# Bounds for the Jensen Gap in terms of Power Means with Applications

Authors: Xuexiao You; Muhammad Adil Khan; Hamid Reza Moradi
Journal: Journal of Function Spaces
 (2021)

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## Abstract

Jensen’s and its related inequalities have attracted the attention of several mathematicians due to the fact that Jensen’s inequality has numerous applications in almost all disciplines of mathematics and in other fields of science. In this article, we propose new bounds for the difference of two sides of Jensen’s inequality in terms of power means. An example has been presented for the importance and support of the main results. Related results have been given in quantum calculus. As consequences, improvements of quantum integral version of Hermite-Hadamard inequality have been derived. The obtained inequalities have been applied for some well-known inequalities such as Hermite-Hadamrd, Hölder, and power mean inequalities. Finally, some applications are given in information theory. The tools performed for obtaining the main results may be applied to obtain more results for other inequalities.


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## Body

## 1. Introduction

There is no doubt that one of the most important classes of functions is the class of convex functions. The beauty of convex functions is due to its unique graphical representation, geometrical interpretation, and developments in the theory of inequalities. There are numerous applicable inequalities which have been established for this class of functions such as Jensen’s, the Jensen-Steffensen, and majorization inequalities [1, 2]. One of the most important and widely applicable inequalities which has attracted the attention of many mathematicians is the Jensen inequality [2 , p. 43]. According to this inequality, if ϕ:α1,α2⟶ℝ is a convex function and xi∈α1,α2,pi≥0 for each i∈1,2,⋯,n with Pn≔∑i=1npi>0, then(1)ϕ∑i=1nxipiPn≤1Pn∑i=1npiϕxi.The inequality in (1) flips when the function is concave.The integral version of the Jensen inequality is presented in the following theorem [3].Theorem 1.
Suppose thatα1,α2 is an interval and g1,g2:a1,a2⟶ℝ be functions such that g1y∈α1,α2 for y∈a1,a2. Let the function ϕ:α1,α2⟶ℝ be convex and the functions g2,g1g2,f∘g1.g2 be integrable. Also, assume that g2y≥0 for all y∈a1,a2 and ∫a1a2g2ydy=Γ>0, then(2)ϕ1Γ∫a1a2g1yg2ydy≤1Γ∫a1a2ϕ∘g1yg2ydy.This inequality has been utlized in Economics [4], Engineering [5], Optimization [6], Finance [7], Statistics [8], and Information theory [3, 9, 10]. For a particular convex function, this inequality provides many other inequalities such as AM-GM, Hölder and Ky Fan inequalities.In the literature, discrepancy between the two sides of Jensen’s inequality has been studied by several mathematicians in different directions which provides error bounds for certain approximations. In 2015, Costarelli and Spigler [11] studied the discrepancy between the right and left sides of Jensen’s inequality by using convex functions from the class of C2 functions as well as the class of merely Lipschitz continuous functions. Some illustrative examples are presented and compared the bounds with some existing bounds. In 2018, Pec˘arić et al. [12] focused to find the bounds for the difference of two sides of Jensen’s inequality. They considered some Green convex functions and their related identities and derived bounds for the Jensen gap for the class of C2 functions without using convexity condition. In particular, they have applied the Hölder inequality. Related discrete results have also been obtained, and several applications in information theory are presented. In 2020, Khan et al. [3] introduced a new method for the derivation of bounds using higher-order convex functions that is 4-convexity. First of all, they obtained an identity for the Jensen difference in terms of Green convex functions and double derivative of a function. Further, they obtained bounds by using 4-convexity and some properties of absolute function. Some examples are considered for their main results and compared with earlier bounds. Also, several applications for some well-known inequalities such as Hölder and Hermite-Hadmard inequalities have been given. At the end, several applications in information theory also presented. Related discrete results are given in [13]. In 2021, Khan et al. [14] further modified the method given in [3, 13] and derived several results for Jensen and related inequalities. In this method, they have used real weights and found the integrals of some functions which pertaining Green functions, in a very simple way with the help of the obtained identity for the Jensen gap. By virtue of this procedure, they were able to obtain bounds for the Jensen-Steffensen and converse of Jensen’s inequalities. For more interesting results related to the celebrated Jensen’s inequality, we recommend [15–17].The main results of this manuscript utilize power means and its related inequalities; therefore, we want to mention them in the following part of this section. The following well-known power means and their monotonicity are given in [18, p.19]:For two positive realn-tuples x=x1,⋯,xn and p=p1,⋯,pn, the power mean of order κ∈ℝ is defined by(3)Mκx;p=1Pn∑i=1npixiκ1/κ,ifκ≠0,∏i=1nxipi1/Pn,ifκ=0,where Pn=∑i=1npi.For the aboven-tuples, the quasiarithmetic mean is defined by(4)M~hx;p=h−11Pn∑i=1npihxi,where h is a strictly monotone and continuous function.The integral power mean can be defined as follows: Ifp,g:a,b⟶R+ and g are integrable functions, then the integral power mean of order κ∈ℝ is defined by(5)M¯κg;p=1∫abpωdω∫abpωgκωdω1/κ,ifκ≠0,exp∫abpωloggωdω∫abpωdω,ifκ=0.Ifκ1≤κ2, then(6)Mκ1x;p≤Mκ2x;p,(7)M¯κ1g;p≤M¯κ2g;p.The main aim of this paper is to obtain new interesting bounds for the discrepancy of the two sides of the Jensen inequality using new tools. The bounds pertain power means. We give an example, which shows that the bounds obtained in this paper are better than the earlier ones. We proved Jensen’s inequality for quantum integrals and also derived its improvements. As applications, Hermite-Hadamard inequality and its improvements have been deduced forq-integrals. We also give applications for some well-known inequalities such as Hermite-Harmard, Hölder, power, and quasiarithmetic mean inequalities. At the end, we focused to give applications for Shannon-entropy, Csiszár, and Zipf-Mandelbrot entropy etc.


## 2. Main Results

We begin by presenting our first major finding.Theorem 2.
Letϕ:a,b⟶ℝ be a convex function, xi∈a,b,pi>0fori=1,2,⋯,n with Pn=∑i=1npi. Then, for s≥1 and r≤1, the following inequalities hold:(8)Mry;p≤1Pn∑i=1npiϕxi−ϕx¯≤Msy;p,where x¯=1/Pn∑i=1npixi and y=y1,y2,⋯,yn with yi=ϕxi−ϕx¯−ϕ′+x¯xi−x¯ for i=1,2,⋯,n.Proof.
It is obvious that(9)1Pn∑i=1npiϕxi−ϕx¯=1Pn∑i=1npiϕxi−ϕx¯−ϕ′+x¯xi−x¯=1Pn∑i=1npiyi.
Since the functionϕ is convex, therefore, we have(10)yi≔ϕxi−ϕx¯−ϕ′+x¯xi−x¯≥0.
Hence, by the power mean inequality (6), we have(11)1Pn∑i=1npiyi≤Msy;p,fors≥1,and similarly,(12)Mry;p≤1Pn∑i=1npiyi,forr≤1.
Using (11) and (12) in (9), we obtain (8).In the theorem below, we give an integral version of the above theorem.Theorem 3.
Consider the intervalI and the convex function ϕ:I⟶ℝ and let g,p:a,b⟶ℝ be integrable functions such that gℓ∈I,pℓ∈ℝ+ for all ℓ∈a,b and P=∫abpℓdℓ. Then, for s≥1 and r≤1, we have(13)M¯rh;p≤1P∫abpℓϕgℓdℓ−ϕg¯≤M¯sh;p,where hℓ=ϕgℓ−ϕg¯−ϕ′+g¯gℓ−g¯ with g¯=1/P∫abpℓgℓdℓ, ℓ∈a,b.The next main result is presented in the following theorem.Theorem 4.
Letϕ:a,b⟶ℝ be a convex function, xi∈a,b and pi>0 for i=1,2,⋯,n with Pn=∑i=1npi. Then, for s≥1 and r≤1, we have(14)y¯PnMrp;y≤1Pn∑i=1npiϕxi−ϕx¯≤y¯PnMsp;y,where y=y1,y2,⋯,yn with yi=ϕxi−ϕx¯−ϕ′+x¯xi−x¯ for i=1,2,⋯,n and x¯=1/Pn∑i=1npixi,y¯=∑i=1nyi.Proof.
By simple calculation, we can write(15)1Pn∑i=1npiϕxi−ϕx¯=1Pn∑i=1npiϕxi−ϕx¯−ϕ′+x¯xi−x¯=1Pn∑i=1npiyi.
Since the functionϕ is convex, therefore, we have(16)yi≔ϕxi−ϕx¯−ϕ′+x¯xi−x¯≥0.
Asy¯≔∑i=1nyi, so from (15), we can write(17)1Pn∑i=1npiϕxi−ϕx¯=y¯Pn∑i=1nyipiy¯.
Now, by considering the weightsyi and applying power mean inequality (6), we have(18)∑i=1nyipiy¯≤Msp;y,fors≥1,and similarly,(19)∑i=1nyipiy¯≥Mrp;y,forr≤1.
Using (18) and (19) in (17), we obtain (14).In the forthcoming theorem, we present the integral version of the above theorem.Theorem 5.
Considerϕ:I⟶ℝ be a convex function and p,g:a,b⟶ℝ be integrable functions such that gω∈I,pω∈ℝ+ for all ω∈a,b and P=∫abpωdω. Then, for s≥1 and r≤1, the following inequalities hold:(20)h¯PM¯rp;h≤1P∫abpωϕgωdω−ϕg¯≤h¯PM¯sp;h,where g¯=1/P∫abpωgωdω and h¯=∫abhωdω with hω=ϕgω−ϕg¯−ϕ′+g¯gω−g¯, ω∈a,b.In the following example, we compare our new bound with earlier bounds of the Jensen gap.Example 1.
Let the functionsϕ,g:0,1⟶ℝ be defined by ϕx=x4, gx=x and px=1 for all x∈0,1. Then,(21)P=1,g¯=1P∫01pxgxdx=∫01xdx=12,ϕ′g¯=12,and hx=x4−x/2+3/16, h¯=∫01x4−x/2+3/16dx=0.1375.
Now, we calculate the right hand side of (13) for s=2:(22)M¯2h;p=∫01x4−x2+3162dx1/2=0.2101.
For the same functions, the bound for the Jensen gap from inequality (5) in [3] is a1≔0.25. Also, in bounds for the Jensen gap from the inequalities (6) and (11) in [11], we have a2=0.8748 and a3=0.4998, respectively.
Hence, we concluded thatM¯2h,p=0.2101<ai for i=1,2,3. That is, for these functions, the bound obtained in (13) is better than the earlier bound obtained in [3, 11].
AsM¯sh;p is increasing with respect to s, therefore for more better estimate, we can find M¯sh,p for 1<s<2. For example, M¯1.5h;p is given by(23)M¯1.5h;p=∫01x4−x2+3161.5dx1/1.5≈0.1753,thatisM¯1.5h;p<M¯2h;p.
For inequality (20), we consider the above functions with px=x. For these functions, the value of the Jensen difference is 0.1358, and the value of the bound for the Jensen gap from the inequality (5) in [3] is: b1≔0.2074. Now, we calculate the right hand side of (20).(24)P=0.2,h¯=∫01x4−1.1854x+0.5927dx=0.2,h¯PM¯2p;h=0.20.510.2∫01x4−1.1854x+0.5927x2dx1/2=0.1878.
So,h¯/PM¯2p;h<b1. Similarly, we can take more better bound by considering s=1/2 that is(25)h¯PM¯1.5p;h=0.20.510.2∫01x4−1.1854x+0.5927x1.5dx1/1.5=0.1634.
Hence, in this case, we concluded that the bound obtained in (20) is better than the earlier bound obtained in the inequality (5) in [3].


## 3. Jensen’s Type Inequalities in Quantum Calculus

Theq-calculus or quantum calculus deals with the study of calculus without utlizing the idea of limits. The popular mathematician Euler proposed the ponder q-calculus within the 18th century, while he introduced the term q in Newton’s work of infinite series. Jackson has begun a symmetric study of q-calculus and presented q-definite integrals in twentieth century [19]. The field of q-calculus has various interesting applications in several branches of Physics, Mathematics, and in other areas [20, 21]. This field has gotten extraordinary attention by numerous researchers, and a lot of research is devoted to this field. In [22], the authors defined q-analogue operator of Ruscheweyh type involving multivalent functions and derived several properties. By using the newly presented Harmonic q-Starlike class of functions, some important problems such as distortion limits, necessary and sufficient conditions, convolutions and convexity, and problems with partial sums have been studied in [23]. Srivastava et al. [24] applied the idea of a particular advanced convolution q-operator together with the concept of convolution and analyzed two new classes of meromorphically harmonic functions.First, we give some preliminaries which are useful in our results. Throughout this section,q belongs to 0,1. Also, we assume that all the series which are used in this section are convergent.Definition 6 (see [25]).
Letψ:a,b⟶ℝ be a continuous function and y∈a,b. Then, the q-derivative of the function ψ at y is denoted by aDqψy and defined by(26)aDqψy=ψy−ψqy+1−qa1−qy−a,y≠a,aDqψa=limy⟶aaDqψy.We say thatψ is q-differentiable on a,b if aDqψy exists for all y∈a,b.Definition 7 (see [25]).
Letψ:a,b⟶ℝ be a continuous function. Then, the q-integral on a,b is defined as(27)∫ayψxadqx=1−qy−a∑k=0∞qkψqky+1−qka,for y∈a,b. Moreover, if c∈a,y, then the q-integral on a,b is defined as(28)∫cyψxadqx=∫ayψxadqx−∫acψxadqx.Remark 8.
From Definitions6 and 7, we make the following remarks:(1)
By takinga=0, the expression in (26) becomes the well-known q-derivative, Dqψw, of the function ψ defined by(29)Dqψw=ψy−ψqy1−qy(2)
Also, ifa=0, then (27) reduces to the classical q-integral of a function ψ:0,∞⟶ℝ defined by(30)∫0yψx0dqx=1−qy∑k=0∞qkψqkyNow, we present Jensen’s inequality forq-integrals.Theorem 9.
Letϕ:I⟶ℝ be a continuous convex function defined on the interval I and ψ:a,b⟶I be a continuous function. Then,(31)ϕ1b−a∫abψxadqx≤1b−a∫abϕψxadqx.Proof.
From the convexity ofϕ, we have(32)ϕx−ϕy≥ϕ′+yx−y,forallx,y∈I.
Takingx=ψqkb+1−qka for k=0,1,2,⋯ and y=y¯=1−q∑k=0∞qkψqkb+1−qka=1/b−ab−a∫abψxadqx, in (32), we obtain(33)ϕψqkb+1−qka−ϕy¯≥ϕ′+y¯ψqkb+1−qka−y¯.
Multiplying both sides of (33) by 1−qqk and then taking summation over k, we get(34)1−q∑k=0∞qkϕψqkb+1−qka−ϕy¯≥ϕ+′y¯1−q∑k=0∞qkψqkb+1−qka−y¯.
Sincey¯=1−q∑k=0∞qkψqkb+1−qka, therefore, from (34), we have(35)1−q∑k=0∞qkϕψqkb+1−qka−ϕy¯≥0,which is equivalent to (31).As a consequence of the above theorem, we deduce Hermite-Hadamard inequality for quantum integral. This inequality has been proved in [26, 27].Corollary 10.
Letϕ:a,b⟶ℝ be a continuous convex function. Then,(36)ϕb+aq1+q≤1b−a∫abϕxadqx.Proof.
Ifψx=x, then(37)∫abψxadqx=∫abxadqx=1−qb−a∑k=0∞qkqkb+1−qka=1−qb−ab∑k=0∞q2k+a∑k=0∞qk−a∑k=0∞q2k=1−qb−ab1−q2+a1−q−a1−q2=b+aqb−a1+q.
Hence, usingψx=x in (31), we obtain (36).Now, we give improvement of quantum integral version of Jensen’s inequality in terms of means.Theorem 11.
Letϕ:I⟶ℝ be a continuous convex function defined on the interval I and ψ:a,b⟶I be a continuous function. Then, for s≥1 and r≤1, the following inequalities hold:(38)1−q∑k=0∞qkzks1/s≤1b−a∫abϕψxadqx−ϕ1b−a∫abψxadqx≤1−q∑k=0∞qkzkr1/r,where zk≔ϕψqkb+1−qka−ϕy¯−ϕ′+y¯ψqkb+1−qka−y¯ with y¯=1/b−ab−a∫abψxadqx.Proof.
From the proof of Theorem9, we see that(39)1b−a∫abϕψxadqx−ϕ1b−a∫abψxadqx=1−q∑k=0∞qkϕψqkb+1−qka−ϕy¯−ϕ′+y¯ψqkb+1−qka−y¯.
By convexity ofϕ, we have zk≔ϕψqkb+1−qka−ϕy¯−ϕ′+y¯ψqkb+1−qka−y¯≥0. Also, as q∈0,1 so by geometric series we have ∑k=0∞qk=1/1−q that is 1−q∑k=0∞qk=1/1−q=1 and hence 1−q∑k=0∞qkzk is the arithmetic mean. Now, by using power mean inequality in (39), we obtain(40)1b−a∫abϕψxadqx−ϕ1b−a∫abψxadqx=1−q∑k=0∞qkzk≤1−q∑k=0∞qkzkr1/r.
Similarly, we can prove the left inequality in (42).As an application of the above theorem, we deduce improvement of quantum integral version of Hermite-Hadamard inequality.Corollary 12.
Letϕ:a,b⟶ℝ be a continuous convex function. Then, for s≥1 and r≤1, the following inequalities hold:(41)1−q∑k=0∞qktks1/s≤1b−a∫abϕxadqx−ϕb+aq1+q≤1−q∑k=0∞qktkr1/r.where tk≔ϕqkb+1−qka−ϕb+aq/1+q−ϕ′+b+aq/1+qqkb+1−qka−b+aq/1+q.Proof.
Usingψx=x in (42), we obtain (41).In the following theorem, we present another improvement of quantum integral version of Jensen’s inequality in terms of means.Theorem 13.
Letϕ:I⟶ℝ be a continuous convex function defined on the interval I and ψ:a,b⟶I be a continuous function. Then, for s≥1 and r≤1, the following inequalities hold:(42)1−q∑k=0∞zk∑k=0∞zkqks∑k=0∞zk1/s≤1b−a∫abϕψxadqx−ϕ1b−a∫abψxadqx≤1−q∑k=0∞zk∑k=0∞zkqkr∑k=0∞zk1/r,where zk≔ϕψqkb+1−qka−ϕy¯−ϕ′+y¯ψqkb+1−qka−y¯ with y¯=1/b−ab−a∫abψxadqx.Proof.
The proof can be given by a similar way as the proof of Theorem11.As an application of Theorem13, we deduce improvement of quantum integral version of Hermite-Hadamard inequality.Corollary 14.
Letϕ:a,b⟶ℝ be a continuous convex function. Then, for s≥1 and r≤1, the following inequalities hold:(43)1−q∑k=0∞qktks1/s≤1b−a∫abϕxadqx−ϕb+aq1+q≤1−q∑k=0∞qk∑k=0∞qkrtkr∑k=0∞qk1/r.where tk≔ϕqkb+1−qka−ϕb+aq/1+q−ϕ′+b+aq/1+qqkb+1−qka−b+aq/1+q.Proof.
Usingψx=x in (42), we obtain (43).


## 4. Applications for Some Well-Known Inequalities

In this section, we demonstrate improvements of some well-known inequalities. We start with the Hermite-Hadamard inequality.Corollary 15.
Letϕ:a,b⟶ℝ be a convex function and s≥1, r≤1, then(44)M¯rψ,I≤1b−a∫abϕxdx−ϕa+b2≤M¯sψ,I,where ψx=ϕx−ϕa+b/2−ϕ′+a+b/2x−a+b/2 and Ix=1 is the constant function.Proof.
Applying Theorem5 for pω=1,gω=ω, we obtain (44).In the corollary below, we give improvements of power means inequality.Corollary 16.
Letx=x1,⋯,xn and p=p1,⋯,pn be positive n-tuples with Pn=∑i=1npi and s,r,k1,k2∈ℝ such that s≥1, r≤1 and k1≤k2.(i)
Ifk2≠0 and k1<0, then(45)Mrw,p≤Mk1k1x,p−Mk2k1x,p≤Msw,p(ii)
Ifk1>0, then the reverse inequalities hold in (45)(iii)
Ifk1≠0 and k2≥0, then(46)Mrz,p≤Mk2k2x,p−Mk1k2x,p≤Msz,p(iv)
Ifk2<0, then the reverse inequalities hold in (46)
wherew=w1,w2,⋯,wn with wi=xik1+k1−k2/k2Mk2k1x,p−k1/k2xik2Mk2k1−k2x,p and z=z1,z2,⋯,zn with zi=xik2+k2−k1/k1Mk2k1x,p−k2/k1xik1Mk1k2−k1x,p for i=1,2,⋯,n.Proof.
(i)
Ifk1<0, k1≤k2 and ϕx=xk1/k2, then ϕ is convex. Therefore, applying Theorem 4 for this function and xi⟶xik2, we obtain (45)(ii)
Ifk1>0, then ϕx=xk1/k2 is a concave function, so applying Theorem 4 for concave function, we deduce the reverse inequality in (45)(iii)
Ifk1≠0, k2≥0 and ϕx=xk2/k1, then ϕ is convex. Therefore, applying Theorem 4 for this function and xi⟶xik1, we obtain (46)
Similarly, we can prove the reverse inequality in (46).Corollary 17.
Letp=p1,⋯,pn and x=x1,⋯,xn be positive n-tuples with Pn=∑i=1npi and s,r,∈ℝ such that s≥1, r≤1. Let the function h be monotone continuous function and f∘h−1 be convex function, then(47)Mrz¯,p≤1Pn∑i=1npifxi−fM~hx,p≤Msz¯,p,where z¯=z¯1,z¯2,⋯,z¯n with z¯i=fxi−fM~hx,p−f∘h−1′+1/Pn∑i=1npihxihxi−1/Pn∑i=1npihxi for i=1,2,⋯,n.Proof.
The inequality (47) can be obtained by using Theorem 2 for ϕ⟶f∘h−1 and xi⟶hxi for i=1,2,⋯,n.The following two results are devoted to improvements of Hölder inequality.Proposition 18.
Letγ1,γ2,⋯,γn,η1,η2,⋯,ηn be positive n-tuples and s,l,m∈ℝbe such that s≥1 and l,m>1 with 1/l+1/m=1, then(48)∑i=1nηim1/m∑i=1nγil1/l−∑i=1nηiγi≤∑i=1nγil1−1/ms∑i=1nγilηimζi−l−∑i=1nγiηi∑i=1nγilm−m∑i=1nγiηi∑i=1nγilm−1ηiγi−l/m−∑i=1nηiγi∑i=1nγils1/ms.Proof.
Letϕz=zm, then clearly ϕ is convex for z>0 and m>1, therefore, using the right inequality in (8) for ϕz=zm,pi=γil, and xi=ηiγi−l/m, and then simplifying, we derive that(49)∑i=1nηim∑i=1nγilm−1−∑i=1nηiγim1/m≤∑i=1nγil1−1/ms∑i=1nγilηimζi−l−∑i=1nγiηi∑i=1nγilm−m∑i=1nγiηi∑i=1nγilm−1ηiγi−l/m−∑i=1nηiγi∑i=1nγils1/ms.
Since the inequalityk1η−k2η≤k1−k2η holds for η∈0,1 and 0≤k2≤k1, therefore by putting k1=∑i=1nηim∑i=1nγilm−1, k2=∑i=1nηiγim, and η=1/m, we obtain(50)∑i=1nηim1/m∑i=1nγil1/l−∑i=1nηiγi≤∑i=1nηim∑i=1nγilm−1−∑i=1nηiγim1/m.
Now, (49) and (50) give (48).Corollary 19.
Letγ1,γ2,⋯,γn,η1,η2,⋯,ηn be positive n-tuples and let s,r,l,m∈ℝ be such that 0<l<1,m=l/l−1 and s≥1,r≤1, then(51)Mrj,ηm∑i=1nηim≤∑i=1nηiγi−∑i=1nηim1/m∑i=1nγil1/l≤Msj,ηm∑i=1nηim,where ηm=η1m,η2m,⋯,ηnm and j=j1,j2,⋯,jn with ji=γiηi−m/l−∑i=1nγil/∑i=1nηim1/l−1/l∑i=1nγil/∑i=1nηim1/l−1γiηil−m−∑i=1nγil/∑i=1nηim.Proof.
As the functionϕx=x1/l is convex for x>0,l∈0,1. So by using (8) for ϕx=x1/l,pi=ηim, and xi=γilηi−m, we get (51).Remark 20.
Similarly, we can present applications of the second main results. Also, we can give integral version of the above results.


## 5. Applications in Information Theory

Information theory studies how to measure, store, and transmit digital information. The field of information theory was initially entrenched by Hartley’s work in 1920 and got worldwide attention by the work of Shannon in 1940s. The field of information theory is in close collaboration with probability theory, statistical mechanics, information, and electrical engineering. A fundamental measure in information theory is entropy that measures the uncertainty involved in the occurrence of a random process. After Shannon’s work, the field attracted the concentration of various scientists and got inflated. Different entropy functional have been introduced that can be elaborated as generalized entropies.Definition 21.
TheCsiszárf-divergence for two positive n-tuples l=l1,l2,⋯,ln, m=m1,m2,⋯,mn is defined by(52)Cfl,m=∑i=1nmiflimi,where f:0,∞⟶ℝ is a convex function.Theorem 22.
Letl=l1,l2,⋯,ln, m=m1,m2,⋯,mn be positive n-tuples and f:0,∞⟶ℝ be a convex function. If s≥1 and r≤1, then(53)Mrz,m∑i=1nmi≤Cfl,m−f∑i=1nli∑i=1nmi∑i=1nmi≤Msz,m∑i=1nmi,where z=z1,z2,⋯,zn with zi=fli/mi−f∑i=1nli/∑i=1nmi−f′+∑i=1nli/∑i=1nmili/mi−∑i=1nli/∑i=1nmi for i=1,2,⋯,n.Proof.
Using (8) for ϕ=f, xi=li/mi, and pi=mi for i∈1,2,⋯,n, we obtain (53).Definition 23 (Shannon entropy).
Letq=q1,q2,⋯,qnbe positive probability distribution, then the Shannon entropy is defined by(54)Sq=−∑i=1nqilogqi.Corollary 24.
Letm=m1,m2,⋯,mn be a positive probability distribution. Ifs≥1 and r≤1, then(55)Mrz,m≤logn−Sm≤Msz,m,where z=z1,z2,⋯,zn with zi=logmi+logn+1/nmi−1 for i=1,2,⋯,n.Proof.
Takingfξ=−logξ,ξ∈0,∞, li=1, for each i∈1,2,⋯,n, in (53), we obtain (55).Corollary 25.
Letm=m1,m2,⋯,mn, l=l1,l2,⋯,ln be positive n-tuples with ∑i=1nli=1. Ifs≥1 and r≤1, then(56)Mrz,m∑i=1nmi≤−Sl−∑i=1nlilogmi+log∑i=1nmi≤Msz,m∑i=1nmi,where z=z1,z2,⋯,zn with zi=li/milogli/mi−1/∑i=1nmilog1/∑i=1nmi−1−log∑i=1nmili/mi−1/∑i=1nmi for i=1,2,⋯,n.
In particular, ifmi=1 for each i=1,2,⋯,n, then(57)nMrz,i≤−Sl−lognn≤nMsz,i,where i=1,1,⋯,1 and z=z1,z2,⋯,zn with zi=lilogli+1/nlogn−1−lognli−1/n for i=1,2,⋯,n.Proof.
Takingfx=x,x∈0,∞, for each i∈1,2,⋯,n, in (53), we obtain (56).Definition 26.
The Kullback-Leibler divergence for two positive probability distributionsl=l1,l2,⋯,ln and m=m1,m2,⋯,mn is defined by(58)Kdl,m=∑i=1nliloglimi.Corollary 27.
Letl=l1,l2,⋯,ln, m=m1,m2,⋯,mn be positive probability distributions, then for s≥1 and r≤1, we have(59)Mrz,m≤Kdl,m≤Msz,m,where z=z1,z2,⋯,zn with zi=li/milogli/mi−2li/mi−1 for i=1,2,⋯,n.Proof.
Takingfx=xlogx,x∈0,∞, in (53), we obtain (59).Corollary 28.
Assume that all the assumptions of Corollary27 hold. Then,(60)Mrz,m≤Kdm,l≤Msz,m,where z=z1,z2,⋯,zn with zi=−logli/mi+li/mi−1 for i=1,2,⋯,n.Proof.
Takingfx=−logx,x∈0,∞, in (53), we obtain (60).Definition 29.
Ifl=l1,l2,⋯,ln and m=m1,m2,⋯,mn are two positive probability distributions, then the variational distance is defined by(61)Vdl,m=∑i=1nli−mi.Corollary 30.
Assume that all the assumptions of Corollary27 hold. Then,(62)Mrz,m≤Vdl,m≤Msz,m,where z=z1,z2,⋯,zn with zi=∣li/mi−1∣−li/mi+1 for i=1,2,⋯,n.Proof.
Using the functionfx=x−1,x∈0,∞, in (53), we obtain (62).Definition 31.
Ifl=l1,l2,⋯,ln and m=m1,m2,⋯,mn are two positive probability distributions, then the Jeffrey distance is defined by(63)Jdl,m=∑i=1nli−miloglimiz.Corollary 32.
Assume that all the assumptions of Corollary27 hold. Then,(64)Mrz,m≤Jdl,m≤Msz,m,where z=z1,z2,⋯,zn with zi=li/mi−1logli/mi for i=1,2,⋯,n.Proof.
Applying the functionfx=x−1logx,x∈0,∞, in (53), we obtain (64).Definition 33.
Ifl=l1,l2,⋯,ln and m=m1,m2,⋯,mn are positive probability distributions, then the Bhattacharyya coefficient is defined by(65)Bdl,m=∑i=1nlimi.Corollary 34.
Assume that all the assumptions of Corollary27 hold, then(66)Mrz,m≤1−Bdl,m≤Msz,m,where z=z1,z2,⋯,zn with zi=−li/mi+li/2mi+1/2 for i=1,2,⋯,n.Proof.
Using the functionfx=−x,x∈0,∞, in (53), we obtain (66).Definition 35.
Ifl=l1,l2,⋯,ln and m=m1,m2,⋯,mn are positive probability distributions, then the Hellinger distance is defined by(67)Hdl,m=∑i=1nli−mi2.Corollary 36.
Under the assumptions of Corollary27, the following inequality holds:(68)Mrz,m≤Hdl,m≤Msz,m,where z=z1,z2,⋯,zn with zi=li/mi−12 for i=1,2,⋯,n.Proof.
Using the functionfx=x−12,x∈0,∞, in (53), we obtain (68).Definition 37.
For two positive probability distributionsm=m1,m2,⋯,mn, l=l1,l2,⋯,ln, the triangular discrimination is defined by(69)Tdl,m=∑i=1nli−mi2li+mi.Corollary 38.
Assume that all the assumptions of Corollary27 hold, then(70)Mrz,m≤Tdl,m≤Msz,m,where z=z1,z2,⋯,zn with zi=li−mi2/mili+mi for i=1,2,⋯,n.Proof.
As the functionϕx=x−12/x+1,x∈0,∞ is convex. Therefore applying the function fx=ϕx, in (53), we obtain (70).Now, we begin to give inequalities for Zipf-Mandelbrot entropy. Before presenting the results, we include some introductory part about Zipf-Mandelbrot entropy.Zipf’s law is a fundamental and helpful law in the field of information science. In the field of social studies, this law was regarded as a valuable statistical distribution procedure. The law demonstrate the relation between the size and the rank of discrete phenomena. Furthermore, this law is also applied to the intensity of solar flares, earth quack magnitude, geology, city populations, the size of moon craters, and website traffic etc. It has had limited success in the realm of geology when it comes to assessing petroleum and other mining.Mandelbrot invented the Zipf-Mandelbrot law in 1966, which is a further enhancement of Zipf’s law [28]. The low-rank words in the corpus are taken into account by this law, where i<100 [29]: li=c/i+εs, if one put ε=0; the law of Zipf will then be deduced. There are numerous applications of Zipf-Mandelbrot law, which are available in linguistics [29, 30] and information sciences [31] and is also generally useful in ecological field studies [32]. The Zipf-Mandelbrot entropy ZeT,ε,s is given by(71)ZeT,ε,s=sTn,ε,s∑i=1nlogi+εi+εs+logTn,ε,s,where ε≥0,n∈ℕ, s>0, Tn,ε,s=∑i=1n1/i+εs and the Zipf-Mandelbrot law is given by: ZMLi,n,ε,s=1/i+εs/Tn,ε,s.Now, we are in position to give inequalities for Zipf-Mandelbrot entropy.Corollary 39.
Letε≥0, s,mi>0, i=1,2,..,n with ∑i=1nmi=1. Ifs≥1 and r≤1, then(72)Mrz,m≤−ZeT,ε,s−∑i=1nlogmii+εsTn,ε,s≤Msz,m,where z=z1,z2,⋯,zn with zi=1/mii+εsTn,ε,slog1/mii+εsTn,ε,s−21/mii+εsTn,ε,s−1 for i=1,2,⋯,n.Proof.
Letli=1/i+εsTn,ε,s, i∈1,2,⋯,n, then(73)∑i=1nlilogli=∑i=1n1i+εsTn,ε,slog1i+εsTn,ε,s=−∑i=1n1i+εsTn,ε,slogi+εsTn,ε,s=−∑i=1nsi+εsTn,ε,slogi+ε−∑i=1nlogTn,ε,si+εsTn,ε,s=−sTn,ε,s∑i=1nlogi+εi+εs−logTn,ε,sTn,ε,s∑i=1n1i+εs=−ZeT,ε,s.SinceTn,ε,s=∑i=1n1/ε+is, therefore, ∑i=1n1/i+εsTn,ε,s=1. Hence, using (59) for li=1/i+εsTn,ε,s, i=1,2,..,n, we obtain (72).Utilizing two Zipf’s law associated to distinct parameters, we derive estimation for Zipf-Mandelbrot entropy.Corollary 40.
Letξ1,ξ2≥0, s1,s2>0. Ifs≥1 and r≤1, then(74)Mrz,m≤−ZeT,ξ1,s1+∑i=1nlogi+ξ2s2Tn,ξ2,s2i+ξ1s1Tn,ξ1,s1≤Msz,m,where z=z1,z2,⋯,zn with zi=i+ξ2s2Tn,ξ2,s2/i+ξ1s1Tn,ξ,s1logi+ξ2s2Tn,ξ2,s2/i+ξ1s1Tn,ξ1,s1−2i+ξ2s2Tn,ξ2,s2/i+ξs1Tn,ξ1,s1−1, i=1,2,⋯,n.Proof.
Letli=1/i+ξ1s1Tn,ξ1,s1 and mi=1/i+ξ2s2Tn,ξ2,s2, i=1,2,⋯,n, then as in the proof of Corollary 39, we have(75)∑i=1nlilogli=∑i=1n1i+ξ1s1Tn,ξ1,s1log1i+ξ1s1Tn,ξ1,s1=−ZeT,ξ1,s1,∑i=1nlilogmi=∑i=1n1i+ξ1s1Tn,ξ1,s1log1i+ξ2s2Tn,ξ2,s2=−∑i=1nlogi+ξ2s2Tn,ξ2,s2i+ξ1s1Tn,ξ1,s1.
Also,∑i=1nli=∑i=1n1/i+ξ1s1Tn,ξ1,s1=1 and ∑i=1nmi=∑i=1n1/i+ξ2s2Tn,ξ2,s2=1.
Therefore using (59) for li=1/i+ξ1s1Tn,ξ1,s1 and mi=1/i+ξ2s2Tn,ξ2,s2, i=1,2,⋯,n, we obtain (74).Remark 41.
Similarly, we can use (60) and derive inequalities for Zipf-Mandelbrot entropy.


## 6. Conclusion

In the literature, there are several results which are devoted to Jensen’s and its related inequalities. In this manuscript, we have used a new approach for the derivation of improvements of Jensen’s inequality. We have given Jensen’s inequality and its improvements in quantum calculus. In particular, we have deduced Hermite-Hadamard type inequalities forq-integrals. We have also given applications of main results for some well-known inequalities and in information theory. The results of this manuscript which are initiated and given in quantum calculus may stimulate further research.



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