# Estimations of the Slater Gap via Convexity and Its Applications in Information Theory

Authors: Muhammad Adil Khan; Hidayat Ullah; Tareq Saeed; Hamed H. Alsulami; Z. M. M. M. Sayed; Ahmed Mohammed Alshehri
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Convexity has played a prodigious role in various areas of science through its properties and behavior. Convexity has booked record developments in the field of mathematical inequalities in the recent few years. The Slater inequality is one of the inequalities which has been acquired with the help of convexity. In this note, we obtain some estimations for the Slater gap while dealing with the notion of convexity in an extensive manner. We acquire the deliberated estimations by utilizing the definition of convex function, Jensen’s inequality for concave functions, and triangular, power mean, and Hölder inequalities. We discuss several consequences of the main results in terms of inequalities for the power means. Moreover, by utilizing the main results, we give estimations for the Csiszár and Kullback–Leibler divergences, Shannon entropy, and the Bhattacharyya coefficient. Furthermore, we present some estimations for the Zipf–Mandelbrot entropy as additional applications of the acquired results. The perception and approaches adopted in this note may pretend more research in this direction.


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## Body

## 1. Introduction

In the diverse fields of science, convex functions are of the greatest importance due to their dominant manners and wealthy structure [1–3]. In recent years, the abundant applicability of convex functions has been observed in engineering [4], differential equations [5], epidemiology [6], information theory [7], statistics [8], optimization [9], and many others. Moreover, convex functions have some unique properties and, due to such properties, it became a focus point for researchers [10, 11]. Furthermore, convex functions have also been generalized, refined, and extended in different directions while utilizing their characteristics and behavior [12]. In a formal way, the convex function can be defined as follows.Definition 1.
Leta,b be an interval in ℝ and Ψ be a real-valued function defined on a,b. Then, the function Ψ is said to be convex, if the inequality(1)Ψβx+1−βy≤βΨx+1−βΨy,holds, for all x,y∈a,b and β∈0,1.
If the inequality (1) is valid in the opposite direction, then the function Ψ is said to be concave.
The field of mathematical inequalities is one of the favorable areas for the class of convex functions, where it has been employed extensively. Many inequalities would not be conceivable to prove without convex functions [13–17]. Majorization [18], Favard [19], Jensen–Mercer [20], and Hermite–Hadamard [21] inequalities are some of the important inequalities which have been acquired with the use of convex functions. The Jensen inequality [22] is one of the most important inequalities among the aforementioned inequalities for the class of convex functions. Jensen’s inequality has a very strong relationship with ordinary convexity in the sense that it generalizes the definition of convex function. Another important fact about this inequality is that it is the origin of many other classical inequalities [23]. The mathematical form of Jensen’s inequality is stated in the next theorem.Theorem 1.
Assume thatΨ is a real-valued convex function defined on a,b and yi∈a,b, qi≥0 for each i∈1,2,…,m with ∑i=1mqi>0; then,(2)Ψ1∑i=1mqi∑i=1mqiyi≤1∑i=1mqi∑i=1mqiΨyi.IfΨ is a concave function, then (2) holds in the opposite direction.The continuous form of (2) is verbalized in the following theorem.Theorem 2.
LetΨ:c,d⟶ℝ be any convex function and assume that g1,g2:a,b⟶c,d are arbitrary integrable functions such that g1y≥0. If Ψ∘g2 is an integrable function and ∫abg1ydy>0, then(3)Ψ∫abg1yg2ydy∫abg1ydy≤∫abg1yΨg2ydy∫abg1ydy.IfΨ is a concave function, then (3) holds in the opposite direction.The Jensen inequality has a variety of interesting properties and also has a very desirable structure [24]. Furthermore, there are a huge number of applications of this inequality in the different fields of science [4, 7, 25, 26]. Due to the huge importance and applicability of this inequality, a lot of work has been carried out on it [8, 27]. In 1980, Slater [28] presented a companion inequality to the aforementioned inequality while using a convex function, which is well known as Slater’s inequality in the literature. The formal form of the Slater inequality is given below.Theorem 3.
LetΨ:a,b⟶ℝ be any function and qi≥0, yi∈a,b for each i∈1,2,…,m such that ∑i=1mqi>0 and ∑i=1mqiΨ+′yi≠0. If the function Ψ is convex and increasing, then(4)∑i=1mqiΨyi∑i=1mqi≤Ψ∑i=1mqiyiΨ+′yi∑i=1mqiΨ+′yi.In 1985, Pečarić [29] generalized the Slater inequality by relaxing the monotonicity condition of the function.Theorem 4.
Assume thatΨ:a,b⟶ℝ is any convex function and qi≥0, yi∈a,b for each i∈1,2,…,m such that ∑i=1mqi>0 and ∑i=1mqiΨ+′yi≠0. If ∑i=1mqiyiΨ+′yi/∑i=1mqiΨ+′yi∈a,b, then (4) is true.
Inequality (4) can also be written in the following form:(5)Ψ∑i=1mqiyiΨ+′yi∑i=1mqiΨ+′yi−∑i=1mqiΨyi∑i=1mqi≥0.Throughout the article, by the Slater gap (difference), we shall mean that the left side of the above inequality.In the recent decades, a lot of work has been dedicated to Slater’s inequality by numerous researchers from different angles by following different techniques and methods. In 1985, Pečarić [30] acquired some extensions of Slater’s inequality by taking convex functions of several real variables. In 2000, Matić and Pečarić [31] established a couple of companion inequalities to Jensen’s inequality in both discrete and integral versions by utilizing convex functions. Then, they used these generalized inequalities and deduced Slater’s as well as some related inequalities. In 2010, Khan and Pečarić [32] obtained an improvement and a reversion of Slater’s inequality by utilizing some earlier established results. In 2013, Khan et al. [33] presented refinements of Slater’s and other related inequalities of the Slater type for convex functions defined on linear spaces. Moreover, they also provided refinements for majorization type inequalities. In 2018, Song et al. [34] established Slater’s inequality for strongly convex functions and also presented some more results of the Jensen type for the aforementioned class of convex functions.


## 2. Estimations of the Slater Gap

In this section, we are going to establish some new estimations for the Slater gap. The proposed estimations will be acquired by utilizing the definition of convex function, Jensen’s inequality for concave functions, Hölder, power mean, and triangular inequalities. First, we state a lemma in which a general inequality is constructed while using a twice differentiable function.Lemma 1.
Presume thatyi∈a,b, qi∈ℝ for each i∈1,2,…,m with ∑i=1mqi≠0, and also let Ψ:a,b⟶ℝ be any function such that Ψ″ exists. If ∑i=1mqiΨ′yi≠0 and ∑i=1mqiyiΨ′yi/∑i=1mqiΨ′yi∈a,b, then(6)Ψ∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi−1∑i=1mqi∑i=1mqiΨyi≤∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi2×∫01tΨ″tyi+1−t∑i=1mqiyiΨ′yi∑i=1mqiΨ′yidt.Proof.
Without loss of generality, letyi≠∑i=1mqiyiΨ′yi/∑i=1mqiΨ′yi for each i∈1,2,…,m. Utilizing the integration by parts rule, we have(7)1∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi2∫01tΨ″tyi+1−t∑i=1mqiyiΨ′yi∑i=1mqiΨ′yidt=1∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi2×tyi−∑i=1mqiyiΨ′yi/∑i=1mqiΨ′yiΨ′tyi+1−t∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi|01−1yi−∑i=1mqiyiΨ′yi/∑i=1mqiΨ′yi∫01Ψ′tyi+1−t∑i=1mqiyiΨ′yi∑i=1mqiΨ′yidt=1∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi21yi−∑i=1mqiyiΨ′yi/∑i=1mqiΨ′yiΨ′yi−1yi−∑i=1mqiyiΨ′yi/∑i=1mqiΨ′yi2Ψ.tyi+1−t∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi|01=−1∑i=1mqi∑i=1mqiΨ′yi+Ψ∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi.
From this, we can write that(8)Ψ∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi−1∑i=1mqi∑i=1mqiΨyi=1∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi2×∫01tΨ′tyi+1−t∑i=1mqiyiΨ′yi∑i=1mqiΨ′yidt.
Instantly, taking absolute of (8) and then applying the triangular inequality, we acquire (6).
In the following theorem, we obtain an estimate for the Slater gap by applying the definition of the convex function.Theorem 5.
Assume that the hypotheses of Lemma1 are true, and further let the function Ψ″p be convex for p>1. Then,(9)Ψ∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi−1∑i=1mqi∑i=1mqiΨyi≤∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi2×q+1Ψ″yip+Ψ″∑i=1mqiyiΨ′yi/∑i=1mqiΨ′yipq+1q+21/p.Proof.
Applying Hölder inequality on the right side of (6), we obtain(10)Ψ∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi−1∑i=1mqi∑i=1mqiΨyi≤∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨyi∑i=1mqiΨ′yi2×∫01tΨ″tyi+1−t∑i=1mqiyiΨ′yi∑i=1mqiΨ′yipdt1/p.
By utilizing the convexity ofΨ″p on the right side of (10), we acquire(11)Ψ∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi−1∑i=1mqi∑i=1mqiΨyi≤∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi2×Ψ′′yip∫01tp+1dt+Ψ″∑i=1mqiyiΨ′yi∑i=1mqiΨ′yip∫01tp−tp+1dt1/p.
Now, evaluating the integrals in (11), we receive (9).
Utilizing Jensen’s inequality for concave functions, we receive an estimate for the Slater gap stated in the following theorem.Theorem 6.
Suppose that the assumptions of Lemma1 are true. Furthermore, if |Ψ″|p is concave for p〉1, then(12)Ψ∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi−1∑i=1mqi∑i=1mqiΨyi≤∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi2×1/p+11/pΨ″p+1yi+∑i=1mqiyiΨ′yi/∑i=1mqiΨ′yip+2.Proof.
By utilizing (10), we acquire(13)Ψ∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi−1∑i=1mqi∑i=1mqiΨyi≤∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi2×1/p+11/p∫01tpΨ″tyi+1−t∑i=1mqiyiΨ′yi/∑i=1mqiΨ′yipdt∫01tpdt1/p.
As the functionΨ″p is concave, therefore by applying Jensen’s inequality to (13), we obtain(14)Ψ∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi−1∑i=1mqi∑i=1mqiΨyi≤∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi2×1/p+11/pΨ″yi∫01tp+1dt+∑i=1mqiyiΨ′yi/∑i=1mqiΨ′yi∫01tp−tp+1dt∫01tpdt.
Inequality (12) can easily be deduced by finding the integrals in (14).
In the following theorem, we construct an inequality which provides an estimate for the Slater gap while using the definition of the convex function and the renowned Hölder inequality.Theorem 7.
Let all the assumptions of Theorem5 be valid. Additionally, if s〉1 such that 1/p+1/s=1, then(15)Ψ∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi−1∑i=1mqi∑i=1mqiΨyi≤1/s+11/s∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi2×Ψ″yip+Ψ″∑i=1mqiyiΨ′yi/∑i=1mqiΨ′yip21/p.Proof.
From (6), we can write that(16)Ψ∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi−1∑i=1mqi∑i=1mqiΨyi≤∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi2×∫01tΨ″tyi+1−t∑i=1mqiyiΨ′yi∑i=1mqiΨ′yidt.
Now, using Hölder inequality on the right side of (16), we obtain(17)Ψ∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi−1∑i=1mqi∑i=1mqiΨyi≤∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi2×1/s+11/s∫01Ψ″tyi+1−t∑i=1mqiyiΨ′yi∑i=1mqiΨ′yipdt1/p.
Instantly, applying the definition of convex function on the right side of (17), we get(18)Ψ∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi−1∑i=1mqi∑i=1mqiΨyi≤∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi2×1/s+11/sΨ″yip∫01tdt+Ψ″∑i=1mqiyiΨ′yi∑i=1mqiΨ′yip∫011−tdt1/p.
Finding integrals in (18), we obtain (15).
By utilizing the Hölder inequality and Jensen’s inequality for concave functions, we obtain an estimate for the Slater gap given in the following theorem.Theorem 8.
Let the assumptions of Theorem6 be hold. Moreover, if s>1 such that 1/p+1/s=1, then(19)Ψ∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi−1∑i=1mqi∑i=1mqiΨyi≤∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi2×1/s+11/sΨ″yi+∑i=1mqiyiΨ′yi/∑i=1mqiΨ′yi2.Proof.
Utilizing the Jensen inequality on the right side of (17), we deduce(20)Ψ∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi−1∑i=1mqi∑i=1mqiΨyi≤∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi2×1/s+11/sΨ″yi∫01tdt+∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi∫011−tdt.
Now, evaluating integrals in (20), we obtain (19).
We receive another estimate for the Slater gap by applying the power mean inequality and the definition of convex function which states the following.Theorem 9.
Let the conditions of Theorem5 be true. Then,(21)Ψ∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi−1∑i=1mqi∑i=1mqiΨyi≤1/21−1/p∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi2×2Ψ″yip+Ψ″∑i=1mqiyiΨ′yi/∑i=1mqiΨ′yip61/p.Proof.
Applying the power mean inequality on the right side of (16), we receive(22)Ψ∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi−1∑i=1mqi∑i=1mqiΨyi≤1/21−1/p∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi2×∫01tΨ″tyi+1−t∑i=1mqiyiΨ′yi∑i=1mqiΨ′yiqdt1/p.
Now, utilizing the convexity ofΨ″p on the right side of (22), we acquire(23)Ψ∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi−1∑i=1mqi∑i=1mqiΨyi≤121−1/p∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi2×|Ψ″yi|p∫01t2dt+Ψ″∑i=1mqiyiΨ′yi∑i=1mqiΨ′yip∫01t−t2dt1/p.
By finding the integrals in (23), we get (21).
In the following theorem, we acquire an estimate for the Slater gap by utilizing the power mean inequality and the famous Jensen’s inequality for concave functions.Theorem 10.
Let all the suppositions of Theorem6 be valid. Then,(24)Ψ∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi−1∑i=1mqi∑i=1mqiΨyi≤12∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi2×Ψ″2yi+∑i=1mqiyiΨ′yi/∑i=1mqiΨ′yi3.Proof.
Since the functionΨ″p is concave, therefore, by applying Jensen’s inequality on the right side of (22), we arrive(25)Ψ∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi−1∑i=1mqi∑i=1mqiΨyi≤12∑i=1mqi∑i=1mqiyi−∑i=1mqiyiΨ′yi∑i=1mqiΨ′yi2×Ψ″yi∫01t2dt+∑i=1mqiyiΨ′yi/∑i=1mqiΨ′yi∫01t−t2dt∫01tdt.
Instantly, simplifying (25), we deduce (24).


## 3. Applications for the Power Means

In this section of the note, we will establish some new relations for the power means. The intended relations will be acquired by putting some particular convex functions in the main results. Now, we recall the definition of the power mean.Definition 2.
For any positivem− tuples s1=γ1,γ2,…,γm and s2=ζ1,ζ2,…,ζm with ∑i=1mγi=γ¯, the power mean of the order u∈ℝ is defined as(26)Mus1,s2=∑i=1mγiζiuγ¯1/u,u≠0,∏i=1mζiγi1/γ¯,u=0.
The following corollary is the consequence of Theorem5 for the power means.Corollary 1.
Assume thats1=γ1,γ2,…,γm and s2=ζ1,ζ2,…,ζm are positive tuples such that ∑i=1mγi=γ¯ and p>1. Also, let u,t∈ℝ with t<u. Then, the following statements are true:(i)
Ift,u>0, then(27)Mtts1,s2−γ¯Mtts1,s2∑i=1mγiζit−ut/u≤tu−tγ¯u2∑i=1mγiζiu−γ¯Mtts1,s2∑i=1mγiζit−u2×p+1ζipt−2u+γ¯Mtts1,s2/∑i=1mγiζit−upt/u−2p+1p+21/p.(ii)
Ift,u<0 with t/u∉2,2+1/p, then(28)γ¯Mtts1,s2∑i=1mγiζit−ut/u−Mtts1,s2≤tt−uγ¯u2∑i=1mγiζiu−γ¯Mtts1,s2∑i=1mγiζit−u2×p+1ζipt−2u+γ¯Mtts1,s2/∑i=1mγiζit−upt/u−2p+1p+21/p.(iii)
Ifu>0 and t<0, then (28) holds.Proof.
(i)
LetΨy=yt/u be defined on 0,∞. Then, Ψ″=t/ut/u−1yt/u−2 and Ψ″p″=t/ut/u−1ppt/u−2pt/u−2−1ypt/u−2−2. Clearly, Ψ″ is negative, and Ψ″p″ is positive on 0,∞ for the mentioned values of t,u, and p, which implies that the function Ψ is concave and the function Ψ″p is convex. Hence, utilizing (9) for Ψy=yt/u, qi=γi, and yi=ζiu, we acquire (27).(ii)
For the given values oft,u, and p, both the functions Ψ and Ψ″p are convex. Therefore, by following the procedure of i, we obtain (28).(iii)
Obviously, the functionsΨ and Ψ″p are both convex on 0,∞ for the specified values of t,u, and p. Therefore, the inequality (28) can easily be deduced by adopting the procedure of i.
We utilized Theorem6 and constructed a relation for the power means stated in the following corollary.Corollary 2.
Lets1=γ1,γ2,…,γm and s2=ζ1,ζ2,…,ζm be any positive tuples with ∑i=1mγi=γ¯ and p>1. Furthermore, assume that u,t<0 such that t<u and t/u∈2,2+1/p. Then,(29)γ¯Mtts1,s2∑i=1mγiζit−uu,t−Mtts1,s2≤1p+11/ptt−uγ¯u2∑i=1mγiζiu−γ¯Mtts1,s2∑i=1mγiζit−u2×p+1ζiu+γ¯Mtts1,s2/∑i=1mγiζit−up+2t/u−2.Proof.
ConsiderΨy=yt/u,y>0. Then, clearly Ψ and Ψ″p are convex and concave functions, respectively, with respect to the given conditions. Therefore, to obtain (29), put Ψy=yt/u, qi=γi, and yi=ζiu in (12).
In the following corollary, we acquire some relations for the power means as a consequence of Theorem7.Corollary 3.
Assume that all hypotheses of Corollary1 are true. Moreover, if s>1 such that 1/p+1/s=1, then we have the following assertions:(i)
Ift,u>0, then(30)Mtts1,s2−γ¯Mtts1,s2∑i=1mγiζit−ut/u≤1s+11/stu−tγ¯u2∑i=1mγiζiu−γ¯Mtts1,s2∑i=1mγiζit−u2×ζipt−2u+γ¯Mtts1,s2/∑i=1mγiζit−upt/u−221/p.(ii)
Ifu,t<0 with t/u∉2,2+1/p, then(31)γ¯Mtts1,s2∑i=1mγiζit−ut/u−Mtts1,s2≤1s+11/stt−uγ¯u2∑i=1mγiζiu−γ¯Mtts1,s2∑i=1mγiζit−u2×ζipt−2u+γ¯Mtts1,s2/∑i=1mγiζit−upt/u−221/p.(iii)
Ifu>0 and t<0, then (31) holds.Proof.
First, we prove (30). For this, consider Ψx=yt/u defined on 0,∞. Then, certainly for t,u>0, and p>1, the function Ψ is concave, and the function Ψ″p is convex. Therefore, inequality (30) can easily be acquired by taking Ψy=yt/u, qi=γi, and yi=ζiu in (15).
Now, we prove (31). Obviously, both the functions Ψ and Ψ″p are convex on 0,∞ for t,u<0, and p>1. Therefore, by taking Ψy=yt/u, qi=γi, and yi=ζiu in (15), we receive (3.29).
For the case, whenu>0 and t<0, follow the procedure of (31). □
We utilized Theorem8 and obtained a relation for power means, which is stated in the next corollary.Corollary 4.
Suppose that all the assumptions of Corollary2 are true. Furthermore, if s>1 such that 1/s+1/p=1, then(32)γ¯Mtts1,s2∑i=1mγiζit−ut/u−Mtts1,s2≤1s+11/stt−uγ¯u2∑i=1mγiζiu−γ¯Mtts1,s2∑i=1mγiζit−u2×ζiu+γ¯Mtts1,s2/∑i=1mγiζit−u2t/u−2.Proof.
Let us takeΨx=yt/u defined 0,∞. Then, Ψ is a convex function, and Ψ″p is a concave function on 0,∞ for t,u<0 with t/u∈2,2+1/p and p>1. Hence, to deduce (32), use Ψy=yt/u, qi=γi, and yi=ζiu in (19).
The following corollary is the consequence of Theorem9 for the power means.Corollary 5.
Let all the suppositions of Corollary1 be held. Then, the following statements are valid:(i)
Ift,u>0, then(33)Mtts1,s2−γ¯Mtts1,s2∑i=1mγiζit−ut/u≤121−1/ptu−tγ¯u2∑i=1mγiζiu−γ¯Mtts1,s2∑i=1mγiζit−u2×2ζipt−2u+γ¯Mtts1,s2/∑i=1mγiζit−upt/u−261/p.(ii)
Ifu,t<0 with t/u∉2,2+1/p, then(34)γ¯Mtts1,s2∑i=1mγiζit−ut/u−Mtts1,s2≤121−1/ptt−uγ¯u2∑i=1mγiζiu−γ¯Mtts1,s2∑i=1mγiζit−u2×2ζipt−2u+γ¯Mtts1,s2/∑i=1mγiζit−upt/u−261/p.(iii)
Ifu>0 and t<0, then (34) holds.Proof.
(i) LetΨy=yt/u,y>0. Then surely, the functions Ψ and Ψ″p are concave and convex, respectively, with respect to the given conditions. Therefore, using Ψy=yt/u, qi=γi, and yi=ζiu in (21), we arrive (33).
ii−iii For the specified conditions on t,u, and p given in the cases ii and iii, respectively, both the functions Ψ and Ψ″p are convex on 0,∞. Therefore, to deduce (3.32), follow the procedure of i.
We receive the following relation for the power means as a consequence of Theorem10.Corollary 6.
Let all the hypotheses of Corollary2 be true. Then,(35)γ¯Mtts1,s2∑i=1mγiζit−ut/u−Mtts1,s2≤tt−u2γ¯u2∑i=1mγiζiu−γ¯Mtts1,s2∑i=1mγiζit−u2×2ζiu+γ¯Mtts1,s2/∑i=1mγiζit−u3t/u−2.Proof.
ConsiderΨy=yt/u defined on 0,∞, then undoubtedly, the functions Ψ and |Ψ″|p are convex and concave for the stated conditions. Therefore, to get (35), assume Ψy=yt/u, qi=γi, and yi=ζiu in (24). □
The following are some more relations for the power means as a consequence of Theorem5.Corollary 7.
Lets1=γ1,γ2,…,γmands2=ζ1,ζ2,…,ζmbem−tuples such that∑i=1mγi=γ¯andp>1. Then,(i)
(36)M0s1,s2M−1s1,s2≤exp1γ¯∑i=1mγiζi−M−1s1,s22×p+1ζi−2p+M−1−2ps1,s2p+1p+21/p.(ii)
(37)expM22s1,s2M1s1,s2−M1s1,s2≤1γ¯∑i=1mγilogζi−M22s1,s2M1s1,s22×p+1ζip+expM22s1,s2/M1s1,s2p+1p+21/p.Proof.
First, we prove (36). For this, let us take Ψy=−logy defined on 0,∞. Then obviously, Ψ and Ψ″p are convex functions with the mentioned condition. Therefore, by putting Ψy=−logy, qi=γi, and yi=ζi in (9), we obtain (36).
Now, we prove (37). For this, consider Ψy=expy defined on −∞,∞. Then, certainly Ψ and Ψ″p are convex on −∞,∞ for the specified value of p. Therefore, to receive (37), use Ψy=expy, qi=γi, and yi=logζi in (9).
The following corollary provides relations for the power means, which can easily be obtained from Theorem7.Corollary 8.
Let the assumptions of Corollary7 be true. Moreover, if s>1 such that 1/p+1/s=1, then(i)
(38)M0s1,s2M−1s1,s2≤exp1s+11/s1γ¯∑i=1mγiζi−M−1s1,s22×ζi−2p+M−1−2ps1,s221/p.(ii)(39)expM22s1,s2M1s1,s2−M1s1,s2≤1s+11/s1γ¯∑i=1mγilogζi−M22s1,s2M1s1,s22×ζip+expM22s1,s2/M1s1,s221/p.Proof.
Inequality (38) can easily be obtained by putting Ψy=−logy,y>0, qi=γi, and yi=ζi in (15).
To acquire (39), we need to consider Ψy=expy,y∈−∞,∞, qi=γi, and yi=logζi in (15).
The next corollary is the consequence of Theorem9 for the power means.Corollary 9.
Suppose that all the assumptions of Corollary7 are valid, then(i)
(40)M0s1,s2M−1s1,s2≤exp121−1/p1γ¯∑i=1mγiζi−M−1s1,s22×2ζi−2p+M−1−2ps1,s261/p.(ii)
(41)expM22s1,s2M1s1,s2−M1s1,s2≤121−1/p1γ¯∑i=1mγilogζi−M22s1,s2M1s1,s22×2ζip+expM22s1,s2/M1s1,s261/p.Proof.
PuttingΨy=−logy,y>0, qi=γi, and yi=ζi in (21), we get (40).
Now, to obtain (41), take Ψy=expy,y∈−∞,∞, qi=γi, and yi=logζi in (21).


## 4. Applications in Information Theory

In this section, we are going to present applications of the main results in the information theory. The applications will provide different estimations for Csiszár and Kullback–Leibler divergences, Shannon entropy, and for the Bhattacharyya coefficient. To proceed to the desired goals, first we state the definition of Csiszár divergence.Definition 3.
LetΨ:0,∞⟶ℝ be any function and s1=γ1,γ2,…,γm, s2=ζ1,ζ2,…,ζm be arbitrary positive m− tuples. Then, the Csiszár divergence is defined by(42)CΨs1,s2=∑i=1mγiΨζiγi.
The following theorem is an application of Theorem5 for the Csiszár divergence.Theorem 11.
Assume thats1=γ1,γ2,…,γm and s2=ζ1,ζ2,…,ζm are positive m− tuples such that ∑i=1mγi=γ¯. Furthermore, let Ψ″p be a convex function on 0,∞ for p>1 and ∑i=1mζiΨ′ζi/γi/∑i=1mγiΨ′ζi/γi>0. Then,(43)Ψ∑i=1mζiΨ′ζi/γiCΨ′s1,s2−CΨs1,s2γ¯≤1γ¯∑i=1mγiζiγi−∑i=1mζiΨ′ζi/γiCΨ′s1,s22×p+1Ψ″ζi/γip+Ψ″∑i=1mζiΨ′ζi/γi/CΨ′s1,s2pp+1p+21/p.Proof.
To receive (42), put qi=γi and yi=ζi/γi in (9).
The following theorem provides a bound for the Csiszár divergence.Theorem 12.
Assume thats1=γ1,γ2,…,γm and s2=ζ1,ζ2,…,ζm are positive m− tuples such that ∑i=1mγi=γ¯. If ∑i=1mζiΨ′ζi/γi/∑i=1mγiΨ′ζi/γi∈0,∞, and Ψ″p is a concave function on 0,∞ for p>1, then(44)Ψ∑i=1mζiΨ′ζi/γiCΨ′s1,s2−CΨs1,s2γ¯≤1γ¯p+11/p∑i=1mγiζiγi−∑i=1mζiΨ′ζi/γiCΨ′s1,s22×Ψ″p+1ζi/γi+∑i=1mζiΨ′ζi/γi/CΨ′s1,s2p+2.Proof.
Inequality (44) can easily be acquired by assuming qi=γi and yi=ζi/γi in (12).
Utilizing Theorem7, we acquire an estimate for the Csiszár divergence.Theorem 13.
Let all the conditions of Theorem11 be true. Additionally, if s>1 such that 1/p+1/s=1, then(45)Ψ∑i=1mζiΨ′ζi/γiCΨ′s1,s2−CΨs1,s2γ¯≤1γ¯s+11/s∑i=1mγiζiγi−∑i=1mζiΨ′ζi/γiCΨ′s1,s22×Ψ″ζi/γip+Ψ″∑i=1mζiΨ′ζi/γi/CΨ′s1,s2p21/p.Proof.
Utilizing (15) by taking qi=γi and yi=ζi/γi, we arrive (45).
The following inequality gives an estimate for the Csiszár divergence, which can be acquired from Theorem8.Theorem 14.
Suppose that all the assumptions of Theorem12 are valid. Furthermore, if s>1 such that 1/p+1/s=1, then(46)Ψ∑i=1mζiΨ′ζi/γiCΨ′s1,s2−CΨs1,s2γ¯≤1γ¯s+11/s∑i=1mγiζiγi−∑i=1mζiΨ′ζi/γiCΨ′s1,s22×Ψ″ζi/γi+∑i=1mζiΨ′ζi/γi/CΨ′s1,s22.Proof.
Puttingqi=γi and yi=ζi/γi in (19), we obtain (46).
With the help of Theorem9, we establish a relation for the Csiszár divergence, which is stated in the next theorem.Theorem 15.
Let all the assumptions of Theorem11 be satisfied. Then,(47)Ψ∑i=1mζiΨ′ζi/γiCΨ′s1,s2−CΨs1,s2γ¯≤121−1/pγ¯∑i=1mγiζiγi−∑i=1mζiΨ′ζi/γiCΨ′s1,s22×2Ψ″ζi/γip+Ψ″∑i=1mζiΨ′ζi/γi/CΨ′s1,s2p61/p.Proof.
To obtain (47), use qi=γi and yi=ζi/γi in (21). □
The following estimate for the Csiszár divergence is deduced from Theorem10.Theorem 16.
Suppose that all the hypotheses of Theorem12 are true, then(48)Ψ∑i=1mζiΨ′ζi/γiCΨ′s1,s2−CΨs1,s2γ¯≤12γ¯∑i=1mγiζiγi−∑i=1mζiΨ′ζi/γiCΨ′s1,s22×Ψ″2ζi/γi+∑i=1mζiΨ′ζi/γi/CΨ′s1,s23.Proof.
Useqi=γi and yi=ζi/γi in (24), we get (48).
Now, recall the definition of Shannon entropy.Definition 4.
For any positive probability distributions1=γ1,γ2,…,γm, the Shannon entropy is defined as(49)Ss1=−∑i=1mγilogγi.
The following corollary gives a bound for the Shannon entropy.Corollary 10.
Lets1=γ1,γ2,…,γm be any probability distribution, and p>1. Then,(50)log∑i=1mγi2+Ss1≤∑i=1mγi1γi−1∑i=1mγi22∑i=1mγi22pp+1p+2p+1γi2p+∑i=1mγi22pp+1p+21/p.Proof.
To acquire the inequality (50), let us consider the function Ψy=−logy, which is defined on 0,∞. Then by successive differentiations, we have Ψ″p=y−2 and Ψ″p=2p2p+1y−2p+2. Clearly, both Ψ″ and Ψ″p are positive on 0,∞ for p>1, which admit that the functions Ψ and Ψ″p are convex. Therefore, by using Ψy=−logy and ζi=1,i=1,2,…,m in (43), we obtain (50).
The following inequality provides an estimate for the Shannon entropy.Corollary 11.
Assume that all the conditions of Corollary10 are valid. Moreover, if s>1 such that 1/p+1/s=1, then(51)log∑i=1mγi2+Ss1≤1s+11/s∑i=1mγi1γi−1∑i=1mγi22γi2p+∑i=1mγi22p21/p.Proof.
TakingΨy=−logy,y>0, and ζi=1,i=1,2,…,m in (45), we receive (51). □
As applications of Theorem9, we receive a relation for Shannon entropy stated in the coming corollary.Corollary 12.
Suppose that the assumptions of Corollary10 are satisfied, then(52)log∑i=1mγi2+Ss1≤121−1/p∑i=1mγi1γi−1∑i=1mγi222γi2p+∑i=1mγi22p61/p.Proof.
Utilizing (47) for Ψy=−logy,y>0, and ζi=1,i=1,2,…,m, we deduce (52).
Instantly, we state the definition of Kullback–Leibler divergence.Definition 5.
For any positive probability distributionss1=γ1,γ2,…,γm and s2=ζ1,ζ2,…,ζm, the Kullback–Leibler divergence is defined as(53)Kds1,s2=∑i=1mγilogγiζi.
The following corollary is an application of Theorem5 for the Kullback–Leibler divergence.Corollary 13.
Assume thats1=γ1,γ2,…,γm and s2=ζ1,ζ2,…,ζm are positive probability distributions, and p>1, then(54)log∑i=1mγi2ζi−Kds1,s2≤∑i=1mγiζiγi−∑i=1mγi2ζi−12×p+1γi/ζi2p+∑i=1mγi2/ζi2pp+1p+21/p.Proof.
To get (54), we need to just put Ψy=−logy,y>0 in (43).
An application of Theorem7 for the Kullback–Leibler divergence is given in the next corollary.Corollary 14.
Let the assumptions of Corollary13 be true. Furthermore, if s>1 such that 1/p+1/s=1, then(55)log∑i=1mγi2ζi−Kds1,s2≤1s+11/s∑i=1mγiζiγi−∑i=1mγi2ζi−12×γi/ζi2p+∑i=1mγi2/ζi2p21/p.Proof.
Applying inequality (45) while taking Ψy=−logy,y>0, we acquire (55). □
The next corollary gives an estimate for the Kullback–Leibler divergence.Corollary 15.
Assume that all the conditions of Corollary13 are satisfied, then(56)log∑i=1mγi2ζi−Kds1,s2≤121−1/p∑i=1mγiζiγi−∑i=1mγi2ζi−12×2γi/ζi2p+∑i=1mγi2/ζi2p61/p.Proof.
UsingΨy=−logy,y>0 in (47), we get (56).
The Bhattacharyya coefficient can be defined as follows.Definition 6.
For any positive probability distributionss1=γ1,γ2,…,γm and s2=ζ1,ζ2,…,ζm, the Bhattacharyya coefficient is defined as(57)Bcs1,s2=∑i=1mγiζi.
The following corollary provides a bound for the Bhattacharyya coefficient.Corollary 16.
Lets1=γ1,γ2,…,γm and s2=ζ1,ζ2,…,ζm be any positive probability distributions, and p>1. Then,(58)Bcs1,s2−Bcs1,s2∑i=1mγi3/2ζi−1/2≤14∑i=1mγiζiγi−Bcs1,s2∑i=1mγi3/2ζi−1/22×p+1γi/ζi2p/3+∑i=1mγi3/2ζi−1/2/Bcs1,s22p/3p+1p+21/p.Proof.
Consider the functionΨy=−y, which is defined on 0,∞. Then, Ψ″y=1/4y−3/2 and Ψ″=1/4p3p/23p/2+1y−3p/2−2, which implies that Ψ‴ and Ψ″ are positive on 0,∞ for p>1. Thus, this shows that both the functions Ψ and Ψ″ are convex. Hence, utilizing (43) for Ψy=−y, we obtain (58).
An estimate for the Bhattacharyya coefficient is stated in the coming corollary.Corollary 17.
Assume that all hypotheses of Corollary16 are valid. Furthermore, if s>1 such that 1/p+1/s=1, then(59)Bcs1,s2−Bcs1,s2∑i=1mγi3/2ζi−1/2≤14s+11/s∑i=1mγiζiγi−Bcs1,s2∑i=1mγi3/2ζi−1/22×γi/ζi2p/3+∑i=1mγi3/2ζi−1/2/Bcs1,s22p/321/p.Proof.
To obtain (59), we need to utilize Ψy=−y,y>0 in (45).
Another estimate for the Bhattacharyya coefficient is stated in the following corollary.Corollary 18.
Suppose that the assumptions of Corollary16 are true, then(60)Bcs1,s2−Bcs1,s2∑i=1mγi3/2ζi−1/2≤123−1/p∑i=1mγiζiγi−Bcs1,s2∑i=1mγi3/2ζi−1/22×2γi/ζi2p/3+∑i=1mγi3/2ζi−1/2/Bcs1,s22p/361/p.Proof.
To deduce (60), use Ψy=−y,y>0 in (47).


## 5. Applications for the Zipf–Mandelbrot Entropy

The present section of the note is dedicated to the Zipf–Mandelbrot entropy. Here, we establish a number of relations for the aforementioned entropy by using the main results. First, we discuss some basics about the aforementioned entropy.For anyθ≥0,v>0, i∈1,2,…,m, and m∈1,2,…, the generalized harmonic number is defined as follows:(61)Mm,θ,v=∑i=1m1i+θv.The relation(62)Φi,m,θ,v=1/i+θvMm,θ,s.represents the probability mass function for the Zipf–Mandelbrot law.The mathematical form of the Zipf–Mandelbrot entropy is given below:(63)ZM,θ,v=sMm,θ,v∑i=1mlogi+θi+θv+logMm,θ,v.The following corollary gives an estimate for the Zipf–Mandelbrot entropy as an application of Theorem5.Corollary 19.
Lets1=ζ1,ζ2,…,ζm be a probability distribution, and p>1. If θ≥0 and v>0, then(64)ZM,θ,v+1Mm,θ,v∑i=1mlogζii+θv+log1Mm,θ,v2∑i=1m1i+θ2vζi≤1Mm,θ,v∑i=1m1i+θvMm,θ,vi+θvζi−1Mm,θ,v2∑i=1m1i+θ2vζi−12×p+1Mm,θ,vi+θvζi−2p+1/Mm,θ,v2∑i=1m1/i+θ2vζi2pp+1p+21/p.Proof.
Takingγi=1/Mm,θ,vi+θv in (54), we acquire (64). □
The following corollary is the application of Theorem5 for the Zipf–Mandelbrot entropy.Corollary 20.
Assume thatθ1,θ2≥0 and v1,v2>0. If p>1, then(65)ZM,θ1,v1−v2Mm,θ1,v1∑i=1mlogi+θ2i+θ1v1−logMm,θ2,v2+logMm,θ2,v2Mm,θ1,v12∑i=1mi+θ2v2i+θ12v1≤1Mm,θ,v1∑i=1m1i+θ1v1Mm,θ1,v1i+θ1v1Mm,θ2,v2i+θ2v2−Mm,θ2,v2Mm,θ1,v12∑i=1mi+θ2v2i+θ12v1−12×p+1Mm,θ2,v2i+θ2v2/Mm,θ1,v1i+θ1v12p+Mm,θ2,v2/Mm,θ1,v12∑i=1mi+θ2v2/i+θ12v12pp+1p+21/p.Proof.
Inequality (65) can easily be deduced by putting γi=1/Mm,θ1,v1i+θ1v1 and ζi=1/Mm,θ2,v2i+θ2v2 in (54).
In the following corollary, we acquire an estimate for the Zipf–Mandelbrot entropy.Corollary 21.
Assume that all the suppositions of Corollary19 are satisfied. Moreover, if s>1 such that 1/p+1/s=1, then(66)ZM,θ,v+1Mm,θ,v∑i=1mlogζii+θv+log1Mm,θ,v2∑i=1m1i+θ2vζi≤1Mm,θ,vs+11/s∑i=1m1i+θv×Mm,θ,vi+θvζi−1Mm,θ,v2∑i=1m1i+θ2vζi−12×Mm,θ,vi+θvζi−2p+1/Mm,θ,v2∑i=1m1/i+θ2vζi2p21/p.Proof.
By utilizing (55) for γi=1/Mm,θ,vi+θv, we obtain (66).
In the following corollary, an estimation for the Zipf–Mandelbrot entropy is obtained as an application of Theorem7.Corollary 22.
Let the conditions of Corollary20 be true. Moreover, if s>1 such that 1/s+1/p=1, then(67)ZM,θ1,v1−v2Mm,θ1,v1∑i=1mlogi+θ2i+θ1v1−logMm,θ2,v2+logMm,θ2,v2Mm,θ1,v12∑i=1mi+θ2v2i+θ12v1≤1Mm,θ1,v1s+11/s∑i=1m1i+θ1v1×Mm,θ1,v1i+θ1v1Mm,θ2,v2i+θ2v2−Mm,θ2,v2Mm,θ1,v12∑i=1mi+θ2v2i+θ12v1−12×Mm,θ2,v2i+θ2v2/Mm,θ1,v1i+θ1v12p+Mm,θ2,v2/Mm,θ1,v12∑i=1mi+θ2v2/i+θ12v12p21/p.Proof.
Takingγi=1/Mm,θ1,v1i+θ1v1 and ζi=1/Mm,θ2,v2i+θ2v2 in (55), we receive (67).
The following corollary is an application of Theorem9.Corollary 23.
Suppose that the assumptions of Corollary19 are valid, then(68)ZM,θ,v+1Mm,θ,v∑i=1mlogζii+θv+log1Mm,θ,v2∑i=1m1i+θ2vζi≤121−1/p1Mm,θ,v∑i=1m1i+θv×Mm,θ,vi+θvζi−1Mm,θ,v2∑i=1m1i+θ2vζi−12×2Mm,θ,vi+θvζi−2p+1/Mm,θ,v2∑i=1m1/i+θ2vζi2p61/p.Proof.
Useγi=1/Mm,θ,vi+θv in (56) to arrive (68). □
The following corollary provides an estimate for the Zipf–Mandelbrot entropy.Corollary 24.
Assume that all the suppositions of Corollary20 are valid, then(69)ZM,θ1,v1−v2Mm,θ1,v1∑i=1mlogi+θ2i+θ1v1−logMm,θ2,v2+logMm,θ2,v2Mm,θ1,v12∑i=1mi+θ2v2i+θ12v1≤121−1/p1Mm,θ,v1∑i=1m1i+θ1v1×Mm,θ1,v1i+θ1v1Mm,θ2,v2i+θ1v2−Mm,θ2,v2Mm,θ1,v12∑i=1mi+θ2v2i+θ12v1−12×2Mm,θ2,v2i+θ2v2/Mm,θ1,v1i+θ1v12p+Mm,θ2,v2/Mm,θ1,v12∑i=1mi+θ2v2/i+θ12v12p61/p.Proof.
To acquire (69), we need to put γi=1/Mm,θ1,v1i+θ1v1 and ζi=1/Mm,θ2,v2i+θ2v2 in (56).


## 6. Conclusions

The mathematical inequalities have a wealthy history with the diverse type of applications in several areas of science. It has been observed that a lot of inventive concepts about mathematical inequalities and their applications can be received with the support of convex functions. The Slater inequality is one of the inequalities which has been established with the help of convex functions. This inequality is actually a companion inequality to the famous Jensen’s inequality. In this note, we obtained some estimations for the Slater inequality by utilizing the definition of the convex function, Jensen’s inequality for concave functions, Hölder, power mean, and triangular inequalities. We gave different relations for the power means as consequences of the acquired estimations. Furthermore, we presented some applications of the received estimations in the form of inequalities for Csiszár and Kullback–Leibler divergence, Shannon entropy, and the Bhattacharyya coefficient. Moreover, we gave some additional applications of the established estimations for Zipf–Mandelbrot entropy.



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