# A Refinement of the Integral Jensen Inequality Pertaining Certain Functions with Applications

Authors: Zaid Mohammed Mohammed Mahdi Sayed; Muhammad Adil Khan; Shahid Khan; Josip Pečarić
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we present a new refinement of the integral Jensen inequality by utilizing certain functions and give its applications to various means. We utilize the refinement to obtain some new refinements of the Hermite-Hadamard and Hölder’s inequalities as well. Also, we present its applications in information theory. At the end of this paper, we give a more general form of the proposed refinement of the Jensen inequality, associated to several functions.


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## Body

## 1. Introduction

Being an important part of modern applied analysis, the field of mathematical inequalities has recorded an exponential growth with significant impact on various parts of science and technology [1–5]. These inequalities are also extended and generalized in various aspects; one can see such results in [6–15]. The Jensen weighted integral inequality is a central tool among them; its basic form is follows as [16].Theorem 1.
Assume a convex functionf:I⟶ℝ and g,h:θ1,θ2⟶ℝ are measurable functions such that gθ∈I and hθ≥0∀θ∈θ1,θ2. Also, suppose that h,gh,f∘g.h are all integrable functions on θ1,θ2 and ∫θ1θ2hθdθ>0, then
(1)f∫θ1θ2gθhθdθ∫θ1θ2hθdθ≤∫θ1θ2f∘gθhθdθ∫θ1θ2hθdθ.The Jensen inequality is one of the fundamental inequalities in modern applied analysis. This inequality is of pivotal importance because various other classical inequalities, for example, the Beckenbach-Dresher, Minkowski’s, the Hermite-Hadamard, Ky-Fan’s, Hölder’s, the arithmetic-geometric, and Levinson’s and Young’s inequalities, can be deduced from this inequality. Also, this inequality can be treated as a problem solving oriented tool in different areas of science and technology, and an extensive literature is dedicated to this inequality regarding its counterparts, generalizations, improvements, and converse results (see, for instance, [17–21]) and the references therein.The Hermite-Hadamard inequality is presented as follows ([22], page 10 in [23]).Theorem 2.
Assume a convex functionf:θ1,θ2⟶ℝ, then the following double inequalities hold:
(2)fθ1+θ22≤1θ2−θ1∫θ1θ2fθdθ≤fθ1+fθ22.The Hölder inequality in its integral form is presented as follows [23].Theorem 3.
Let1<p,q be such that 1/p+1/q=1 and assume two measurable functions say h,g:θ1,θ2⟶ℝ such that the functions hθp and gθq are integrable on θ1,θ2. Then
(3)∫θ1θ2hθgθdθ≤∫θ1θ2hθpdθ1/p∫θ1θ2gθqdθ1/q.The remaining paper is organized in the following manner: Section2 proposes a new refinement of the integral Jensen inequality, associated to four functions whose sum is equal to unity in pairs. Utilizing this refinement, we derive some new refinements of the Hölder and Hermite-Hadamard inequalities and some new inequalities for power and quasi-arithmetic means. In Section 3, we focus to deduce inequalities for the Csiszâr divergence, variational distance, Shannon entropy, and Kullback-Leibler divergence. In the last section, we present a more general form of the proposed refinement concerning several certain functions.


## 2. Main Results

Assuming a real valued convex functionf defined on the interval I and suppose that h,u,v,w,z,g:θ1,θ2⟶ℝ are some integrable functions with the following conditions gθ∈I,hθ,uθ,vθ,wθ,zθ∈ℝ+ with uθ+wθ=1, vθ+zθ=1 for all θ∈θ1,θ2 and H≔∫θ1θ2hθdθ. Also, let for a nonempty subinterval J of θ1,θ2, we put J¯=θ1,θ2\J; then, for the above facts, we can define the following functional
(4)ℤ∫f,u,v,h,w,z,g;J=1H∫Juθhθdθf∫Juθhθgθdθ∫Juθhθdθ+1H∫Jwθhθdθf∫Jwθhθgθdθ∫Jwθhθdθ+1H∫J¯vθhθdθf∫J¯vθhθgθdθ∫J¯vθhθdθ+1H∫J¯zθhθdθf∫J¯zθhθgθdθ∫J¯zθhθdθ.We give the following refinement of the integral Jensen inequality associated to four functions.Theorem 4.
Letf be a convex function defined on the interval I. Also, let h,u,v,w,z,g:θ1,θ2⟶ℝ be some integrable functions with the following conditions such as gθ∈I,hθ,uθ,vθ,wθ,zθ∈ℝ+ with uθ+wθ=1, vθ+zθ=1 for all θ∈θ1,θ2 and H≔∫θ1θ2hθdθ. Then, for any nonempty subinterval J of θ1,θ2 with J¯=θ1,θ2\J, the following inequalities hold
(5)1H∫θ1θ2hθfgθdθ≥ℤ∫f,u,v,h,w,z,g;J≥f1H∫θ1θ2hθgθdθ.
For a concave functionf, the reverse inequalities hold in (5).Proof.
Sinceuθ+wθ=1, vθ+zθ=1 for all θ∈θ1,θ2, therefore for the subinterval J of θ1,θ2 with J¯=θ1,θ2\J, we have
(6)∫θ1θ2hθgθdθ=∫Jhθgθdθ+∫J¯hθgθdθ=∫Juθ+wθhθgθdθ+∫J¯vθ+zθhθgθdθ.
Multiplying equation (6) by 1/H and assigning it to the function f, then by convexity of f, we have
(7)f1H∫θ1θ2hθgθdθ=f1H∫Juθhθgθdθ+1H∫Jwθhθgθdθ+1H∫J¯vθhθgθdθ+1H∫J¯zθhθgθdθ=f∫JuθhθdθH.∫Juθhθgθdθ∫Juθhθdθ+∫JwθhθdθH.∫Jwθhθgθdθ∫Jwθhθdθ+∫J¯vθhθdθH.∫J¯vθhθgθdθ∫J¯vθhθdθ+∫J¯zθhθdθH.∫J¯zθhθgθdθ∫J¯zθhθdθ≤1H∫Juθhθdθf∫Juθhθgθdθ∫Juθhθdθ+1H∫Jwθhθdθf∫Jwθhθgθdθ∫Jwθhθdθ+1H∫J¯vθhθdθf∫J¯vθhθgθdθ∫J¯vθhθdθ+1H∫J¯zθhθdθf∫J¯zθhθgθdθ∫J¯zθhθdθ=ℤ∫f,u,v,h,w,z,g;J.
Also, by making use of the integral Jensen inequality, one has(8)ℤ∫f,u,v,h,w,z,g;J=1H∫Juθhθdθf∫Juθhθgθdθ∫Juθhθdθ+1H∫Jwθhθdθf∫Jwθhθgθdθ∫Jwθhθdθ+1H∫J¯vθhθdθf∫J¯vθhθgθdθ∫J¯vθhθdθ+1H∫J¯zθhθdθf∫J¯zθhθgθdθ∫J¯zθhθdθ≤∫JuθhθdθH.∫Juθhθfgθdθ∫Juθhθdθ+∫JwθhθdθH.∫Jwθhθfgθdθ∫Jwθhθdθ+∫J¯vθhθdθH.∫J¯vθhθfgθdθ∫J¯vθhθdθ+∫J¯zθhθdθH.∫J¯zθhθfgθdθ∫J¯zθhθdθ=1H∫Juθhθfgθdθ+1H∫Jwθhθfgθdθ+1H∫J¯vθhθfgθdθ+1H∫J¯zθhθfgθdθ=1H∫θ1θ2hθfgθdθ.
From (7) and (8), we get (5).Remark 5.
The following is an equivalent form of the inequality (5)
(9)infϕ≠J⊂θ1,θ2ℤ∫f,u,v,h,w,z,g;J≥f1H∫θ1θ2hθgθdθ,(10)1H∫θ1θ2hθfgθdθ≥supϕ≠J⊂θ1,θ2ℤ∫f,u,v,h,w,z,g;J.From Theorem4, we obtain a new refinement of the H-H inequality as follows.Corollary 6.
Letf be a convex function defined on θ1,θ2. Also, let u,v,w,z:θ1,θ2⟶ℝ be integrable functions such that uθ,vθ,wθ,zθ∈ℝ+ with uθ+wθ=1 and vθ+zθ=1 for all θ∈θ1,θ2. Then, for a subinterval J of θ1,θ2 with J¯=θ1,θ2\J, the following inequalities hold
(11)1θ2−θ1∫θ1θ2fθdθ≥1θ2−θ1∫Juθdθf∫Jθuθdθ∫Juθdθ+1θ2−θ1∫Jwθdθf∫Jθwθdθ∫Jwθdθ+1θ2−θ1∫J¯vθdθf∫J¯θvθdθ∫J¯vθdθ+1θ2−θ1∫J¯zθdθf∫J¯θzθdθ∫J¯zθdθ≥fθ1+θ22.
The direction of the inequalities reverses in (11), when the function f becomes concave.Proof.
Utilizing Theorem4 for hθ=1,gθ=θ for all θ∈θ1,θ2, we get (11).From Theorem4, we deduce the following refinement of Hölder’s inequality.Corollary 7.
Ifp,q∈ℝ and the functions u,v,w,z,h1,g1, and g2 defined on θ1,θ2 are nonnegative such that the functions h1g1p,h1g2q,uh1g2q,wh1g2q,vh1g2q,zh1g2q,uh1g1g2 and wh1g1g2,vh1g1g2,zh1g1g2,h1g1g2 are integrable with uθ+wθ=1 and vθ+zθ=1 for all θ∈θ1,θ2. Also, if J is a subinterval of θ1,θ2 with J¯=θ1,θ2\J, then
(A)
Forp,q>1 such that 1/p+1/q=1, the following inequalities hold(12)∫θ1θ2h1θg1pθdθ1/p∫θ1θ2h1θg2qθdθ1/q≥∫θ1θ2h1θg2qθdθ1/q∫Juθh1θg2qθdθ1−p∫Juθh1θg1θg2θdθp+∫Jwθh1θg2qθdθ1−p∫Jwθh1θg1θg2θdθp+∫J¯vθh1θg2qθdθ1−p∫J¯vθh1θg1θg2θdθp+∫J¯zθh1θg2qθdθ1−p∫J¯zθh1θg1θg2θdθp1/p≥∫θ1θ2h1θg1θg2θdθ(B)
For0<p<1 and q=p/p−1 with ∫θ1θ2h1θg2qθdθ>0, the following inequalities hold(13)∫θ1θ2h1θg1θg2θdθ≥∫Juθh1θg2qθdθ1/q∫Juθh1θg1pθdθ1/p+∫Jwθh1θg2qθdθ1/q∫Jwθh1θg1pθdθ1/p+∫J¯vθh1θg2qθdθ1/q∫J¯vθh1θg1pθdθ1/p+∫J¯zθh1θg2qθdθ1/q∫J¯zθh1θg1pθdθ1/p≥∫θ1θ2h1θg1pθdθ1/p∫θ1θ2h1θg2qθdθ1/q(C)
Forp<0 and q=p/p−1 with ∫θ1θ2h1θg1pθdθ>0, the inequalities in (13) holdProof.
(A)
In the case whenp,q>1, let ∫θ1θ2h1θg2qθdθ>0. Then, by using Theorem 4 for fθ=θp, θ>0, hθ=h1θg2qθ, and gθ=g1θg2−q/pθ, we obtain (12). Also, let ∫θ1θ2h1θg1pθdθ>0; then, applying the same procedure as above and replacing p,q,g1,g2 by q,p,g2,g1, respectively, we obtain (12)
For∫θ1θ2h1θg2qθdθ=0 and ∫θ1θ2h1θg1pθdθ=0, the inequalities in (12) also hold. This can be proved as follows, since we know that
(14)0≤h1θg1θg2θ≤1ph1θg1pθ+1qh1θg2qθ.
Taking integral, then with the proposed conditions, we obtain∫θ1θ2h1θg1θg2θdθ=0, which concludes the result.
(B)
In the case when0<p<1, we have q<0. So, applying (12) for p⟶1/p>1, q⟶1−p−1, g1θ⟶g1θg2θp, and g2θ⟶g2θ−p, we obtain (13)(C)
In the case whenp<0, we have 0<q<1. So, applying the arguments of part B with replacing p,q,g1θ,g2θ by q,p,g2θ,g1θ, respectively, we get (13)The Hölder inequality is refined by the following corollary.Corollary 8.
Letp,q∈ℝ and u,v,w,z,h1,g1,g2 be nonnegative functions defined on θ1,θ2 such that h1g1p,h1g2q,uh1g2q,uh1g1p,wh1g2q,wh1g1p,vh1g2q,vh1g1p,zh1g2q and zh1g1p,h1g1g2 are integrable functions with uθ+wθ=1 and vθ+zθ=1 for all θ∈θ1,θ2. Also, assume that J is a subinterval of θ1,θ2 with J¯=θ1,θ2\J, then
(A)
Forp>1,q=p/p−1 and ∫θ1θ2h1θg2qθdθ>0, the following inequalities hold(15)∫θ1θ2h1θg1pθdθ1/p∫θ1θ2h1θg2qθdθ1/q≥∫Juθh1θg2qθdθ1/q∫Juθh1θg1pθdθ1/p+∫Jwθh1θg2qθdθ1/q∫Jwθh1θg1pθdθ1/p+∫J¯vθh1θg2qθdθ1/q∫J¯vθh1θg1pθdθ1/p+∫J¯zθh1θg2qθdθ1/q∫J¯zθh1θg1pθdθ1/p≥∫θ1θ2h1θg1θg2θdθ(B)
For0<p<1 and q=p/p−1 with ∫θ1θ2h1θg2qθdθ>0, the following inequalities hold(16)∫θ1θ2h1θg1pθdθ1/p∫θ1θ2h1θg2qθdθ1/q≤∫θ1θ2h1θg2qθdθ1/q∫Juθh1θg2qθdθ1−p∫Juθh1θg1θg2θdθp+∫Jwθh1θg2qθdθ1−p∫Jwθh1θg1θg2θdθp+∫J¯vθh1θg2qθdθ1−p∫J¯vθh1θg1θg2θdθp+∫J¯zθh1θg2qθdθ1−p∫J¯zθh1θg1θg2θdθp1/p≤∫θ1θ2h1θg1θg2θdθ(C)
Forp<0 and q=p/p−1 with ∫θ1θ2h1θg1pθdθ>0, the result in (16) holdsProof.
(A)
Letfθ=θ1/p, θ>0 which is clearly a concave function for p>1. Thus, by using Theorem 4 for fθ=θ1/p, hθ=h1θg2qθ, and gθ=g1pθg2−qθ, we obtain (15). If ∫θ1θ2h1θg1pθdθ>0, then adopting the same procedure and replacing p,q,g1,g2 by q,p,g2,g1, respectively, we obtain (15)
For∫θ1θ2h1θg2qθdθ=0 and ∫θ1θ2h1θg1pθdθ=0, the inequalities in (12) also hold. This can be proved as follows, since we know that
(17)0≤h1θg1θg2θ≤1ph1θg1pθ+1qh1θg2qθ.
Hence, taking integral and using the proposed conditions, we get∫θ1θ2h1θg1θg2θdθ=0, which verifies the result.
(B)
In the case when0<p<1, applying (15) for p⟶1/p>1, q⟶1−p−1, g1θ⟶g1θg2θp, and g2θ⟶g2θ−p, we obtain (16)(C)
In the case when0>p, we have q∈0,1, which shows that this case reflects case B; therefore, applying the arguments of case B with replacing p,q,g1θ,g2θ by q,p,g2θ,g1θ, respectively, we get (16)Leth and g be positive integrable functions defined on θ1,θ2 and J be any nonempty subinterval of θ1,θ2, then the integral power means of order r∈ℝ are defined as
(18)Mr;Jh;g=∫Jhθgrθdθ∫Jhθdθ1/r,ifr≠0,exp∫Jhθloggθdθ∫Jhθdθ,ifr=0,Mr;Ju.h;g=∫Juθhθgrθdθ∫Juθhθdθ1/r,ifr≠0,exp∫Juθhθloggθdθ∫Juθhθdθ,ifr=0,(19)Mr;θ1,θ2h;g=∫θ1θ2hθgrθdθ∫θ1θ2hθdθ1/r,ifr≠0,exp∫θ1θ2hθloggθdθ∫θ1θ2hθdθ,ifr=0.Corollary 9.
Assume some positive integrable functionsh,u,v,w,z and g defined on θ1,θ2 with uθ+wθ=1 and vθ+zθ=1 for θ∈θ1,θ2. Also, assume that α,β∈ℝ such that α≤β, then
(A)
Forα/β∈ℝ−−1,0,β≠0, the following inequalities hold(20)Mα;θ1,θ2h;g≤1∫θ1θ2hθdθ1/α∫JuθhθdθMβ;Jαu.h;g+∫JwθhθdθMβ;Jαw.h;g+∫J¯vθhθdθMβ;J¯αv.h;g+∫J¯zθhθdθMβ;J¯αz.h;g1/α≤Mβ;θ1,θ2h;g,α≠0,(21)Mα;θ1,θ2h;g≤exp∫Juθhθdθ∫θ1θ2hθdθlogMβ;Ju.h;g+∫Jwθhθdθ∫θ1θ2hθdθlogMβ;Jw.h;g+∫J¯vθhθdθ∫θ1θ2hθdθlogMβ;J¯v.h;g+∫J¯zθhθdθ∫θ1θ2hθdθlogMβ;J¯z.h;g≤Mβ;θ1,θ2h;g,α=0(B)
Forβ/α∈ℝ−−1,0,α≠0, the following inequalities hold(22)Mβ;θ1,θ2h;g≥1∫θ1θ2hθdθ1/β∫JuθhθdθMα;Jβu.h;g+∫JwθhθdθMα;Jβw.h;g+∫J¯vθhθdθMα;J¯βv.h;g+∫J¯zθhθdθMα;J¯βz.h;g1/β≥Mα;θ1,θ2h;g,β≠0,(23)Mβ;θ1,θ2h;g≥exp∫Juθhθdθ∫θ1θ2hθdθlogMα;Ju.h;g+∫Jwθhθdθ∫θ1θ2hθdθlogMα;Jw.h;g+∫J¯vθhθdθ∫θ1θ2hθdθlogMα;J¯v.h;g+∫J¯zθhθdθ∫θ1θ2hθdθlogMα;J¯z.h;g≥Mα;θ1,θ2h;g,β=0Proof.
(A)
Letfθ=θα/β for θ>0, then the following possible cases can be discussed:
Case 1. Ifα,β∈ℝ− with α≤β, then α/β≥1, and the function fθ is convex. Therefore, utilizing (5) for fθ and gθ⟶gβθ, after that taking power 1/α, we obtain (20)
Case 2. Ifα,β∈ℝ+ with α<β, then 0<α/β<1, and the function fθ is concave. Therefore, utilizing (5) for fθ and gθ⟶gβθ and then taking power 1/α, we obtain (20)
Case 3. Ifβ∈ℝ+, α∈ℝ− with α≤β, then α/β≤0, and the function fθ is convex. Therefore, utilizing (5) for fθ and gθ⟶gβθ and then taking power 1/α, we obtain (20)
For the case whenα=0, taking limα⟶0 of (20), we get (21).
(B)
Letfθ=θβ/α for θ>0, then the following possible cases can be discussed:
Case 1. Ifα,β∈ℝ+ with α≤β, then β/α≥1, and the function fθ is convex. Hence, using (5) for fθ and gθ⟶gαθ and then taking power 1/β, we obtain (22)
Case 2. Ifα,β∈ℝ− with α≤β, then 0<β/α<1, and the function fθ is concave. Hence, using (5) for fθ and gθ⟶gαθ and then taking power 1/β, we obtain (22)
Case 3. Similarly, ifα∈ℝ−, β∈ℝ+ with α≤β, then β/α≤−1, and the function fθ is convex function. Hence, using (5) for fθ and gθ⟶gαθ and taking power 1/β, we obtain (22)
For the case whenβ=0, taking limβ⟶0 in (22), we get (23).Leth and u be some positive integrable functions and g be an arbitrary integrable function defined on θ1,θ2. Further, if p is considered as a strictly monotone and continuous function whose domain is the image of g, then for any nonempty subinterval J of θ1,θ2, the quasi-arithmetic mean is defined by
(24)MpJh;g=p−1∫Jhθpgθdθ∫Jhθdθ,MpJu.h;g=p−1∫Juθhθpgθdθ∫Juθhθdθ,(25)Mpθ1,θ2h;g=p−1∫θ1θ2hθpgθdθ∫θ1θ2hθdθ.Some inequalities are given for the quasi-arithmetic mean as follows.Corollary 10.
Assume some positive integrable functionsh,u,v,w,z defined on θ1,θ2 such that uθ+wθ=1 and vθ+zθ=1 for θ∈θ1,θ2 and further assume that g is an arbitrary integrable function defined on θ1,θ2. Also, suppose that p is a strictly monotone continuous function whose domain is the image of g. Then, for Ψ∘p−1θ as a convex function, the following inequalities hold
(26)1∫θ1θ2hθdθ∫θ1θ2hθΨgθdθ≥1∫θ1θ2hθdθ∫JuθhθdθΨMpJu.h;g+∫JwθhθdθΨMpJw.h;g+∫J¯vθhθdθΨMpJ¯v.h;g+∫J¯zθhθdθΨMpJ¯z.h;g≥ΨMpθ1,θ2h;g.
The direction of the inequalities reverses in (26), when the function Ψ∘p−1θ, becomes concave.Proof.
The desired inequalities can be calculated by utilizing (5) for g⟶p∘g and f⟶Ψ∘p−1.


## 3. Applications in Information Theory

In this section, we use the main result to obtain some new and interesting estimates for various divergences and Shannon entropy in information theory. A literature about the inequalities related to these divergences can be found in [24].Definition 11 (Csiszâr divergence [25]).
Assume thatΨ:I+⟶ℝ is a function defined on a positive interval I+. Also, assume that p,q:θ1,θ2⟶0,∞ are two integrable functions such that qθ/pθ∈I+ for all θ∈θ1,θ2, then the integral form of Csiszâr-divergence is defined by
(27)CΨp,q=∫θ1θ2pθΨqθpθdθ.Theorem 12.
LetΨ:I+⟶ℝ be a convex function defined on a positive interval I+ and assume that u,v,w,z,p,q:θ1,θ2⟶ℝ+ are integrable functions such that qθ/pθ∈I+, uθ+wθ=1, and vθ+zθ=1 for all θ∈θ1,θ2. Then, for a subinterval J of θ1,θ2 with J¯=θ1,θ2\J, the following inequalities hold
(28)CΨp,q≥∫JuθpθdθΨ∫Juθqθdθ∫Juθpθdθ+∫JwθpθdθΨ∫Jwθqθdθ∫Jwθpθdθ+∫J¯vθpθdθΨ∫J¯vθqθdθ∫J¯vθpθdθ+∫J¯zθpθdθΨ∫J¯zθqθdθ∫J¯zθpθdθ≥Ψ∫θ1θ2qθdθ∫θ1θ2pθdθ∫θ1θ2pθdθ.Proof.
Using Theorem4 for f⟶Ψ, gθ=qθ/pθ, and hθ=pθ, we obtain (28).Definition 13 (Shannon entropy).
The Shannon entropy for a positive probability density functionpθ defined on θ1,θ2 is given by
(29)Sp=−∫θ1θ2pθlogpθdθ.Corollary 14.
Letu,v,w,z:θ1,θ2⟶ℝ+ be integrable functions such that uθ+wθ=1 and vθ+zθ=1 for all θ∈θ1,θ2. Also, let p,q be two positive probability density functions defined on θ1,θ2, then for a subinterval J of θ1,θ2 with J¯=θ1,θ2\J, the following inequalities hold
(30)∫θ1θ2pθlogqθdθ+Sp≤∫Juθpθdθlog∫Juθqθdθ∫Juθpθdθ+∫Jwθpθdθlog∫Jwθqθdθ∫Jwθpθdθ+∫J¯vθpθdθlog∫J¯vθqθdθ∫J¯vθpθdθ+∫J¯zθpθdθlog∫J¯zθqθdθ∫J¯zθpθdθ≤log∫θ1θ2qθdθ.Proof.
Using (28) for the function Ψζ=−logζ, ζ∈I+, we obtain (30).Remark 15.
Forqθ=1, the result (30) becomes
(31)Sp≤∫Juθpθdθlog∫Juθdθ∫Juθpθdθ+∫Jwθpθdθlog∫Jwθdθ∫Jwθpθdθ+∫J¯vθpθdθlog∫J¯vθdθ∫J¯vθpθdθ+∫J¯zθpθdθlog∫J¯zθdθ∫J¯zθpθdθ≤logθ2−θ1.Definition 16 (Kullback-Leibler divergence).
The Kullback-Leibler divergence for two positive probability densitiespθ and qθ defined on θ1,θ2 is given by
(32)KLq,p=∫θ1θ2qθlogqθpθdθ.Corollary 17.
Letu,v,w,z,p,q:θ1,θ2⟶ℝ+ be integrable functions such that pθ and qθ are positive probability density functions and uθ+wθ=1 and vθ+zθ=1 for all θ∈θ1,θ2. Then, for a subinterval J of θ1,θ2 with J¯=θ1,θ2\J, the following inequalities hold
(33)KLq,p≥∫Juθqθdθlog∫Juθqθdθ∫Juθpθdθ+∫Jwθqθdθlog∫Jwθqθdθ∫Jwθpθdθ+∫J¯vθqθdθlog∫J¯vθqθdθ∫J¯vθpθdθ+∫J¯zθqθdθlog∫J¯zθqθdθ∫J¯zθpθdθ≥0.Proof.
Using (28) for the convex function Ψζ=ζlogζ, ζ∈I+, we obtain (33).Definition 18 (variational distance).
The variational distance for two positive probability densitiespθ and qθ defined on θ1,θ2 is given by
(34)Vq,p=∫θ1θ2qθ−pθdθ.Corollary 19.
Letu,v,w,z,p,q and J,J¯ be interpreted as in Corollary 17, then
(35)Vq,p≥∫Juθqθ−pθdθ+∫Jwθqθ−pθdθ+∫J¯vθqθ−pθdθ+∫J¯zθqθ−pθdθ.Proof.
Using the convex functionΨζ=ζ−1 for ζ∈I+ in (28), we obtain (35).Definition 20 (Jeffrey’s distance).
Jeffrey’s distance for two positive probability density functionspθ and qθ defined on θ1,θ2 is given by
(36)Jq,p=∫θ1θ2qθ−pθlogqθpθdθ.Corollary 21.
Letu,v,w,z,p,q and J,J¯ be interpreted as in Corollary 17, then
(37)Jq,p≥∫Juθqθ−pθdθlog∫Juθqθdθ∫Juθpθdθ+∫Jwθqθ−pθdθlog∫Jwθqθdθ∫Jwθpθdθ+∫J¯vθqθ−pθdθlog∫J¯vθqθdθ∫J¯vθpθdθ+∫J¯zθqθ−pθdθlog∫J¯zθqθdθ∫J¯zθpθdθ≥0.Proof.
Using the convex functionΨζ=ζ−1logζ,ζ∈I+ in (28), we obtain (37).Definition 22 (Bhattacharyya coefficient).
The Bhattacharyya coefficient for two positive probability density functionspθ and qθ defined on θ1,θ2 is given by
(38)Bp,q=∫θ1θ2pθqθdθ.Corollary 23.
Letu,v,w,z,p,q and J,J¯ be interpreted as in Corollary 17, then
(39)Bp,q≤∫Juθpθdθ∫Juθqθdθ+∫Jwθpθdθ∫Jwθqθdθ+∫J¯vθpθdθ∫J¯vθqθdθ+∫J¯zθpθdθ∫J¯zθpθdθ≤0.Proof.
Using the convex functionΨζ=−ζ, ζ∈I+ in (28), we obtain (39).Definition 24 (Hellinger distance).
The Hellinger distance for two positive probability density functionspθ and qθ defined on θ1,θ2 is given by
(40)Hq,p=∫θ1θ2qθ−pθ2dθ.Corollary 25.
Letu,v,w,z,p,q and J,J¯ be defined as in Corollary 17, then
(41)Hp,q≥∫Juθqθdθ−∫Juθpθdθ2+∫Jwθqθdθ−∫Jwθpθdθ2+∫Jvθqθdθ−∫Jvθpθdθ2+∫Jzθqθdθ−∫Jzθpθdθ2≥0.Proof.
Using (28) for the convex function Ψζ=ζ−12,ζ∈I+, we obtain (41).Definition 26 (triangular discrimination).
The triangular discrimination for two positive probability density functionspθ and qθ defined on θ1,θ2 is given by
(42)Δq,p=∫θ1θ2qθ−pθ2pθ+qθdθ.Corollary 27.
Letu,v,w,z,p,q and J,J¯ be as stated in Corollary 17, then
(43)Δq,p≥∫Juθqθ−pθdθ2∫Juθpθ+qθdθ+∫Jwθqθ−pθdθ2∫Jwθpθ+qθdθ+∫Jvθqθ−pθdθ2∫Jvθpθ+qθdθ+∫Jzθqθ−pθdθ2∫Jzθpθ+qθdθ≥0.Proof.
Utilizing the convex functionΨζ=ζ−12/ζ+1, ζ∈I+ in (28), we obtain (43).


### 3.1. Further Generalization

In this section, we give more general form of the proposed refinement of the integral Jensen inequality concerning several certain functions.Theorem 28.
Letf:I⟶ℝ be a convex function defined on the interval I. Also, let g,h,uℓi:θ1,θ2⟶ℝ be integrable functions for each i=1,2,⋯,s where gθ∈I,hθ,uℓiθ∈ℝ+ for all θ∈θ1,θ2ℓ=1,2,⋯,n,i=1,2,⋯,s,s∈N and H≔∫θ1θ2hθdθ,∑ℓ=1nuℓiθ=1, for each i. Suppose that L1,L2,⋯,Ls be some nonempty subsets of 1,2,⋯,n such that Lk∩Lt=∅ for k≠t and ∪i=1sLi=1,2,⋯,n. Furthermore, if J1,J2,⋯,Js are some nonempty subintervals of θ1,θ2 such that Jk∩Jt=∅ for k≠t and ∪i=1sJi=θ1,θ2, then the following inequalities hold:
(44)1H∫θ1θ2hθfgθdθ≥1H∫J1∑ℓ∈L1uℓ1θhθdθf∫J1∑ℓ∈L1uℓ1θhθgθdθ∫J1∑ℓ∈L1uℓ1θhθdθ+⋯+1H∫J1∑ℓ∈Lsuℓ1θhθdθf∫J1∑ℓ∈Lsuℓ1θhθgθdθ∫J1∑ℓ∈Lsuℓ1θhθdθ+1H∫J2∑ℓ∈L1uℓ2θhθdθf∫J2∑ℓ∈L1uℓ2θhθgθdθ∫J2∑ℓ∈L1uℓ2θhθdθ+⋯+1H∫J2∑ℓ∈Lsuℓ2θhθdθf∫J2∑ℓ∈Lsuℓ2θhθgθdθ∫J2∑ℓ∈Lsuℓ2θhθdθ+⋮+1H∫Js∑ℓ∈L1uℓsθhθdθf∫Js∑ℓ∈L1uℓsθhθgθdθ∫Js∑ℓ∈L1uℓsθhθdθ+⋯+1H∫Js∑ℓ∈Lsuℓsθhθdθf∫Js∑ℓ∈Lsuℓsθhθgθdθ∫Js∑ℓ∈Lsuℓsθhθdθ≥f1H∫θ1θ2hθgθdθ.
If the functionf is concave, then the reverse inequalities hold in (44).Proof.
Since∑ℓ=1nuℓiθ=∑ℓ∈∪i=1sLiuℓiθ=1 for all θ∈θ1,θ2 and each i=1,2,⋯,s, therefore for the subintervals Ji of θ1,θ2, we have
(45)∫θ1θ2hθfgθdθ=∫J1∑ℓ∈∪i=1sLiuℓ1θhθfgθdθ+∫J2∑ℓ∈∪i=1sLiuℓ2θhθfgθdθ+⋯+∫Js∑ℓ∈∪i=1sLiuℓsθhθfgθdθ=∫J1∑ℓ∈L1uℓ1θhθfgθdθ+⋯+∫J1∑ℓ∈Lsuℓ1θhθfgθdθ+∫J2∑ℓ∈L1uℓ2θhθfgθdθ+⋯+∫J2∑ℓ∈Lsuℓ2θhθfgθdθ+⋮+∫Js∑ℓ∈L1uℓsθhθfgθdθ+⋯+∫Js∑ℓ∈Lsuℓsθhθfgθdθ.
Applying the integral Jensen inequality to all terms on the right hand side of (45), we obtain
(46)1H∫θ1θ2hθfgθdθ≥1H∫J1∑ℓ∈L1uℓ1θhθdθf∫J1∑ℓ∈L1uℓ1θhθgθdθ∫J1∑ℓ∈L1uℓ1θhθdθ+⋯+∫J1∑ℓ∈Lsuℓ1θhθdθf∫J1∑ℓ∈Lsuℓ1θhθgθdθ∫J1∑ℓ∈Lsuℓ1θhθdθ+∫J2∑ℓ∈L1uℓ2θhθdθf∫J2∑ℓ∈L1uℓ2θhθgθdθ∫J2∑ℓ∈L1uℓ2θhθdθ+⋯+∫J2∑ℓ∈Lsuℓ2θhθdθf∫J2∑ℓ∈Lsuℓ2θhθgθdθ∫J2∑ℓ∈Lsuℓ2θhθdθ+⋮+∫Js∑ℓ∈L1uℓsθhθdθf∫Js∑ℓ∈L1uℓsθhθgθdθ∫Js∑ℓ∈L1uℓsθhθdθ+⋯+∫Js∑ℓ∈Lsuℓsθhθdθf∫Js∑ℓ∈Lsuℓsθhθgθdθ∫Js∑ℓ∈Lsuℓsθhθdθ≥f1H∫J1∑ℓ∈L1uℓ1θhθgθdθ+⋯+1H∫J1∑ℓ∈Lsuℓ1θhθgθdθ+1H∫J2∑ℓ∈L1uℓ2θhθgθdθ+⋯+1H∫J2∑ℓ∈Lsuℓ2θhθgθdθ+⋯+1H∫Js∑ℓ∈L1uℓsθhθgθdθ+⋯+1H∫Js∑ℓ∈Lsuℓsθhθgθdθ=f1H∫θ1θ2hθgθdθ,which concludes the result (44).Remark 29.
Using Theorem28 for n=s=2, we obtain Theorem 4. Also, one can present similar applications of Theorem 28 as in the previous sections.


## 3.1. Further Generalization

In this section, we give more general form of the proposed refinement of the integral Jensen inequality concerning several certain functions.Theorem 28.
Letf:I⟶ℝ be a convex function defined on the interval I. Also, let g,h,uℓi:θ1,θ2⟶ℝ be integrable functions for each i=1,2,⋯,s where gθ∈I,hθ,uℓiθ∈ℝ+ for all θ∈θ1,θ2ℓ=1,2,⋯,n,i=1,2,⋯,s,s∈N and H≔∫θ1θ2hθdθ,∑ℓ=1nuℓiθ=1, for each i. Suppose that L1,L2,⋯,Ls be some nonempty subsets of 1,2,⋯,n such that Lk∩Lt=∅ for k≠t and ∪i=1sLi=1,2,⋯,n. Furthermore, if J1,J2,⋯,Js are some nonempty subintervals of θ1,θ2 such that Jk∩Jt=∅ for k≠t and ∪i=1sJi=θ1,θ2, then the following inequalities hold:
(44)1H∫θ1θ2hθfgθdθ≥1H∫J1∑ℓ∈L1uℓ1θhθdθf∫J1∑ℓ∈L1uℓ1θhθgθdθ∫J1∑ℓ∈L1uℓ1θhθdθ+⋯+1H∫J1∑ℓ∈Lsuℓ1θhθdθf∫J1∑ℓ∈Lsuℓ1θhθgθdθ∫J1∑ℓ∈Lsuℓ1θhθdθ+1H∫J2∑ℓ∈L1uℓ2θhθdθf∫J2∑ℓ∈L1uℓ2θhθgθdθ∫J2∑ℓ∈L1uℓ2θhθdθ+⋯+1H∫J2∑ℓ∈Lsuℓ2θhθdθf∫J2∑ℓ∈Lsuℓ2θhθgθdθ∫J2∑ℓ∈Lsuℓ2θhθdθ+⋮+1H∫Js∑ℓ∈L1uℓsθhθdθf∫Js∑ℓ∈L1uℓsθhθgθdθ∫Js∑ℓ∈L1uℓsθhθdθ+⋯+1H∫Js∑ℓ∈Lsuℓsθhθdθf∫Js∑ℓ∈Lsuℓsθhθgθdθ∫Js∑ℓ∈Lsuℓsθhθdθ≥f1H∫θ1θ2hθgθdθ.
If the functionf is concave, then the reverse inequalities hold in (44).Proof.
Since∑ℓ=1nuℓiθ=∑ℓ∈∪i=1sLiuℓiθ=1 for all θ∈θ1,θ2 and each i=1,2,⋯,s, therefore for the subintervals Ji of θ1,θ2, we have
(45)∫θ1θ2hθfgθdθ=∫J1∑ℓ∈∪i=1sLiuℓ1θhθfgθdθ+∫J2∑ℓ∈∪i=1sLiuℓ2θhθfgθdθ+⋯+∫Js∑ℓ∈∪i=1sLiuℓsθhθfgθdθ=∫J1∑ℓ∈L1uℓ1θhθfgθdθ+⋯+∫J1∑ℓ∈Lsuℓ1θhθfgθdθ+∫J2∑ℓ∈L1uℓ2θhθfgθdθ+⋯+∫J2∑ℓ∈Lsuℓ2θhθfgθdθ+⋮+∫Js∑ℓ∈L1uℓsθhθfgθdθ+⋯+∫Js∑ℓ∈Lsuℓsθhθfgθdθ.
Applying the integral Jensen inequality to all terms on the right hand side of (45), we obtain
(46)1H∫θ1θ2hθfgθdθ≥1H∫J1∑ℓ∈L1uℓ1θhθdθf∫J1∑ℓ∈L1uℓ1θhθgθdθ∫J1∑ℓ∈L1uℓ1θhθdθ+⋯+∫J1∑ℓ∈Lsuℓ1θhθdθf∫J1∑ℓ∈Lsuℓ1θhθgθdθ∫J1∑ℓ∈Lsuℓ1θhθdθ+∫J2∑ℓ∈L1uℓ2θhθdθf∫J2∑ℓ∈L1uℓ2θhθgθdθ∫J2∑ℓ∈L1uℓ2θhθdθ+⋯+∫J2∑ℓ∈Lsuℓ2θhθdθf∫J2∑ℓ∈Lsuℓ2θhθgθdθ∫J2∑ℓ∈Lsuℓ2θhθdθ+⋮+∫Js∑ℓ∈L1uℓsθhθdθf∫Js∑ℓ∈L1uℓsθhθgθdθ∫Js∑ℓ∈L1uℓsθhθdθ+⋯+∫Js∑ℓ∈Lsuℓsθhθdθf∫Js∑ℓ∈Lsuℓsθhθgθdθ∫Js∑ℓ∈Lsuℓsθhθdθ≥f1H∫J1∑ℓ∈L1uℓ1θhθgθdθ+⋯+1H∫J1∑ℓ∈Lsuℓ1θhθgθdθ+1H∫J2∑ℓ∈L1uℓ2θhθgθdθ+⋯+1H∫J2∑ℓ∈Lsuℓ2θhθgθdθ+⋯+1H∫Js∑ℓ∈L1uℓsθhθgθdθ+⋯+1H∫Js∑ℓ∈Lsuℓsθhθgθdθ=f1H∫θ1θ2hθgθdθ,which concludes the result (44).Remark 29.
Using Theorem28 for n=s=2, we obtain Theorem 4. Also, one can present similar applications of Theorem 28 as in the previous sections.


## 4. Conclusion

Jensen’s inequality and its refinements can help in various aspects; for example, it helps to authenticate the positivity of Kullback-Leibler divergence, it can be used to obtain some useful estimates for Shannon and Zipf-Mandelbrot entropies and for various divergences in information theory, its gap can be utilized to obtain error bounds in the estimation of certain parameters, and it is useful in the stability analysis of discrete and continuous-time systems with time-varying delay. In this paper, a new refinement of the integral Jensen inequality is proposed with the help of four special type of functions. The refinement is utilized to obtain improved inequalities for various means. Also, new refinements of the Hölder and the Hermite-Hadamard inequalities are obtained. New estimates for several divergences and Shannon entropy are presented. At the end of this paper, more general form of the proposed refinement of the Jensen inequality associated to several special type of functions is established.



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