# Refinements of Jensen’s Inequality via Majorization Results with Applications in the Information Theory

Authors: Yongping Deng; Hidayat Ullah; Muhammad Adil Khan; Sajid Iqbal; Shanhe Wu
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this study, we present some new refinements of the Jensen inequality with the help of majorization results. We use the concept of convexity along with the theory of majorization and obtain refinements of the Jensen inequality. Moreover, as consequences of the refined Jensen inequality, we derive some bounds for power means and quasiarithmetic means. Furthermore, as applications of the refined Jensen inequality, we give some bounds for divergences, Shannon entropy, and various distances associated with probability distributions.


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## Body

## 1. Introduction

The Jensen inequality is one of the most significant and fundamental inequalities in the current literature of mathematical inequalities. The Jensen inequality got more importance due to the fact that many inequalities such as Hölder inequality, Minkowski’s inequality, Ky Fan’s inequality, Levinson’s inequality, and Hermite–Hadamard inequality can be obtained from this inequality. Also, many problems arising in different fields of science can be solved with the help of the Jensen inequality [1–3]. Moreover, several applications of the Jensen inequality have been observed in the estimations of several divergences and the well-known Zipf–Mandelbrot entropy [4–9]. Due to the wide significance of the Jensen inequality, many researchers gave several extensions, improvements, generalizations, and refinements of this inequality. Furthermore, this inequality has been also used as a foundation for exploring the generalized convex functions such as s-convex function, η-convex function, coordinate convex function, and h-convex function.The discrete form of the Jensen inequality is given in the following theorem.Theorem 1.
Assume thatΨ:I⟶ℝ is a convex function (I is an arbitrary interval), ζk∈I and ηk≥0, for all k∈1,2,…,n, with η¯=∑k=1nηk>0, then(1)Ψ1η¯∑k=1nηkζk≤1η¯∑k=1nηkΨζk.If a functionΨ is concave, then (1) is valid in the contrary direction.One of the most attractive features of the Jensen inequality is that it generalized the definition of the convex function. Recently, many researchers devoted their attention to the Jensen inequality and contributed to it in various directions. In 2006, Dragomir [10] presented new refinements of the Jensen inequality by using a convex function defined on the linear space. He also gave some applications of the obtained results in the information theory. Pečarić and Perić [11] established some improvements of the converse of the Jensen inequality for linear functional, and some applications of the obtained results were also presented therein. Xiao et al. [12] obtained a new extension and refinement of the Jensen inequality for midconvex functions by using the arithmetic means of the given function.In the remaining portion of the present section, we introduce some fundamental theory and essential results of majorization, which will help us in the obtaining our new results.Majorization is a connection between two vectors, which shows that the components of one vector is “more nearly equal” than the components of the second vector. Due to this property, the concept of majorization becomes a focus point for the researchers of different fields. Many mathematical inequalities have been established with the help of majorization [13]. Over the years, the concept of majorization has been applied widely as a powerful tool in the area of pure and applied mathematics, as well as some other related fields. Majorization also performed an extremely significant character in the theory of matrices. Several matrix inequalities and norm inequalities have been deduced by the majorization relation among the Eigen values and singular values of matrices [14]. For some literature related to majorization theory, refer [15–18].For fixedn≥2, let(2)s1=ζ1,ζ2,…,ζn,s2=ϑ1,ϑ2,…,ϑn,be two n-tuples of real numbers, and we denote the components of s1 and s2 in the decreasing order by(3)ζ1≥ζ2≥⋯≥ζn,ϑ1≥ϑ2≥⋯≥ϑn.Definition 1.
The tuples2 is said to be majorized by the tuple s1, namely, s1 majorizes s2, if(4)∑k=1lϑk≤∑k=1lζk,forl=1,2,…,n−1,∑k=1nϑk=∑k=1nζk.
Symbolically, it can be written ass2≺s1.
The following theorem is very famous in the literature as majorization theorem, and it can be found in [13].Theorem 2.
Lets1=ζ1,ζ2,…,ζn and s2=ϑ1,ϑ2,…,ϑn be two n-tuples, such that ζk,ϑk∈Ik=1,2,…,n. Then, the inequality(5)∑k=1nΨϑk≤∑k=1nΨζk,is true for each continuous convex function Ψ:I⟶ℝ if and only if s2≺s1.The generalization of the above majorization inequality is given in following theorem [19, 20].Theorem 3.
Let a functionΨ:I⟶ℝ be convex and s1=ζ1,ζ2,…,ζnands2=ϑ1,ϑ2,…,ϑn be two n-tuples, such that ζk,ϑk∈Ik=1,2,…,n and s3=η1,η2,…,ηn be nonnegative n-tuples satisfying(6)∑k=1lηkϑk≤∑k=1lηkζk,forl=1,…,n−1,(7)∑k=1nηkϑk=∑k=1nηkζk.If the tuples2 is decreasing, then the following inequality holds:(8)∑k=1nηkΨϑk≤∑k=1nηkΨζk.In the following theorem, Dragomir [21] obtained a weighted majorization inequality with the help of support line inequality and Chebyshev’s inequality.Theorem 4.
Suppose that a functionΨ:I⟶ℝ is convex and s1=ζ1,ζ2,…,ζn and s2=ϑ1,ϑ2,…,ϑn are any two n-tuples, such that ζk,ϑk∈Ik=1,2,…,n, and s3=η1,η2,…,ηn is a nonnegative real n-tuple. Also, suppose that condition (7) is satisfied. If s2 and s1-s2 are monotonic in the parallel direction, then(9)∑k=1nηkΨϑk≤∑k=1nηkΨζk.Remark 1.
Dragomir also obtained inequality (9) by using strict condition that Ψ is an increasing convex function while using the relax condition that ∑k=1nηkϑk≤∑k=1nηkζk instead of using (7).
The majorization inequalities are generalized, extended, and refined in many directions. Adil Khan et al. [22] extended the majorization type inequalities from the convex functions defined on the intervals to the convex functions defined on the rectangles. They used the main result to obtain some extensions for the weighted Favard’s inequality. Wu et al. [23] obtained some refinements of majorization-type inequalities with the help of Taylor theorem with the mean value form of the remainder and also presented an application. Ullah et al. [24] established some improvements and generalizations of majorization-type inequalities with the help of a strongly convex function and also gave some applications of the obtained results.
The aim of this study is to establish some refinements of the famous Jensen equality with the help of majorization theory. We use some basic and fundamental results of majorization to obtain some new refinements of the Jensen inequality. We also establish refined inequalities for different means. As applications, we obtain new bounds for the Csiszár divergence, Kullback–Leibler divergence, and Shannon entropy. Furthermore, we also establish bounds for the various distances, Bhattacharyya coefficient, and triangular discrimination.


## 2. Main Results

We begin this section with the following result, in which we obtain a refinement of the Jensen inequality with the help of Theorem2.Theorem 5.
LetΨ:I⟶ℝ be a convex function, and let s1=ζ1,ζ2,…,ζn and s2=ϑ1,ϑ2,…,ϑn be two n-tuples, such that ζk,ϑk∈I for all k∈1,2,…,n. Also, let ℓ∈0,1 and ζ¯=1/n∑k=1nζk. If s2≺s1, then(10)Ψζ¯≤1n∑k=1nΨ1−ℓζ¯+ℓϑk≤1−ℓΨζ¯+ℓn∑k=1nΨϑk≤1n∑k=1n1−ℓΨζk+ℓn∑k=1nΨϑk≤1n∑k=1nΨζk.IfΨ is a concave function, then the inequalities given in (10) are true in the opposite direction.Proof.
Sinces2≺s1, using the second condition of majorization, we have(11)Ψ1n∑k=1nζk=Ψ1n∑k=1n1−ℓζ¯+ℓϑk.
Now, applying Jensen’s inequality to the right hand side of (11), we get(12)Ψ1n∑k=1n1−ℓζ¯+ℓϑk≤1n∑k=1nΨ1−ℓζ¯+ℓϑk.
Applying the definition of the convex function to the right hand side of (12), we obtain(13)1n∑k=1nΨ1−ℓζ¯+ℓϑk≤1−ℓΨζ¯+ℓn∑k=1nΨϑk.
Instantly, applying Jensen’s inequality to the first term on the right hand side of (13), we acquire(14)1−ℓΨζ¯+ℓn∑k=1nΨϑk≤1−ℓn∑k=1nΨζk+ℓn∑k=1nΨϑk.
Ass2≺s1, using majorization theorem, we have(15)∑k=1nΨϑk≤∑k=1nΨζk.
Now, using (15) in (14), we obtain(16)1−ℓn∑k=1nΨζk+ℓn∑k=1nΨϑk≤1−ℓn∑k=1nΨζk+ℓn∑k=1nΨζk=1n∑k=1nΨζk.
From (12)–(14) and (16), we obtain (10).
ReplacingΨ with −Ψ, we deduce the opposite inequalities for the case that Ψ is a concave function. This completes the proof of Theorem 5.Corollary 1.
Assume that all the assumptions of Theorem5 are fulfilled; then, we have(17)Ψζ¯≤1n∑k=1n1ϑk−ζ¯∫ζ¯ϑkΨυdυ≤12Ψζ¯+12n∑k=1nΨϑk≤12n∑k=1nΨζk+12n∑k=1nΨϑk≤1n∑k=1nΨζk.Proof.
Integrating (10) over ℓ∈0,1, we obtain (17).Remark 2.
Sinces1≺s1, applying Theorem 5 and Corollary 1 for ϑk=ζk, where k∈1,2,…,n, we have the following inequalities:(18)Ψζ¯≤1n∑k=1nΨ1−ℓζ¯+ℓζk≤1−ℓΨζ¯+ℓn∑k=1nΨζk≤1n∑k=1nΨζk,Ψζ¯≤1n∑k=1n1ζk−ζ¯∫ζ¯ζkΨυdυ≤12Ψζ¯+12n∑k=1nΨζk≤1n∑k=1nΨζk.Theorem 6.
LetΨ:I⟶ℝ be a convex function; let s1=ζ1,ζ2,…,ζnands2=ϑ1,ϑ2,…,ϑn be two n-tuples, such that their entries are from the interval I, and let s3=η1,η2,…,ηn be the nonnegative n-tuple with η¯=∑k=1nηk>0. Suppose that the conditions (6) and (7) are satisfied, ζ¯=1/η¯∑k=1nηkζk and ℓ∈0,1. If s2 is a decreasing n-tuple, then the following inequalities hold:(19)Ψζ¯≤1η¯∑k=1nηkΨ1−ℓζ¯+ℓϑk≤1−ℓΨζ¯+ℓη¯∑k=1nηkΨϑk≤1−ℓη¯∑k=1nηkΨζk+ℓη¯∑k=1nηkΨϑk≤1η¯∑k=1nηkΨζk.IfΨ:I⟶ℝ is a concave function, then the inequalities given in (19) hold in the opposite direction.Proof.
By using (7), we may write(20)Ψ1η¯∑k=1nηkζk=Ψ1η¯∑k=1nηk1−ℓζ¯+ℓϑk.
Using the Jensen inequality on the right hand side of (20), we obtain(21)Ψ1η¯∑k=1nηk1−ℓζ¯+ℓϑk≤1η¯∑k=1nηkΨ1−ℓζ¯+ℓϑk.
Now, applying the definition of the convex function on the right hand side of (21), we get(22)1η¯∑k=1nηkΨ1−ℓζ¯+ℓϑk≤1−ℓΨζ¯+ℓη¯∑k=1nηkΨϑk.
Now, applying the Jensen inequality to the first term on the right hand side of (22), we acquire(23)1−ℓΨζ¯+ℓη¯∑k=1nηkΨϑk≤1−ℓη¯∑k=1nηkΨζk+ℓη¯∑k=1nηkΨϑk.
Since all of the conditions of Theorem3 are satisfied in the present assumptions, using Theorem 3 gives(24)∑k=1nηkΨϑk≤∑k=1nηkΨζk.
Now, using (24), we obtain(25)1−ℓη¯∑k=1nηkΨζk+ℓη¯∑k=1nηkΨϑk≤1η¯∑k=1n1−ℓηkΨζk+ℓη¯∑k=1nηkΨζk=1η¯∑k=1nηkΨζk.
From (21)–(23) and (25), it follows (19).
ReplacingΨ with −Ψ, we deduce the opposite inequalities for the case that Ψ is a concave function. The proof of Theorem 6 is complete.Theorem 7.
Under the assumptions of Theorem4, if η¯=∑k=1nηk>0 and ℓ∈0,1, then the inequalities in (19) hold.Proof.
Theorem7 can be proved easily by using the idea of Theorem 6 along with the result of Theorem 4.Corollary 2.
Under the assumptions of Theorem6, the following inequalities hold:(26)Ψζ¯≤1η¯∑k=1nηkϑk−ζ¯∫ζ¯ϑkΨυdυ≤12Ψζ¯+12η¯∑k=1nηkΨϑk≤12η¯∑k=1nηkΨζk+12η¯∑k=1nηkΨϑk≤1η¯∑k=1nηkΨζk.Proof.
Integrating (19) over ℓ∈0,1, we obtain (26).Remark 3.
Ifs1=s2, then obviously the conditions (6) and (7) are satisfied. So, in this case, the inequalities (19) and (26) become(27)Ψζ¯≤1η¯∑k=1nηkΨ1−ℓζ¯+ℓζk≤1−ℓΨζ¯+ℓη¯∑k=1nηkΨζk≤1η¯∑k=1nηkΨζk,(28)Ψζ¯≤1η¯∑k=1nηkζk−ζ¯∫ζ¯ζkΨυdυ≤12Ψζ¯+12η¯∑k=1nηkΨζk≤1η¯∑k=1nηkΨζk.
The inequalities (18) and (28) can also be found in Remark 2 described in [25].


## 3. Applications to Inequalities Involving Special Means

In this section, we give some applications of Theorem6 for power means and quasiarithmetic means.Definition 2.
Lets1=ζ1,ζ2,…,ζn and s3=η1,η2,…,ηn be two nonnegative n-tuples and η¯=∑k=1nηk>0. Then, the power mean of order ρ∈ℝ is defined by(29)ℳρs1;s3=1η¯∑k=1nηkζkρ1/ρ,ifρ≠0,∏k=1nζkηk1/η¯,ifρ=0.
We establish the refinement of inequalities involving power means as follows.Corollary 3.
LetS1=ζ1ρ,ζ2ρ,…,ζnρ,S2=ϑ1ρ,ϑ2ρ,…,ϑnρ, s1=ζ1,ζ2,…,ζn, s2=ϑ1,ϑ2,…,ϑn, and s3=η1,η2,…,ηn be the nonnegative n-tuples, such that(30)∑k=1lηkϑkρ≤∑k=1lηkζkρ,forl=1,2,…,n−1,∑k=1nηkϑkρ=∑k=1nηkζkρ.If then-tuple S3=ϑ1ρ,ϑ2ρ,…,ϑnρ is decreasing and ρ∈ℝ∖0, such that ρ≤ξ, then the following inequalities hold:(31)ℳρs1;s3≤1η¯∑k=1nηk1−ℓℳρρs1;s3+ℓϑkρξ/ρ1/ξ≤1−ℓℳρξs1;s3+ℓℳξξs2;s31/ξ≤1−ℓℳξξs1;s3+ℓℳξξs2;s31/ξ≤ℳξs1;s3,ξ≠0,(32)ℳρs1;s3≤exp1ρη¯∑k=1nηklog1−ℓℳρρs1;s3+ℓϑkρ≤exp1−ℓlogℳρs1;s3+ℓlogℳ0s2;s3≤exp1−ℓlogℳ0s1;s3+ℓlogℳ0s2;s3≤ℳ0s1;s3,ξ=0,where ℓ∈0,1 and η¯=∑k=1nηk>0.Proof.
Note that the functionΨυ=υξ/ρ is convex on 0,∞ for ρ,ξ|0<ρ<ξ or ρ,ξ|ρ<0,ξ>0. Therefore, utilizing (19) and substituting ζk=ζkρ,ϑk=ϑkρ, and Ψυ=υξ/ρ, we get(33)1η¯∑k=1nηkζkρξ/ρ≤1η¯∑k=1nηk1−ℓ1η¯∑k=1nηkζkρ+ℓϑkρξ/ρ≤1−ℓ1η¯∑k=1nηkζkρξ/ρ+ℓη¯∑k=1nηkϑkρξ/ρ≤1−ℓη¯∑k=1nηkζkρξ/ρ+ℓη¯∑k=1nηkϑkρξ/ρ≤1η¯∑k=1nηkζkρξ/ρ.
Takingξth root of (33), we obtain(34)1η¯∑k=1nηkζkρ1/ρ≤1η¯∑k=1nηk1−ℓ1η¯∑k=1nηkζkρρ/ρ+ℓϑkρξ/ρ1/ξ≤1−ℓ1η¯∑k=1nηkζkρξ/ρ+ℓ1η¯∑k=1nηkϑkξξ/ξ1/ξ≤1−ℓ1η¯∑k=1nηkζkξξ/ξ+ℓ1η¯∑k=1nηkϑkξξ/ξ1/ξ≤1η¯∑k=1nηkζkξ1/ξ,which is equivalent to (31).
Since the functionΨυ=υξ/ρ is concave on 0,∞ for ρ,ξ|ρ≤ξ<0, utilizing Theorem 6 and choosing Ψυ=υξ/ρ, ζk=ζkρ and ϑk=ϑkρ, we obtain(35)1η¯∑k=1nηkζkρξ/ρ≥1η¯∑k=1nηk1−ℓ1η¯∑k=1nηkζkρ+ℓϑkρξ/ρ≥1−ℓ1η¯∑k=1nηkζkρξ/ρ+ℓη¯∑k=1nηkϑkρξ/ρ≥1−ℓη¯∑k=1nηkζkρξ/ρ+ℓη¯∑k=1nηkϑkρξ/ρ≥1η¯∑k=1nηkζkρξ/ρ.
Therefore, taking power1/ξξ<0 of (35), we get(36)1η¯∑k=1nηkζkρ1/ρ≤1η¯∑k=1nηk1−ℓ1η¯∑k=1nηkζkρ+ℓϑkρξ/ρ1/ξ≤1−ℓ1η¯∑k=1nηkζkρξ/ρ+ℓ1η¯∑k=1nηkϑkξξ/ξ1/ξ≤1−ℓ1η¯∑k=1nηkζkξξ/ξ+ℓ1η¯∑k=1nηkϑkξξ/ξ1/ξ≤1η¯∑k=1nηkζiξ1/ξ.
Clearly, (36) is equivalent to (31). The inequalities in (31) are proved.
Taking the logarithm (log) of (31) and letting ξ⟶0, we have(37)logℳρs1;s3≤limξ⟶0log1/η¯∑k=1nηk1−ℓℳρρs1;s3+ℓϑkρξ/ρξ≤limξ⟶0log1−ℓℳρξs1;s3+ℓℳξξs2;s3ξ≤limξ⟶0log1−ℓℳξξs1;s3+ℓℳξξs2;s3ξ≤limξ⟶0logℳξs1;s3.
Applying L′ Hospital’s rule for limit, we obtain(38)logℳρs1;s3≤1ρη¯∑k=1nηklog1−ℓℳρρs1;s3+ℓϑkρ≤1−ℓlogℳρs1;s3+ℓlogℳ0s2;s3≤1−ℓlogℳ0s1;s2+ℓlogℳ0s2;s3≤logℳ0s1;s3.
Now, taking the exponent (exp) of (38), we get(39)ℳρs1;s3≤exp1ρη¯∑k=1nηklog1−ℓℳρρs1;s3+ℓϑkρ≤exp1−ℓlogℳρs1;s3+ℓlogℳ0s2;s3≤exp1−ℓlogℳ0s1;s3+ℓlogℳ0s2;s3≤ℳ0s1;s3,which is the required inequality (32).Corollary 4.
Assume thatS1=ζ1ξ,ζ2ξ,…,ζnξ,S2=ϑ1ξ,ϑ2ξ,…,ϑnξ,s1=ζ1,ζ2,…,ζn, s2=ϑ1,ϑ2,…,ϑn, and s3=η1,η2,…,ηn are nonnegative n-tuples, such that(40)∑k=1lηkϑkξ≤∑k=1lηkζkξ,forl=1,2,…,n−1,∑k=1nηkϑkξ=∑k=1nηkϑkξ.If the tupleS2=ϑ1ξ,ϑ2ξ,…,ϑnξ is decreasing and ξ∈ℝ∖0, such that ρ≤ξ, then we have the following inequalities:(41)ℳξs1;s3≥1η¯∑k=1nηk1−ℓℳξξs1;s3+ℓϑkξρ/ξ1/ρ≥1−ℓℳξρs1;s3+ℓℳρρs2;s31/ρ≥1−ℓℳρρs1;s3+ℓℳρρs2;s31/ρ≥ℳρs1;s3,ρ≠0,(42)ℳξs1;s3≥exp1ξη¯∑k=1nηklog1−ℓℳξξs1;s3+ℓϑkξ≥exp1−ℓlogℳξs1;s3+ℓlogℳξs2;s3≥exp1−ℓlogℳ0s1;s3+ℓlogℳ0s2;s3≥ℳ0s1;s3,ρ=0,where η¯=∑k=1nηk>0 and ℓ∈0,1.Proof.
Since the functionΨυ=υρ/ξ is convex on 0,∞ for ρ,ξ|ρ≤ξ<0 or ρ,ξ|ρ<0,ξ>0, utilizing (19) and choosing ζk=ζkξ,ϑk=ϑkξ, and Ψυ=υρ/ξ, we get(43)1η¯∑k=1nηkζkξρ/ξ≤1η¯∑k=1nηk1−ℓ1η¯∑k=1nηkζkξ+ℓϑkξρ/ξ≤1−ℓ1η¯∑k=1nηkζkξρ/ξ+ℓη¯∑k=1nηkϑkξρ/ξ≤1−ℓη¯∑k=1nηkζkξρ/ξ+ℓη¯∑k=1nηkϑkξρ/ξ≤1η¯∑k=1nηkζkξρ/ξ.
Now, takingρth root of (43), we acquire(44)1η¯∑k=1nηkakξ1/ξ≥1η¯∑k=1nηk1−ℓ1η¯∑k=1nηkζkξ+ℓϑkξρ/ξ1/ρ≥1−ℓ1η¯∑k=1nηkζkξρ/ξ+ℓη¯∑k=1nηkϑkρ1/ρ≥1−ℓη¯∑k=1nηkζkρ+ℓη¯∑k=1nηkϑkρ1/ρ≥1η¯∑k=1nηkζkρ1/ρ.
Forρ,ξ|0<ρ≤ξ, one has(45)1η¯∑k=1nηkζkξ1/ξ≥1η¯∑k=1nηk1−ℓ1η¯∑k=1nηkζkξξ/ξ+ℓϑkξρ/ξ1/ρ≥1−ℓ1η¯∑k=1nηkζkξρ/ξ+ℓ1η¯∑k=1nηkϑkρρ/ρ1/ρ≥1−ℓ1η¯∑k=1nηkζkρρ/ρ+ℓ1η¯∑k=1nηkϑkρρ/ρ1/ρ≥1η¯∑k=1nηkζkρ1/ρ.
Inequalities (45) and (44) imply the required inequality (41). Inequality (42) can be obtained by taking the limit of (41) as ρ⟶0.Definition 3.
Lets1=ζ1,ζ2,…,ζn and s3=η1,η2,…,ηn be two nonnegative n-tuples and η¯=∑k=1nηk>0. If a function h is continuous and strictly monotonic, then the quasiarithmetic mean is defined as(46)Mhs1;s3=h−11η¯∑k=1nηkhζk.Corollary 5.
Suppose thath is continuous and strictly monotonic on 0,∞. Also, suppose that s1=ζ1,ζ2,…,ζn, s2=ϑ1,ϑ2,…,ϑn, and s3=η1,η2,…,ηn are the positive n-tuples with(47)∑k=1lηkhϑk≤∑k=1lηkhζk,l=1,2,…,n−1,∑k=1nηkhϑk=∑k=1nηkhζk.Ifhϑ1,hϑ2,…,hϑn is decreasing and the function Φ∘h−1 is convex on 0,∞, then(48)ΦMhs1;s3≤1η¯∑k=1nηkΦ∘h−11−ℓhMhs1;s3+ℓhϑk≤1−ℓΦMhs1;s3+ℓη¯∑k=1nηkΦϑk≤1−ℓη¯∑k=1nηkΦζk+ℓη¯∑k=1nηkΦϑk≤1η¯∑k=1nηkΦζk.Ifhϑ1,hϑ2,…,hϑn is decreasing and the function Φ∘h−1 is concave, then the inequality (48) holds in the opposite direction.Proof.
Using (19) for ζk=hζk,ϑk=hϑk, and Ψυ=Φ∘h−1υ, we get(49)Φ∘h−11η¯∑k=1nηkhζk≤1η¯∑k=1nηkΦ∘h−11−ℓ1η¯∑k=1nηkhζk+ℓhϑk≤1−ℓΦ∘h−11η¯∑k=1nηkhζk+ℓη¯∑k=1nηkΦ∘h−1hϑk≤1−ℓη¯∑k=1nηkΦ∘h−1hζk+ℓη¯∑k=1nηkΦ∘h−1hϑk≤1η¯∑k=1nηkΦ∘h−1hζk,this implies that(50)Φh−11η¯∑k=1nηkhζk≤1η¯∑k=1nηiΦ∘h−11−ℓ×hh−11η¯∑k=1nηkhζk+ℓhϑk≤1−ℓΦh−11η¯∑k=1nηkhζk+ℓη¯∑k=1nηkΦϑk≤1−ℓη¯∑k=1nηkΦζk+ℓη¯∑k=1nηkΦϑk≤1η¯∑k=1nηkΦζk,which is equivalent to (48).


## 4. Applications to Information Theory

The information theory is the mathematical treatment of the concepts, framework, and rule governing the spread of messages through communication systems. The information theory has been used very extensively in the telecommunication systems, and it has also a great role in the study of linguistic studies such as in the sense of speed of reading, length of words, and frequency of words. In this section, we give some meaningful applications of our main results in the area of information theory. We will establish bounds for the different divergences and Shannon entropy. We also establish bounds for various distances, Bhattacharyya coefficient, and triangular discrimination.Definition 4 (See [26]).
Lets1=θ1,θ2,…,θn and s3=τ1,τ2,…,τn be two positive n-tuples, and let Φ:0,∞⟶ℝ be a convex function. Then, the Csiszár Φ-divergence functional is defined as(51)MΦs1,s3=∑k=1nτkΦθkτk.Theorem 8.
Suppose thatΦ:0,∞⟶ℝ is a convex function, suppose also that s1=θ1,θ2,…,θn, s2=ϑ1,ϑ2,…,ϑn, and s3=τ1,τ2,…,τn are three positive n-tuples, such that(52)∑k=1lθk≥∑k=1lτkϑk,forl=1,2,…,n−1,(53)∑k=1nθk=∑k=1nτkϑk.
Ifϑ1,ϑ2,…,ϑn is a decreasing tuple, ℓ∈0,1, then(54)MΦs1,s3≥1−ℓMΦs1,s3+ℓ∑k=1nτkΦϑk≥1−ℓΦ∑k=1nθk∑k=1nτk∑k=1nτk+ℓ∑k=1nτkΦϑk≥∑k=1nτkΦ1−ℓ∑k=1nθk∑k=1nτk+ℓϑk≥Φ∑k=1nθk∑k=1nτk∑k=1nτk.Proof.
Note that the functionΦ is convex, and the conditions of Theorem 6 are satisfied. Therefore, applying Theorem 6 by choosing Ψ=Φ, ζi=θk/τk, and ηk=τk, for all k∈1,2,…,n, we obtain (54).Definition 5 (See [27]).
Assume thats3=τ1,τ2,…,τn is a probability distribution; then, the Shannon entropy is defined by(55)Ss3=−∑k=1nτklogτk.Corollary 6.
Suppose thats3=τ1,τ2,…,τn is a probability distribution with ∑k=1nτk=1, s1=θ1,θ2,…,θn and s2=ϑ1,ϑ2,…,ϑn are two n-tuples, and τk,ϑk,θk>0k=1,2,…,n. If s2 is a decreasing tuple and the conditions (52) and (53) are satisfied with θk=1 for all k∈1,2,…,n, ℓ∈0,1, then(56)Ss3≤1−ℓSs3+ℓ∑k=1nτklogϑk≤1−ℓlogn+ℓ∑k=1nτklogϑk≤∑k=1nτklog1−ℓn+ℓϑk≤logn.Proof.
Since the functionΦυ=−logυ is convex on 0,∞, by considering θk=1 for all k∈1,2,…,n, the conditions of the present corollary become parallel to Theorem 8. Hence, applying Theorem 8 by Φυ=−logυ and θk=1 for all k∈1,2,…,n, we get (56).Definition 6 (See [28]).
Lets1=θ1,θ2,…,θn and s3=τ1,τ2,…,τn be two probability distributions. Then, the Kullback–Leibler divergence is defined as(57)Kds1,s3=∑k=1nθklogθkτk.Corollary 7.
Suppose thats1=θ1,θ2,…,θn and s3=τ1,τ2,…,τn are the probability distributions with ∑k=1nθk=1 and ∑k=1nτk=1. If ϑ1,ϑ2,…,ϑn is a decreasing tuple, such that (52) and (53) are satisfied, and τk,ϑk,θk>0k=1,2,…,n, ℓ∈0,1, then(58)Kds1,s3≥1−ℓKds1,s3+ℓ∑k=1nτkϑklogϑk≥ℓ∑k=1nτkϑklogϑk≥∑k=1nτk1−ℓ+ℓϑklog1−ℓ+ℓϑk.Proof.
Since the functionΦυ=υlogυ is convex on 0,∞, applying Theorem 8 by selecting Φυ=υlogυ, with the help of ∑k=1nθk=1 and ∑k=1nτk=1 given by the probability distributions, we obtain (58).Definition 7.
(See [29]). Let s1=θ1,θ2,…,θn and s3=τ1,τ2,…,τn be two probability distributions. Then, the variational distance is defined as(59)Vds1,s3=∑k=1nθk−τk.
By virtue of Theorem8, choose the convex function Φυ=υ−1υ>0 and use the conditions of probability distributions ∑k=1nθk=1 and ∑k=1nτk=1, and we get Corollary 8.Corollary 8.
Suppose that all conditions of Corollary7 are satisfied. Then,(60)Vds1,s3≥1−ℓVds1,s3+ℓ∑k=1nτkϑk−1≥ℓ∑k=1nτkϑk−1.Definition 8.
(See [30]). Let s1=θ1,θ2,…,θn and s3=τ1,τ2,…,τn be two probability distributions. Then, the Jeffrey distance is defined by(61)Jds1,s3=∑k=1nθk−τklogθkτk.
By mean of Theorem8, take the convex function Φυ=υ−1logυυ>0 and use the conditions of probability distributions ∑k=1nθk=1 and ∑k=1nτk=1, and we get Corollary 9.Corollary 9.
Suppose that all conditions of Corollary7 are satisfied. Then,(62)Jds1,s3≥1−ℓJds1,s3+ℓ∑k=1nτkϑk−1logϑk≥ℓ∑k=1nτkϑk−1logϑk≥ℓ∑k=1nτkϑk−1log1+ℓϑk−1.Definition 10 (See [31]).
Lets1=θ1,θ2,…,θn and s3=τ1,τ2,…,τn be two probability distributions. Then, the Bhattacharyya coefficient is defined as(63)Bds1,s3=∑k=1nθkτk.
By the use of Theorem8, choose the convex function Φυ=−υυ>0 and utilize the conditions of probability distributions ∑k=1nθk=1 and ∑k=1nτk=1, and we get Corollary 10.Corollary 10.
Suppose that all conditions of Corollary7 are satisfied. Then,(64)Bds1,s3≤1−ℓBds1,s3+ℓ∑k=1nτkθk≤1+ℓ∑k=1nτkϑk−1≤∑k=1nτk1+ℓϑk−1≤1.Definition 9.
(See [32]). Let s1=θ1,θ2,…,θn and s3=τ1,τ2,…,τn be two probability distributions. Then, the Hellinger distance is defined as(65)Hds1,s3=∑k=1nθk−τk2.
By means of Theorem8, take the convex function Φυ=υ−12υ>0 and use the conditions of probability distributions ∑k=1nθk=1 and ∑k=1nτk=1, and we get Corollary 11.Corollary 11.
Suppose that all conditions of Corollary7 are satisfied. Then,(66)Hds1,s3≥1−ℓHds1,s3+ℓ∑k=1nτkϑk−12≥ℓ∑k=1nτkϑk−12≥∑k=1nτk1+ℓϑk−1−12.Definition 11.
(See [33]). Let s1=θ1,θ2,…,θn and s3=τ1,τ2,…,τn be two probability distributions. Then, the triangular discrimination is defined by(67)Tds1,s3=∑k=1nθk−τk2θk+τk.
By employing Theorem8, choose the convex function Φυ=υ−12/υ+1υ>0 and utilize the conditions of probability distributions ∑k=1nθk=1 and ∑k=1nτk=1, and we get Corollary 12.Corollary 12.
Suppose that all conditions of Corollary7 are satisfied. Then,(68)Tds1,s3≥1−ℓTds1,s3+ℓ∑k=1nτkϑk−12ϑk+1≥ℓ∑k=1nτkϑk−12ϑk+1≥∑k=1nτkℓ2ϑk−12ℓϑk−1+2.


## 5. Conclusion

The Jensen inequality is one of the most powerful and attractive inequalities among the mathematical inequalities. This inequality generalized the definition of convex function and generated many new inequalities. Nowadays, it became an engaging source for many researchers. The Jensen inequality has been refined, extended, improved, and generalized for convex functions in various directions. Also, it is used for developing some generalized convex functions such ass-convex, η-convex, strongly convex, and coordinate convex. Aside from the mathematics, the Jensen inequality has performed a key role for solving serval problems in multiform fields of science such as economics, statistics, medical science, and physics. In this study, we worked on the refinements of Jensen inequality. By using convex functions and certain types of tuples with majorization conditions, we obtained some refinements of Jensen inequality. Furthermore, we established some bounds for the power mean as well as quasiarithmetic means with the help of the obtained results. We also gave bounds for the Csiszár divergence, Kullback–Leibler divergence, and Shannon entropy. Moreover, we obtained bounds for various distances associated with probability distributions, Bhattacharyya coefficient, and triangular discrimination. The results given in this study would provide a significant supplement to the theory of mathematical inequalities.



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