Information Theory for Jensen’s Discrete Inequality

Jensen’s inequality plays a key role in information theory to construct lower bounds for some notable inequalities, but here we will use it to make connections between inequalities in information theory.LetΨ:ℝ+⟶ℝ+ be a convex function and let p≔p1,…,pm and q≔q1,…,qm be positive probability distributions; then, Ψ-divergence functional is defined (in [38]) as follows:(58)IΨp,q=∑J=1mqJΨpJqJ.Horváth et al. in [39] defined the generalized Csiszár divergence functional as follows.Definition 2.
LetI be an interval in ℝ and Ψ:I⟶ℝ be a function. Also, let p≔p1,…,pm∈ℝm and q≔q1,…,qm∈0,∞m such that(59)pJqJ∈I,J=1,…,m.
Then, let(60)I˜Ψp,q=∑J=1mqJΨpJqJ.
In this section, we write Jensen’s difference here that we use in upcoming results:(61)Fp,xJ,Ψ=Ψx¯−1Pm∑J=1mpJΨxJ.Theorem 13.
Under the assumptions of Theorem9 (ii), let (51) hold and Ψ be n−convex. Also, let p≔p1,…,pminℝm and q≔q1,…,qm∈0,∞m; then, we have the following results:(62)I˜Ψp,q≥PmΨ1−Pm∑ℓ=2n−21ℓ!ℓ+2β−α×Ψℓ+1αFq,pJqJ,x−αℓ+2−Ψℓ+1βFq,pJqJ,x−βℓ+2.Proof.
From Theorem9 by following Jensen’s difference (61), we can rearrange (34) as(63)Ψx¯−1Pm∑J=1mpJΨxJ≤∑ℓ=2n−21ℓ!ℓ+2β−α×Ψℓ+1αFp,xJ,x−αℓ+2−Ψℓ+1βFp,xJ,x−βℓ+2.
Now replacepJ with qJ and xJ with pJ/qJ, and we get (62).For positiven-tuple q=q1,…,qm such that ∑J=1mqJ=1, the Shannon entropy is defined by(64)Sq=−∑J=1mqJlnqJ.Corollary 10.
Under the assumptions of Theorem9 (ii), let (51) hold and Ψ be n−convex.(i)
Ifq≔q1,…,qm∈0,∞m, then(65)∑J=1mqJlnqJ≤Pm∑ℓ=2n−21ℓ!ℓ+2β−α×−1ℓ+1ℓ!αℓ+1Fq,1qJ,−ln⋅−−1ℓ+1ℓ!βℓ+1Fq,1qJ,−ln⋅.(ii)
We can get bounds for the Shannon entropy ofq, if we choose q≔q1,…,qn to be a positive probability distribution.(66)Sq≤Pm∑ℓ=2n−21ℓ!ℓ+2β−α×−1ℓ+1ℓ!αℓ+1Fq,1qJ,−ln⋅−−1ℓ+1ℓ!βℓ+1Fq,1qJ,−ln⋅.Proof.
(i)
SubstitutingΨx≔−lnx and using p≔1,1,…,1 in Theorem 13, we get (65).(ii)
Since we have∑J=1mqJ=1, by multiplying −1 on both sides of (65) and taking into account (64), we get (66).
The Kullback–Leibler distance [40] between the positive probability distributions p=p1,…,pm and q=q1,…,qm is defined by(67)Dqp=∑J=1mqJlnqJpJ.Corollary 11.
Under the assumptions of Corollary10,(i)
Ifq≔q1,…,qm,p≔p1,…,pm∈0,∞m, then(68)∑J=1mqJlnqJpJ≤Pm∑ℓ=2n−21ℓ!ℓ+2β−α×−1ℓ+1ℓ!αℓ+1Fq,pJqJ,−ln⋅−−1ℓ+1ℓ!βℓ+1Fq,pJqJ,−ln⋅.(ii)
Ifq≔q1,…,qm,p≔p1,…,pm are positive probability distributions, then we have(69)Dqp≤Pm∑ℓ=2n−21ℓ!ℓ+2β−α×−1ℓ+1ℓ!αℓ+1Fq,pJqJ,−ln⋅−−1ℓ+1ℓ!βℓ+1Fq,pJqJ,−ln⋅.Proof.
(i)
UsingΨx≔−lnx (which is n-convex for even n) in Theorem 13, we get (68) after simplification.(ii)
It is a special case of (i).


### 3.4. Results for Zipf and Hybrid Zipf–Mandelbrot Entropy

One of the basic laws in information science is Zipf’s law [41,42] which is highly applied in linguistics. Let c≥0, d>0, and N∈1,2,…; Zipf–Mandelbrot entropy can be given as(70)ZMH,c,d=dHc,dN∑J=1NlnJ+cJ+cd+lnHc,dN,where(71)Hc,dN=∑σ=1N1σ+cd.Consider(72)qJ=ΨJ;N,c,d=1J+cdHc,dN,where ΨJ;m,c,d is discrete probability distribution known as Zipf–Mandelbrot law. Zipf–Mandelbrot law has many application in linguistics and information sciences. Some of the recent study about Zipf–Mandelbrot law can be seen in the listed references (see [39, 43]). Now we state our results involving entropy introduced by Mandelbrot law by establishing the relationship with Shannon and relative entropies.Theorem 14.
Letq be Zipf–Mandelbrot law as defined in (72) with parameters c≥0, d>0, and N∈1,2,…, and we have(73)ZMH,c,d=Sq≤N×∑ℓ=2n−21ℓ!ℓ+2β−α×−1ℓ+1ℓ!αℓ+1Fq,J+cdHc,dN,−ln⋅−−1ℓ+1ℓ!βℓ+1Fq,J+cdHc,dN,−ln⋅.Proof.
It is interesting to see that forqJ defined in (72), ∑J=1NqJ=1. Therefore, using above qJ in Shannon entropy (64), we get Mandelbrot entropy (70):(74)Sq=−∑J=1NqJlnqJ=−∑J=1N1J+cdHc,dNln1J+cdHc,dN=dHc,dN∑J=1NlnJ+cJ+cd+lnHc,dN.
Finally, substituting thisqJ=1/J+cdHc,dN in Corollary 10 (ii), we get the desired result.Corollary 12.
Letq and p be Zipf–Mandelbrot law with parameters c1,c2∈0,∞,d1,d2>0, and let Hc1,d1N=∑σ=1N1/σ+c1d1 and Hc2,d2N=∑σ=1N1/σ+c2d2. Now using qJ=1/J+c1d1Hc1,d1N and pJ=1/J+c2d2Hc2,d2N in Corollary 11 (ii), the following holds:(75)Dqp=∑J=1N1J+c1d1Hc1,d1NlnJ+c2d2Hc2,d2NJ+c1d1Hc1,d1m=−ZH,c1,d1+d2Hc1,d1N∑J=1NlnJ+c2J+c1d1+lnHc2,d2m≤N∑ℓ=2n−21ℓ!ℓ+2β−α×−1ℓ+1ℓ!αℓ+1Fq,J+c1d1Hc1,d1NJ+c2d2Hc2,d2N,−ln⋅−−1ℓ+1ℓ!βℓ+1Fq,J+c1d1Hc1,d1NJ+c2d2Hc2,d2N,−ln⋅.
The Next Result for Hybrid Zipf–Mandelbrot Entropy. Further generalization of Zipf–Mandelbrot entropy is Hybrid Zipf–Mandelbrot entropy. Let N∈1,2,…, c≥0ω>0; then, Hybrid Zipf–Mandelbrot entropy can be given as(76)Z^MH∗,c,d,ω=1Hc,d,ω∗∑J=1NωJJ+cdlnJ+cdωJ+lnHc,d,ω∗,where(77)Hc,d,ω∗=∑J=1NωJJ+cd.
Consider(78)qJ=ΨJ;N,c,d,ω=ωJJ+cdHc,d,ω∗,which is called Hybrid Zipf–Mandelbrot law. There is a unified approach, maximization of Shannon entropy [44], that naturally follows the path of generalization from Zipf’s to Hybrid Zipf’s law. Extending this idea, Jakšetic et al. in [45] presented a transition from Zipf–Mandelbrot to Hybrid Zipf–Mandelbrot law by employing maximum entropy technique with one additional constraint. It is interesting that examination of its densities provides some new insights of Lerch’s transcendent.Theorem 15.
Letq be Hybrid Zipf–Mandelbrot law as defined in (78) with parameters c≥0, d,ω>0, and N∈1,2,…, and we have(79)Z^MH∗,c,d,ω=Sq≤N∑ℓ=2n−21ℓ!ℓ+2β−α×−1ℓ+1ℓ!αℓ+1Fq,J+cdHc,d,ω∗ωJ,−ln⋅−−1ℓ+1ℓ!βℓ+1Fq,J+cdHc,d,ω∗ωJ,−ln⋅.Proof.
It is interesting to see that forqJ defined in (78), ∑J=1NqJ=1. Therefore, using above qJ in Shannon entropy (64), we get Hybrid Zipf–Mandelbrot entropy (76):(80)Sq=−∑J=1NqJlnqJ=−∑J=1NωJJ+cdHc,d,ω∗lnωJJ+cdHc,d,ω∗=−1Hc,d,ω∗∑J=1NωJJ+cdlnωJJ+cd+ln1Hc,d,ω∗=1Hc,d,ω∗∑J=1NωJJ+cdlnJ+cdωJ+lnHc,d,ω∗=1Hc,d,ω∗∑J=1NωJJ+cdlnJ+cdωJ+lnHc,d,ω∗.
Finally, substituting thisqJ=ωJ/J+cdHc,d,ω∗ in Corollary 10 (ii), we get the desired result.Corollary 13.
Letq and p be Hybrid Zipf–Mandelbrot law with parameters c1,c2∈0,∞,ω1,ω2,d1,d2>0. Now using qJ=ω1J/J+c1d1Hc1,d1,ω1∗ and pJ=ω2J/J+c2d2Hc2,d2,ω2∗ in Corollary 11 (ii), the following holds:(81)Dq∥p=∑J=1Nω1JJ+c1d1Hc1,d1,ω1∗lnω1JJ+c2d2Hc2,d2,ω2∗ω2JJ+c1d1Hc1,d1,ω1∗=−Z^MH∗,c1,d1,ω1+1Hc1,d1,ω1∗∑J=1Nω1JJ+c1d1lnJ+c2d2ω2J+lnHc2,d2,ω2∗≤N∑ℓ=2n−21ℓ!ℓ+2β−α×−1ℓ+1ℓ!αℓ+1Fq,ω2JJ+c1d1Hc1,d1,ω1∗ω1JJ+c2d2Hc2,d2,ω2∗,−ln⋅−−1ℓ+1ℓ!βℓ+1Fq,ω2JJ+c1d1Hc1,d1,ω1∗ω1JJ+c2d2Hc2,d2,ω2∗,−ln⋅.Remark 9.
Similarly, we can give results for Shannon entropy, Kullback–Leibler distance, Zipf–Mandelbrot entropy, and Hybrid Zipf–Mandelbrot entropy by using generalized Giaccardi inequality defined in (54) on the same steps.


## 3.1. Generalization of Jensen’s Discrete Inequality for Real Weights

In discrete case, we have thatpJ>0 for all J=1,2,…,m. Here we give generalizations of results allowing pJ to be negative real numbers. Also, with usual notations for pJxJJ=1,2,…,n, we notate(29)x=x1,x2,…,xmandp=p1,p2,…,pmto be m−tuples.(30)Pv=∑J=1vpJ,P¯v=Pm−Pv−1v=1,2,…,m,and(31)x¯=1Pm∑J=1mpJxJ.Using Montgomery identity (4), we obtain the following representations of Jensen’s discrete inequality.Theorem 7.
LetΨ:α,β⟶ℝ be such that for n≥1, Ψn−1 is absolutely continuous. Also, let xJ∈a,b⊆α,β, pJ∈ℝJ=1,…,m be such that Pm≠0 and x¯∈α,β.(i)
Then, the following generalized identity holds:(32)Ψx¯−1Pm∑J=1mpJΨxJ=∑ℓ=2n−21ℓ!ℓ+2β−α×Ψℓ+1αx¯−αℓ+2−1Pm∑J=1mpJxJ−αℓ+2−Ψℓ+1βx¯−βℓ+2−1Pm∑J=1mpJxJ−βℓ+2+1n−1!∫αβRnx¯,s−1Pm∑J=1mpJRnxJ,sΨnsds,DGI.1,whereRn⋅,s is defined in (5).(ii)
Moreover, ifΨ is n−convex and the inequality(33)Rnx¯,s≤1Pm∑J=1mpJRnxJ,sholds, then we have the following generalized inequality:(34)Ψx¯−1Pm∑J=1mpJΨxJ≤∑ℓ=2n−21ℓ!ℓ+2β−α×Ψℓ+1αx¯−αℓ+2−1Pm∑J=1mpJxJ−αℓ+2−Ψℓ+1βx¯−βℓ+2−1Pm∑J=1mpJxJ−βℓ+2.
If inequality (33) holds in reverse direction, then (34) also holds reversely.Proof.
Similar to that of Theorem3.
In the later part of this section, we will vary our conditions onpJxJJ=1,2,…,n to obtain generalized discrete variants of Jensen–Steffensen, Jensen’s, and Jensen–Petrovic type inequalities. We start with the following generalization of Jensen–Steffensen discrete inequality for n−convex functions.Theorem 8.
LetΨ be as defined in Theorem 7. Also, let x be monotonic n−tuple, xJ∈a,b⊆α,β, and p be a real n−tuple such that(35)0≤Pv≤Pm,v=1,2,…,m−1,Pm>0is satisfied.(i)
IfΨ is n−convex, then for even n≥3, (33) is valid.(ii)
Moreover, if (33) is valid and the function H⋅ defined in (12) is convex, then we get the following generalized Jensen–Steffensen discrete inequality:(36)Ψx¯≤1Pm∑J=1mpJΨxJ.Proof.
It is interesting to see that under the assumed conditions on tuplesx and p, we have that x¯∈a,b. For x1≥x2≥…≥xm,(37)Pmx1−x¯=∑J=2mpJx1−xJ=∑v=2mxv−1−xvPm−Pv−1≥0.
This shows thatx1≥x¯. Also, x¯≥xn, since we have(38)Pmx¯−xm=∑J=1m−1pJxJ−xm=∑v=1m−1xv−xv−1Pv≥0.
For further details, see the proof of Jensen–Steffensen discrete inequality ([24], p. 57). The idea of the rest of the proof is similar to that of Theorem 3, but here we employ Theorem 7 and Jensen–Steffensen discrete inequality.Corollary 5.
LetΨ be as defined in Theorem 7 and let xJ∈a,b⊆α,β with p being a positive n−tuple.(i)
IfΨ is n−convex, then for even n≥3, (34) is valid.(ii)
Moreover, if (33) is valid and the function H⋅ defined in (12) is convex, then again we get (36) which is called Jensen’s inequality for n−convex functions.Proof.
ForpJ>0,xJ∈a,bJ=1,2,3,…,m ensures that x¯∈a,b. So, by applying classical Jensen’s discrete inequality (1) and idea of Theorem 8, we will get the required results.Remark 5.
Under the assumptions of Corollary5, if we choose Pm=1, then Corollary 5ii gives the following inequality for n−convex functions:(39)Ψ∑J=1mpJxJ≤∑J=1mpJΨxJ.
Now we give following reverses of Jensen–Steffensen and Jensen-type inequalities.Corollary 6.
LetΨ be as defined in Theorem 7. Also, let x be monotonic m−tuple, xJ∈a,b⊆α,β, and p be a real m−tuple such that there exist m∈1,2,…,m such that(40)0≥Pv,forv<mand0≥P¯v,forv>m,where Pm>0 and x¯∈α,β.(i)
If isn−convex, then for even n≥3, then reverse of inequality (33) holds.(ii)
Moreover if (33) holds reversely and the function H⋅ defined in (12) is convex, then we get reverse of generalized Jensen–Steffensen inequality (36) for n−convex functions.Proof.
We follow the idea of Theorem8, but according to our assumed conditions, we employ reverse of Jensen–Steffensen inequality to obtain results.In the next corollary, we give explicit conditions on real tuplep such that we get reverse of classical Jensen inequality.Corollary 7.
LetΨ be as defined in Theorem 7 and let xJ∈a,b⊆α,β such that x¯∈α,β. Let p be a real n−tuple such that(41)0<p1,0≥p2,p3,…,pm,0<Pmis satisfied.(i)
IfΨ is n−convex, then for even n≥3, the reverse of inequality (33) is valid.(ii)
Also, if reverse of (33) is valid and the function H⋅ defined in (12) is convex, then we get reverse of (36).Proof.
We follow the idea of Theorem8, but according to our assumed conditions, we employ reverse of Jensen inequality to obtain results.In [33] (see also [30]), one can find the result which is equivalent to the Jensen–Steffensen and the reverse Jensen–Steffensen inequality together. It is the so-called Jensen–Petrović inequality. Here, without the proof, we give the adequate corollary which uses that result. The proof goes the same way as in the previous corollaries.Corollary 8.
LetΨ be as defined in Theorem 7 and let xi∈a,b⊆α,β be such that xm≥xm−1,…,x2≤x1. Let p be a real m−tuple with Pm=1 such that(42)0≤Pv,for1≤v<m−1and0≤P¯v,for2≤v<m,is satisfied. Then, we get the equivalent results given in Theorem 8i and ii, respectively.Remark 6.
Under the assumptions of Corollary8, if there exist m∈1,2,…,n such that(43)0≥Pv,forv<mand0≥P¯v,forv>m,and x¯∈α,β, then we get the equivalent results for reverse Jensen–Steffensen inequality given in Corollary 6i and ii, respectively.Remark 7.
It is interesting to see that the conditions onpJ,J=1,2,…,m given in Corollary 8 and Remark 6 are coming from Jensen–Petrović inequality which become equivalent to conditions for pJ,J=1,2,…,m for Jensen–Steffensen results given in Theorem 8 and Corollary 6, respectively, when Pm=1.
Now we give results for Jensen and its reverses forn−tuples x and p when n is an odd number.Corollary 9.
LetΨ be as defined in Theorem 7 and let xJ∈a,b⊆α,β for J=1,2,…,m be such that x, p be real m−tuples, m=2m+1,m∈ℕ and x^=1/∑J=12k+1pJ∑J=12k+1pJxJ∈α,β for all k=1,2,…,m. If for every k=1,2,…,m, we havei∗p1>0,p2k≤0,p2k+p2k+1≤0,∑J=12kpJ≥0,∑J=12k+1pJ>0ii∗x2k≤x2k+1,∑J=12k+1pJxJ−x2k+1≥0,then we have the following statements to be valid.(i)
IfΨ is n−convex, then for even n≥3, the inequality(44)Rnx^,s≥1P2m+1∑J=12m+1pJRnxJ,s.(ii)
Also if (44) is valid and the function H⋅ defined in (12) is convex, then we get the following generalized inequality:(45)Ψx^≥1P2m+1∑J=12m+1pJΨxJ.Proof.
We employ the idea of the proofs of Theorems7 and 8 for n=odd along with inequality of Vasić and Janic [34].Remark 8.
We can also discuss the following important cases by considering the explicit conditions given in [34].
We conclude this section by giving the following important cases:(Case 1)Let the conditioni∗ hold and the reverse inequalities in condition ii∗ hold. Then, again we can give inequalities (44) and (45), respectively, given in Corollary 9.(Case 2)If in case of conditionsi∗ and ii∗, the following are valid:iii∗p1>0,p2k+1≥0,p2k+p2k+1≥0,∑J=12kpJ≥0,∑J=12k+1pJ>0iv∗x2k≤x2k+1,∑J=12k−1pJxJ−x2k≤0,then we can give reverses of inequalities (44) and (45), respectively, given in Corollary 9.(Case 3)Finally, we can also give reverses of inequalities (44) and (45), respectively, given in Corollary 9 provided that the condition iii∗ holds and the reverse inequalities in condition iv∗ hold.
The result given inCase3 is type of generalization of inequality by Szegö [35].


## 3.2. Generalization of Converse Jensen’s Discrete Inequality for Real Weights

In this section, we give the results for converse of Jensen’s inequality in discrete case by using the Montgomery identity.LetxJ∈a,b⊆α,β, a≠b, pJ∈ℝJ=1,…,n be such that Pm≠0. Then, we have the following difference of converse of Jensen’s inequality for Ψ:α,β⟶ℝ:(46)CJdisΨ=1Pm∑J=1mpJΨxJ−b−x¯b−aΨa−x¯−ab−aΨb.Similarly, we assume the Giaccardi difference [36] given as(47)GcardiΨ=∑J=1mpJΨxJ−AΨ∑J=1mpJxJ−B∑J=1mpJ−1Ψx0,where(48)A=∑J=1mpJxJ−x0∑J=1mpJxJ−x0,B=∑J=1mpJxJ∑J=1mpJxJ−x0and∑J=1mpJxJ≠x0.Theorem 9.
LetΨ:α,β⟶ℝ be such that for n≥1, Ψn−1 is absolutely continuous. Also, let x0,xJ∈a,b⊆α,β, pJ∈ℝJ=1,…,m, be such that ∑J=1mpJxJ≠x0.(i)
Then, the following generalized identity holds:(49)CJdisΨ=∑ℓ=0n−21ℓ!ℓ+2β−αΨℓ+1αCJdisxJ−αℓ+2−Ψℓ+1βCJdisxJ−βℓ+2+1n−1!∫αβCJdisRnxJ,sΨnsds,DC.GI,whereRn⋅,s is defined in (5).(ii)
Moreover, ifΨ is n−convex and the inequality(50)CJdisRnxJ,s≤0holds, then we have the following generalized inequality:(51)CJdisΨ≤∑ℓ=0n−21ℓ!ℓ+2β−αΨℓ+1αCJdisxJ−αℓ+2−Ψℓ+1βCJdisxJ−βℓ+2.
If inequality (50) holds in reverse direction, then (51) also holds reversely.Theorem 10.
LetΨ:α,β⟶ℝ be such that for n≥1, Ψn−1 is absolutely continuous. Also, let xJ∈a,b⊆α,β, pJ∈ℝJ=1,…,m, be such that Pm≠0 and x¯∈α,β.(i)
Then, the following generalized Giaccardi identity holds:(52)GcardiΨ=∑ℓ=0n−21ℓ!ℓ+2β−αΨℓ+1αGcardixJ−αℓ+2−Ψℓ+1βGcardixJ−βℓ+2+1n−1!∫αβGcardiRnxJ,sΨnsds,GIA.GI,whereRn⋅,s is defined in (5).(ii)
Moreover, ifΨ is n−convex and the inequality(53)GcardiRnxJ,s≤0holds, then we have the following generalized Giaccardi inequality:(54)GcardiΨ≤∑ℓ=0n−21ℓ!ℓ+2β−αΨℓ+1αGcardixJ−αℓ+2−Ψℓ+1βGcardixJ−βℓ+2.
If inequality (53) holds in reverse direction, then (54) also holds reversely.In the later part of this section, we will vary our conditions onpJxJJ=1,2,…,m to obtain generalized converse discrete variants of Jensen’s inequality and Giaccardi inequality for n−convex functions.Theorem 11.
LetΨ be as defined in Theorem 9. Also, let xJ∈a,b⊆α,β and p be a positive m−tuple.(i)
IfΨ is n−convex, then for even n≥3, (50) is valid.(ii)
Moreover, if (50) is valid and the function H⋅ defined in (12) is convex, then we get the following generalized converse of Jensen’s inequality:(55)1Pm∑J=1mpJΨxJ≤b−x¯b−aΨa+x¯−ab−aΨb.Proof.
We follow the idea of Theorem8, but according to our assumed conditions, we employ converse of Jensen’s inequality (see [32] or [30], p. 98) to obtain results.Finally, in this section, we give Giaccardi inequality for higher-order convex functions.Theorem 12.
LetΨ be as defined in Theorem 9. Also, let x0,xJ∈a,b⊆α,β and p be a positive m−tuple such that(56)∑J=1mpJxJ≠x0andxv−x0∑J=1mpJxJ−xv≥0,v=1,…,m.(i)
IfΨ is n−convex, then for even n≥3, (53) is valid.(ii)
Moreover, if (53) is valid and the function H⋅ defined in (12) is convex, then we get the following generalized Giaccardi inequality:(57)∑J=1mpJΨxJ≤AΨ∑J=1mpJxJ+B∑J=1mpJ−1Ψx0,where A and B are defined in (47).Proof.
We follow the idea of Theorem8, but according to our assumed conditions, we employ Giaccardi inequality (see [36] or [37], p. 11) to obtain results.


## 3.3. Applications in Information Theory for Jensen’s Discrete Inequality

Jensen’s inequality plays a key role in information theory to construct lower bounds for some notable inequalities, but here we will use it to make connections between inequalities in information theory.LetΨ:ℝ+⟶ℝ+ be a convex function and let p≔p1,…,pm and q≔q1,…,qm be positive probability distributions; then, Ψ-divergence functional is defined (in [38]) as follows:(58)IΨp,q=∑J=1mqJΨpJqJ.Horváth et al. in [39] defined the generalized Csiszár divergence functional as follows.Definition 2.
LetI be an interval in ℝ and Ψ:I⟶ℝ be a function. Also, let p≔p1,…,pm∈ℝm and q≔q1,…,qm∈0,∞m such that(59)pJqJ∈I,J=1,…,m.
Then, let(60)I˜Ψp,q=∑J=1mqJΨpJqJ.
In this section, we write Jensen’s difference here that we use in upcoming results:(61)Fp,xJ,Ψ=Ψx¯−1Pm∑J=1mpJΨxJ.Theorem 13.
Under the assumptions of Theorem9 (ii), let (51) hold and Ψ be n−convex. Also, let p≔p1,…,pminℝm and q≔q1,…,qm∈0,∞m; then, we have the following results:(62)I˜Ψp,q≥PmΨ1−Pm∑ℓ=2n−21ℓ!ℓ+2β−α×Ψℓ+1αFq,pJqJ,x−αℓ+2−Ψℓ+1βFq,pJqJ,x−βℓ+2.Proof.
From Theorem9 by following Jensen’s difference (61), we can rearrange (34) as(63)Ψx¯−1Pm∑J=1mpJΨxJ≤∑ℓ=2n−21ℓ!ℓ+2β−α×Ψℓ+1αFp,xJ,x−αℓ+2−Ψℓ+1βFp,xJ,x−βℓ+2.
Now replacepJ with qJ and xJ with pJ/qJ, and we get (62).For positiven-tuple q=q1,…,qm such that ∑J=1mqJ=1, the Shannon entropy is defined by(64)Sq=−∑J=1mqJlnqJ.Corollary 10.
Under the assumptions of Theorem9 (ii), let (51) hold and Ψ be n−convex.(i)
Ifq≔q1,…,qm∈0,∞m, then(65)∑J=1mqJlnqJ≤Pm∑ℓ=2n−21ℓ!ℓ+2β−α×−1ℓ+1ℓ!αℓ+1Fq,1qJ,−ln⋅−−1ℓ+1ℓ!βℓ+1Fq,1qJ,−ln⋅.(ii)
We can get bounds for the Shannon entropy ofq, if we choose q≔q1,…,qn to be a positive probability distribution.(66)Sq≤Pm∑ℓ=2n−21ℓ!ℓ+2β−α×−1ℓ+1ℓ!αℓ+1Fq,1qJ,−ln⋅−−1ℓ+1ℓ!βℓ+1Fq,1qJ,−ln⋅.Proof.
(i)
SubstitutingΨx≔−lnx and using p≔1,1,…,1 in Theorem 13, we get (65).(ii)
Since we have∑J=1mqJ=1, by multiplying −1 on both sides of (65) and taking into account (64), we get (66).
The Kullback–Leibler distance [40] between the positive probability distributions p=p1,…,pm and q=q1,…,qm is defined by(67)Dqp=∑J=1mqJlnqJpJ.Corollary 11.
Under the assumptions of Corollary10,(i)
Ifq≔q1,…,qm,p≔p1,…,pm∈0,∞m, then(68)∑J=1mqJlnqJpJ≤Pm∑ℓ=2n−21ℓ!ℓ+2β−α×−1ℓ+1ℓ!αℓ+1Fq,pJqJ,−ln⋅−−1ℓ+1ℓ!βℓ+1Fq,pJqJ,−ln⋅.(ii)
Ifq≔q1,…,qm,p≔p1,…,pm are positive probability distributions, then we have(69)Dqp≤Pm∑ℓ=2n−21ℓ!ℓ+2β−α×−1ℓ+1ℓ!αℓ+1Fq,pJqJ,−ln⋅−−1ℓ+1ℓ!βℓ+1Fq,pJqJ,−ln⋅.Proof.
(i)
UsingΨx≔−lnx (which is n-convex for even n) in Theorem 13, we get (68) after simplification.(ii)
It is a special case of (i).


## 3.4. Results for Zipf and Hybrid Zipf–Mandelbrot Entropy

One of the basic laws in information science is Zipf’s law [41,42] which is highly applied in linguistics. Let c≥0, d>0, and N∈1,2,…; Zipf–Mandelbrot entropy can be given as(70)ZMH,c,d=dHc,dN∑J=1NlnJ+cJ+cd+lnHc,dN,where(71)Hc,dN=∑σ=1N1σ+cd.Consider(72)qJ=ΨJ;N,c,d=1J+cdHc,dN,where ΨJ;m,c,d is discrete probability distribution known as Zipf–Mandelbrot law. Zipf–Mandelbrot law has many application in linguistics and information sciences. Some of the recent study about Zipf–Mandelbrot law can be seen in the listed references (see [39, 43]). Now we state our results involving entropy introduced by Mandelbrot law by establishing the relationship with Shannon and relative entropies.Theorem 14.
Letq be Zipf–Mandelbrot law as defined in (72) with parameters c≥0, d>0, and N∈1,2,…, and we have(73)ZMH,c,d=Sq≤N×∑ℓ=2n−21ℓ!ℓ+2β−α×−1ℓ+1ℓ!αℓ+1Fq,J+cdHc,dN,−ln⋅−−1ℓ+1ℓ!βℓ+1Fq,J+cdHc,dN,−ln⋅.Proof.
It is interesting to see that forqJ defined in (72), ∑J=1NqJ=1. Therefore, using above qJ in Shannon entropy (64), we get Mandelbrot entropy (70):(74)Sq=−∑J=1NqJlnqJ=−∑J=1N1J+cdHc,dNln1J+cdHc,dN=dHc,dN∑J=1NlnJ+cJ+cd+lnHc,dN.
Finally, substituting thisqJ=1/J+cdHc,dN in Corollary 10 (ii), we get the desired result.Corollary 12.
Letq and p be Zipf–Mandelbrot law with parameters c1,c2∈0,∞,d1,d2>0, and let Hc1,d1N=∑σ=1N1/σ+c1d1 and Hc2,d2N=∑σ=1N1/σ+c2d2. Now using qJ=1/J+c1d1Hc1,d1N and pJ=1/J+c2d2Hc2,d2N in Corollary 11 (ii), the following holds:(75)Dqp=∑J=1N1J+c1d1Hc1,d1NlnJ+c2d2Hc2,d2NJ+c1d1Hc1,d1m=−ZH,c1,d1+d2Hc1,d1N∑J=1NlnJ+c2J+c1d1+lnHc2,d2m≤N∑ℓ=2n−21ℓ!ℓ+2β−α×−1ℓ+1ℓ!αℓ+1Fq,J+c1d1Hc1,d1NJ+c2d2Hc2,d2N,−ln⋅−−1ℓ+1ℓ!βℓ+1Fq,J+c1d1Hc1,d1NJ+c2d2Hc2,d2N,−ln⋅.
The Next Result for Hybrid Zipf–Mandelbrot Entropy. Further generalization of Zipf–Mandelbrot entropy is Hybrid Zipf–Mandelbrot entropy. Let N∈1,2,…, c≥0ω>0; then, Hybrid Zipf–Mandelbrot entropy can be given as(76)Z^MH∗,c,d,ω=1Hc,d,ω∗∑J=1NωJJ+cdlnJ+cdωJ+lnHc,d,ω∗,where(77)Hc,d,ω∗=∑J=1NωJJ+cd.
Consider(78)qJ=ΨJ;N,c,d,ω=ωJJ+cdHc,d,ω∗,which is called Hybrid Zipf–Mandelbrot law. There is a unified approach, maximization of Shannon entropy [44], that naturally follows the path of generalization from Zipf’s to Hybrid Zipf’s law. Extending this idea, Jakšetic et al. in [45] presented a transition from Zipf–Mandelbrot to Hybrid Zipf–Mandelbrot law by employing maximum entropy technique with one additional constraint. It is interesting that examination of its densities provides some new insights of Lerch’s transcendent.Theorem 15.
Letq be Hybrid Zipf–Mandelbrot law as defined in (78) with parameters c≥0, d,ω>0, and N∈1,2,…, and we have(79)Z^MH∗,c,d,ω=Sq≤N∑ℓ=2n−21ℓ!ℓ+2β−α×−1ℓ+1ℓ!αℓ+1Fq,J+cdHc,d,ω∗ωJ,−ln⋅−−1ℓ+1ℓ!βℓ+1Fq,J+cdHc,d,ω∗ωJ,−ln⋅.Proof.
It is interesting to see that forqJ defined in (78), ∑J=1NqJ=1. Therefore, using above qJ in Shannon entropy (64), we get Hybrid Zipf–Mandelbrot entropy (76):(80)Sq=−∑J=1NqJlnqJ=−∑J=1NωJJ+cdHc,d,ω∗lnωJJ+cdHc,d,ω∗=−1Hc,d,ω∗∑J=1NωJJ+cdlnωJJ+cd+ln1Hc,d,ω∗=1Hc,d,ω∗∑J=1NωJJ+cdlnJ+cdωJ+lnHc,d,ω∗=1Hc,d,ω∗∑J=1NωJJ+cdlnJ+cdωJ+lnHc,d,ω∗.
Finally, substituting thisqJ=ωJ/J+cdHc,d,ω∗ in Corollary 10 (ii), we get the desired result.Corollary 13.
Letq and p be Hybrid Zipf–Mandelbrot law with parameters c1,c2∈0,∞,ω1,ω2,d1,d2>0. Now using qJ=ω1J/J+c1d1Hc1,d1,ω1∗ and pJ=ω2J/J+c2d2Hc2,d2,ω2∗ in Corollary 11 (ii), the following holds:(81)Dq∥p=∑J=1Nω1JJ+c1d1Hc1,d1,ω1∗lnω1JJ+c2d2Hc2,d2,ω2∗ω2JJ+c1d1Hc1,d1,ω1∗=−Z^MH∗,c1,d1,ω1+1Hc1,d1,ω1∗∑J=1Nω1JJ+c1d1lnJ+c2d2ω2J+lnHc2,d2,ω2∗≤N∑ℓ=2n−21ℓ!ℓ+2β−α×−1ℓ+1ℓ!αℓ+1Fq,ω2JJ+c1d1Hc1,d1,ω1∗ω1JJ+c2d2Hc2,d2,ω2∗,−ln⋅−−1ℓ+1ℓ!βℓ+1Fq,ω2JJ+c1d1Hc1,d1,ω1∗ω1JJ+c2d2Hc2,d2,ω2∗,−ln⋅.Remark 9.
Similarly, we can give results for Shannon entropy, Kullback–Leibler distance, Zipf–Mandelbrot entropy, and Hybrid Zipf–Mandelbrot entropy by using generalized Giaccardi inequality defined in (54) on the same steps.


## 4. Concluding Remarks

In this paper, we gave generalization of Jensen’s inequality as well as converse of Jensen’s inequality by using Montgomery identity. We also formulate results for other inequalities like Jensen–Steffensen inequality, Jensen–Boas inequality, and Jensen–Brunk inequality. We can obtain Jensen–Steffensen inequality, Jensen–Boas inequality, and Jensen–Brunk inequality by changing the assumption of Jensen’s inequality. At the end, we gave applications in information theory for our obtained results, especially we gave results for Hybrid Zipf–Mandelbrot entropy for our obtained results [46].



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