# Some Improvements of Jensen’s Inequality via 4-Convexity and Applications

Authors: Hidayat Ullah; Muhammad Adil Khan; Tareq Saeed; Zaid Mohammed Mohammed Mahdi Sayed
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

The intention of this note is to investigate some new important estimates for the Jensen gap while utilizing a 4-convex function. We use the Jensen inequality and definition of convex function in order to achieve the required estimates for the Jensen gap. We acquire new improvements of the Hölder and Hermite–Hadamard inequalities with the help of the main results. We discuss some interesting relations for quasi-arithmetic and power means as consequences of main results. At last, we give the applications of our main inequalities in the information theory. The approach and techniques used in the present note may simulate more research in this field.


---

## Body

## 1. Introduction

The theory of convex functions performs an extremely significant and consequential role in several areas of pure and applied sciences. Due to its numerous and extensive applications, the concept of convex functions has been extended and generalized in many directions. The most important and elegant aspect of the class of convex functions, which attracted many researchers, is its deep relation with theory of inequalities [1–3]. In the literature, there are several well-known inequalities which are the direct consequences and applications of convexity [4, 5]. In this respect, some of the noted inequalities associated with the class of convex functions are majorization, Hermite–Hadamard and Jensen–Mercer inequalities [6]. Among these inequalities, one of the considerable and vital inequalities which are studied very widely in the literature is the Jensen inequality. This celebrated inequality reads as follows:Theorem 1.
Assume thatI is an interval of real numbers and Ψ is a convex function on I. If yj∈I and wj>0forj=1,2,⋯,n with W=∑j=1nwj, then
(1)Ψ1W∑j=1nwjyj≤1W∑j=1nwjΨyj.
Inequality (1) will be true in the reverse direction, if the function Ψ is concave on I.The Jensen inequality has multitudinous applications in Mathematics [7–11], Statistics [12], Economics [13] and Information Theory [14], etc. The most interesting and attractive applications of this inequality is that it generalized the classical convexity. Moreover, there are several inequalities which are the direct consequences of this inequality such as Ky Fan, Cauchy, Hermite–Hadamard and Hölder inequalities. Due to the vast applications of the Jensen inequality, many researchers dedicated their work to this inequality. This inequality has been extended, improved, and refined in multidirections by using different techniques and principals. For some more extensive literature concerning to the Jensen inequality, see [15, 16].


## 2. Main Results

In the present part, we discuss the main results. Let us begin this section with the following theorem, in which we acquire an upper bound for the Jensen gap.Theorem 2.
Assume thatI is an interval in ℝ,xi∈Iandpi>0fori=1,2,⋯,n with Pn≔∑i=1npi and x¯=1/Pn∑pixi. If Ψ is a twice differentiable function such that Ψ is 4-convex on I, then
(2)1Pn∑i=1npiΨxi−Ψx¯≤16Pn∑i=1npix¯−xi22Ψ″x¯+Ψ″xi.
Inequality (2) will be true in the opposite direction, if the function Ψ is 4-concave.Proof.
Without misfortune of sweeping statement, assume thatx¯≠xifori=1,2,⋯,n. Utilizing integration by parts, we have
(3)1Pn∑i=1npix¯−xi2∫01tΨ″tx¯+1−txidt=1Pn∑i=1npix¯−xi2tx¯−xiΨ′tx¯+1−txi01−1x¯−xi∫01Ψ′tx¯+1−txidt=1Pn∑i=1npix¯−xi2Ψ′x¯x¯−xi−tx¯−xi2Ψtx¯+1−txi01=1Pn∑i=1npix¯−xi2Ψ′x¯x¯−xi−tx¯−xi2Ψx¯−Ψxi=1Pn∑i=1npix¯−xiΨ′x¯−1Pn∑i=1npiΨx¯−Ψxi=1Pn∑i=1npiΨxi−Ψx¯,which implies that
(4)1Pn∑i=1npiΨxi−Ψx¯=1Pn∑i=1npixi−x¯2∫01tΨ″tx¯+1−txidt.
Since, the functionΨ is 4-convex on I. Therefore, using the definition of convex function on the right hand side of (4), we receive
(5)1Pn∑i=1npiΨxi−Ψx¯≤1Pn∑i=1npixi−x¯2∫01t2Ψ″x¯+t1−tΨ″xidt=1Pn∑i=1npix¯−xi2Ψ″x¯∫01t2dt+Ψ″xi∫01t−t2dt=1Pn∑i=1npix¯−xi213Ψ″x¯+16Ψ″xi,which is equivalent to (2).The integral version of (2) is stated in the following theorem.Theorem 3.
Assume thatI is an interval in ℝ,Ψ:I⟶ℝ that is a twice differentiable function such that Ψ is 4-convex and ϕ,φ:a,b⟶I are integrable functions with φ≥0ona,b. Also, assume that Ψ∘ϕ:a,b⟶ℝ is an integrable function, φ¯≔∫abφxdx>0andϕ¯≔1¯¯φ∫abφxϕxdx. Then
(6)1¯¯φ∫abφxΨ∘ϕxdx−Ψϕ¯≤16φ¯∫abφxϕ¯−ϕx22Ψ″ϕ¯+Ψ″ϕxdx.
Inequality (6) will be true in the reverse sense, if  Ψ is a 4-concave function.In the next theorem, we acquire a lower bound for the Jensen gap while utilizing the Jensen inequality.Theorem 4.
Assume that all the suppositions of Theorem2 are valid, then
(7)1Pn∑i=1npiΨxi−Ψx¯≥12Pn∑i=1npix¯−xi2Ψ″2x¯+xi3.
Inequality (7) will become true in the reverse direction, if the function Ψ is 4-concave.Proof.
Since, the functionΨ is 4-convex on I. Therefore, applying integral Jensen’s inequality on the right of (4), we get
(8)1Pn∑i=1npiΨxiΨx¯=1Pn∑i=1npix¯−xi2∫01tΨ″tx¯+1−txidt∫01tdt≥12Pn∑i=1npix¯−xi2Ψ″∫01ttx¯+1−txidt∫01tdt=12Pn∑i=1npix¯−xi2Ψ″x¯∫01t2dt+xi∫01t−t2dt∫01tdt=12Pn∑i=1npix¯−xi2Ψ″2x¯+xi3,which is the required inequality.The analogous inequality of (7) is given in the following theorem.Theorem 5.
Suppose that all the hypotheses of Theorem3 are true. Then
(9)1φ¯∫abφxΨ∘ϕxdx−Ψϕ¯≥12φ¯∫abφxϕ¯−ϕx2Ψ″2ϕ¯+ϕx3dx.
If the functionΨ is 4-concave, then the inequality (9) holds in the opposite direction.


## 3. Numerical Experiments

In this section, we are going to provide some simple examples to show how sharp our estimates for the Jensen gap.Example 6.
Consider the functionsΨx=x4,ϕx=x and φx=1forallx∈0,1. Then, Ψ″x=12x2≥0andΨ‴′x=24>0on0,1. This verifies that the function Ψ is convex as well as 4-convex. Now, utilizing (6) for Ψx=x4, ϕx=x, φx=1, a=0, and b=1, we get
(10)0.1375<0.15.
Using above functions with the given interval in inequality (4) in [17], we acquire
(11)0.1375<0.25.
From inequalities (10) and (11) it is clear that the bounds given in (6) provide a good and better estimate for the Jensen gap. Moreover, the inequality (10) shows that the value of the obtained estimate for the Jensen gap given in (6) is very close to the value of the Jensen gap.Example 7.
Consider the functionsΨx=1−x5, φx=1, and ϕx=1 for all x∈0,1. Then, Ψ″x=201−x3andΨ″″x=1201−x. Clearly, both Ψ″andΨ″″ are nonnegative on [0,1]. This shows that the function Ψx=1−x5 is convex as well as 4-convex. Utilizing Ψx=1−x5,Ψ″x=201−x3,φx=1 and ϕx=1 in (6), we obtain
(12)0.1345<0.1666.
Now, using the chosen functions in the inequality (4) in [17], we acquire
(13)0.1345<0.4166.
From (12) and (13) it is clear that the inequality (6) provides an efficient and superior estimate as compared to the inequality (4) in [17].Example 8.
Assume that the functionsΨx=expx, φx=1, and ϕx=x are defined on [0,1]. Then, Ψ′′x=expx and Ψ′′′′x=expx. Obviously, both the functions Ψ′′ and Ψ′′′′ are nonnegative on ∈0,1. This confirms the convexity and 4-convexity of the function Ψx=expx. Choosing Ψx=expx,Ψ′′x=expx,φx=1andϕx=x in (6), we obtain
(14)0.0695<0.0704.
Now, using the given functions in the inequality (4) in [17], we acquire
(15)0.0695<0.0996.
Again, from (14) and (15), it is obvious that the estimate provided by inequality (6) for the Jensen gap is better than the estimate provided by inequality (4) in [17]. Moreover, the value of the estimate for the Jensen gap in (6) is very close to the value of the Jensen gap.Remark 9.
The authors in [17] compared the value of estimate for the Jensen gap in the inequality (4) with the value of the estimates for the Jensen gap in inequalities (5) and (8) in [18]. From the comparison, the authors declared that the estimate for the Jensen gap in inequality (4) in [17] is better than the estimates for the Jensen gap in inequalities (5) and (8) in [18]. Hence from this, we can also conclude that our estimate for the Jensen gap may be better than the estimates for the Jensen gap in (5) and (8) in [18].


## 4. Applications for Classical Inequalities

This section is devoted to the consequences of main results. In this section, we obtain some improvements for the Hölder and Hermite–Hadamard inequalities with the help of our main results. Furthermore, we acquire different relations for the power and quasi-arithmetic means with the utilization of our obtained results.In the following proposition, we give an improvement for the Hölder inequality with the help of Theorem2.Proposition 10.
Letm1=ζ1,ζ2,⋯,ζn and m2=γ1,γ2,⋯,γn be two positive n-tuples and p,q>1, such that p∉2,3. If 1/p+1/q=1, then
(16)∑i=1nζip1/p∑i=1nγiq1/q−∑i=1nγiζi≤pp−16∑i=1nγiq∑i=1nγiζi∑i=1nγiq−ζiγi−q/p2×∑i=1nγiζi∑i=1nγiq−ζiγi−q/p1/p∑i=1nγiq1q.Proof.
Since the functionΨ=xp is convex as well as 4-convex on (0,∞) for all p>1,p∉2,3. Therefore, utilizing (2) by choosing Ψx=xp and pi=γiq,xi=ζiγi−q/p for all i∈1,2,⋯,n and then taking power 1/p, we get
(17)∑i=1nζip∑i=1nγiqp−1−∑i=1nγiζip1/p≤pp−16∑i=1nγiq∑i=1nγiζi∑i=1nγiq−ζiγi−q/p2×∑i=1nγiζi∑i=1nγiq−ζiγi−q/p1/p∑i=1nγiq1/q.
As the inequality(18)al−bl≤a−blholds, for all a,b≥0 and l∈0,1, thus using (18) for a=∑i=1nζip∑i=1nγiqp−1,b=∑i=1nγiζipandl=1/p, we obtain
(19)∑i=1nζip1/l∑i=1nγiq1/q−∑i=1nγiζi≤∑i=1nζip∑i=1nγiqp−1−∑i=1nγiζi1/p.
Now, comparing (17) and (19), we acquire (16).Another consequence of Theorem2 is given in the following corollary, in which we provide a relation for the Hölder inequality.Corollary 11.
Letm1=ζ1,ζ2,⋯,ζn and m2=γ1,γ2,⋯,γn be two positive n-tuples, 0<p<1 and q=p/p−1 such that 1/p∉2,3. If 1/p+1/q=1, then
(20)∑i=1nγiζi−∑i=1nζip1/p∑i=1nγiq1/q≤1−p6p2∑i=1nγiq∑i=1nζip∑i=1nγiq−ζipγi−q2×∑i=1nζip∑i=1nγiq−ζipγi−q.Proof.
Forp∈0,1 such that 1/p∉2,3, the function Ψx=x1/p is convex as well as 4-convex on [0,∞]. Therefore, utilizing (2) by choosing Ψx=x1/p,pi=γiq,andxi=γi−qζip, we get (20).As a consequence of Theorem4, we acquire another improvement for the Hölder inequality which is stated in the following corollary.Corollary 12.
Assume thatm1=γ1,γ2,⋯,γn and m2=ζ1,ζ2,⋯,ζn are two n-tuples such that γi,ζi>0 for all i∈1,2,⋯,n. If p∈0,1 and q=p/p−1 such that 1/p∉2,3, then
(21)∑i=1nγiζi−∑i=1nζip1/p∑i=1nγiq1/q≥1−p2p2∑i=1nγiq∑i=1nζip∑i=1nγiq−ζipγi−q2×2∑i=1nζip+ζipγi−q∑i=1nγiq3∑i=1nγiq.Proof.
The functionΨx=x1/p is both convex and 4-convex for x≥0 with p∈0,1 such that 1/p∉2,3. Therefore, inequality (21) can easily be acquired by putting Ψx=x1/p,pi=γiq,andxi=γi−qζip in (7).Now, we recall the definition of power mean.Definition 13.
Letm1=γ1,γ2,⋯,γn and m2=ζ1,ζ2,⋯,ζn be two n-tuples such that γi,ζi∈0,∞ for all i∈1,2,⋯,n with γ¯=∑i=1nγi. Then, the power mean of order p∈ℝ is defined by
(22)Mpm1,m2=1¯¯γ∑i=1nγiζip1/p,p≠0∏i=1nζiγi1¯¯γ,p=0.As a consequence of Theorem2, in the following corollary, we give bound for the power mean.Corollary 14.
Letm1=γ1,γ2,⋯,γn and m2=ζ1,ζ2,⋯,ζn be two positive n-tuples such that γ¯=∑i=1nγi and r and t be nonzero real numbers. Then the following statements are true:
(i)
Ifr>0 with t≥3r or 2r≥t≥r or 0>t, then(23)Mrrm1,m2−Mrtm1,m2≤tt−r6r2γ¯∑i=1nγiMrrm1,m2−ζir2×Mrt−2rm1,m2+ζit−2.(ii)
Ifr<0 with t≤3r or 2r≤t≤r or 0<t, then (23) holds.(iii)
Ifr>0 with 3r>t>2r or r<0 with 3r<t<2r, then (23) holds in the reverse direction.Proof.
(i)
Forx>0, the function Ψx=xt/r is 4-convex with the given conditions.
Therefore, using (2) by taking Ψx=xt/r,pi=γi,andxi=ζir, we obtain (23)
(ii)
If the given conditions are hold, then the functionΨx=xt/r will be 4-convex on (0,∞). Thus, (23) can easily be obtained by adopting the procedure of (i)(iii)
The functionΨx=xt/r is 4-concave on (0,∞) for the given values of r and t. Therefore, we can get the reverse inequality in (23) by adopting the procedure of (i).In the following result, we present an application of Theorem4.Corollary 15.
Letm1, m2, and γ¯ be the same as that of Corollary 14 and r,t∈ℝ−0. Then
(A)
If the conditions given in (i) and (ii) are satisfied, then(24)Mttm1,m2−Mrtm1,m2≥tt−r2γ¯r2∑i=1nγiMrrm1,m2−ζir22Mrrm1,m2+ζit−23t/r−2.(B)
If the conditions in (iii) are fulfilled, then (24) holds in the reverse direction.Proof.
(A)
Since, the functionΨx=xt/r is 4-convex on (0,∞) for the conditions given in (i) and (ii) of Corollary 14. Therefore, using (7) for Ψx=xt/r,pi=γi,andxi=ζir, we obtain (24)
(B)
If the condition onr and t mentioned in (iii) of Corollary 14 is true, then the function Ψx=xt/r will be 4-concave for x>0. Thus, utilizing (7) while choosing Ψx=xt/r,pi=γi,andxi=ζir, we obtain the reverse inequality in (24).In the following corollary, we obtain an interesting relation for different means as a consequence of Theorem2.Corollary 16.
Letm1=γ1,γ2,⋯,γn and m2=ζ1,ζ2,⋯,ζn be positive n-tuples with γ¯=∑i=1nγi. Then
(25)M1m1,m2M0m1,m2≤exp16γ¯∑i=1nγiM1m1,m2−ζi2M1−2m1,m2+ζi−2.Proof.
LetΨx=−lnx, x>0. Then, Ψ″x=1/x2andΨ″x=6/x4. Clearly, both Ψ″xandΨ″″x are positive for all x>0. This confirms that the function Ψx is convex as well as 4-convex on (0,∞). Therefore, putting Ψx=−lnx,pi=γi,andxi=ζi in (2), we acquire (25).In the following corollary, a relation for distinct means is obtain with the help of Theorem2.Corollary 17.
Let hypotheses of Corollary16 hold. Then
(26)Mttm1,m2−M0m1,m2≤16γ¯∑i=1nγiM1m1,m2−Inζi22M0m1,m2+ζi.Proof.
Consider functionΨx=expx,x∈ℝ. Then clearly, Ψ″x=expx>0andΨ″″x=expx>0. This shows that the given function is convex as well as 4-convex. Thus, applying (2) by choosing pi=γi, xi=lnζi, and Ψx=expx, we get (26).An application of Theorem4 is acquired in the below corollary.Corollary 18.
Suppose that all the assumptions of Corollary16 are true, then
(27)M1m1,m2M0m1,m2≥exp12γ¯∑i=1nγiM1m1,m2−ζi232M1m1,m2+ζi2.Proof.
PutΨx=−lnx, pi=γi, and xi=ζi in (7), we get (27).The following is another relation for distinct means which is the consequence of Theorem4.Corollary 19.
Let the hypotheses of Corollary16 be fulfilled. Then
(28)M1m1,m2−M0m1,m2≥12γ¯∑i=1nγiM0m1,m2−Inζi2exp2M0m1,m2+Inζi3.Proof.
Utilizing (7) for pi=γi, xi=lnζi, and Ψx=expx, we get (28).
Now, we give the definition of quasiarithmetic mean.Definition 20.
Letm1=γ1,γ2,⋯,γn and m2=ζ1,ζ2,⋯,ζn be positive n-tuples with γ¯=∑i=1nγiandφ be strictly monotonic continuous function. Then the quasi-arithmetic mean is defined as
(29)Mφm1,m2=φ−11¯¯γ∑i=1nγiφζi.In the following corollary, we obtain a relation for the quasi-arithmetic mean with the help of Theorem2.Corollary 21.
Letm1=γ1,γ2,⋯,γn and m2=ζ1,ζ2,⋯,ζn be positive n-tuples with γ¯=∑i=1nγi. Also, let φ be strictly monotonic continuous function and Ψ°φ−1 be 4-convex on (0,∞). Then
(30)1¯¯γ∑i=1nγiΨζi−ΨMφm1,m2≤16γ¯∑i=1nγiφMφm1,m2−φζi2×2Ψ∘φ−1″φMφm1,m2+Ψ∘φ−1″φζi.Proof.
Since, the functionΨ∘φ−1 is 4-convex on (0,∞). Therefore, choosing Ψ=Ψ∘φ−1, pi=γi, and xi=ϕζi in (2), we obtain (30).As an application of Theorem4, in the following corollary, we present a relation for the quasi-arithmetic mean.Corollary 22.
Let the hypotheses of Corollary21 hold. Then
(31)1¯¯γ∑i=1nγiΨζi−ΨMφm1,m2≥12γ¯∑i=1nγiφMφm1,m2−φζi2×Ψ∘φ−1′′2φMφm1,m2+φζi3.Proof.
Using (7) for Ψ=Ψ∘φ−1, pi=γi, and xi=φζi, we obtain (31).In the following corollaries, we present some improvements for the Hermite–Hadamard inequalities with the support of our main results.Corollary 23.
LetΨ:a,b⟶ℝ be a 4-convex function. Then
(32)1b−a∫abΨxdx−Ψa+b2≤16b−a∫aba+b2−x22Ψ″a+b2+Ψ″xdx.If the functionΨ is 4-concave, then the inequality (32) holds in the reverse direction.Proof.
Since, the functionΨ is 4-convex on a,b. Therefore, using (6) for φx=1 and ϕx=x, we get (32).Corollary 24.
Assume that the functionΨ:a,b⟶R is 4-convex, then
(33)1b−a∫abΨxdx−Ψa+b2≥12b−a∫aba+b2−x2Ψ″a+b+x3dx.
Inequality (33) will be true in the opposite sense, if the function Ψ is 4-concave.Proof.
Inequality (33) can easily be deduced by choosing ϕx=x and φx=1 in (9).Remark 25.
The integral version of the above discrete improvements of Hölder and Hermite–Hadamard inequalities and relations for different means can easily be achieved by using Theorems3 and 5.


## 5. Applications in Information Theory

In this part of the note, we are going to discuss some applications of the main inequalities in information theory. These applications involve some bounds for different divergences, the Bhattacharyya coefficient and the Shannon entropy.Definition 26.
LetΦ be a real valued function defined on a,b⊂R and m1=γ1,γ2,⋯,γn and m2=ζ1,ζ2,⋯,ζn be two n-tuples such that ζi/γi∈a,b and γi>0 for all i∈1,2,⋯,n. Then the Csisźar divergence is defined by
(34)Dcm1,m2=∑i=1nγiΦζiγi.Theorem 27.
Assume that the functionΦ:a,b⟶R is 4-convex and m1=γ1,γ2,⋯,γn and m2=ζ1,ζ2,⋯,ζn are two n-tuples such that ∑i=1nζi/∑i=1nγi,ζi/γi∈a,b and γi>0 for all i∈1,2,⋯,n, then
(35)Dcm1,m2−Φ∑i=1nζi∑i=1nγi∑i=1nγi≤16∑i=1nγi∑i=1nζi∑i=1nγi−ζiγi22Φ″∑i=1nζi∑i=1nγi+Φ″ζiγi.Proof.
All the hypotheses of this theorem are same as that of Theorem2.
Thus, using (2) by taking Ψ=Φ,pi=γi/∑i=1nγi,andxi=ζi/γi, we obtain (35).Theorem 28.
Suppose that all the assumptions of Theorem27 are satisfied, then
(36)Dcm1−m2−Φ∑i=1nζi∑i=1nγi∑i=1nγi≥12∑i=1nγi∑i=1nζi∑i=1nγi−ζiγi2Φ″2∑i=1nζi3∑i=1nγi+ζi3γi.Proof.
Since, the functionΦ is 4-convex. Therefore, utilizing (7) for Ψ=Φ,pi=γi/∑i=1nγi,andxi=ζi/γi, we get (36).Definition 29.
For anyδ∈0,∞ such that δ≠1 and arbitrary positive probability distributions m1=γ1,γ2,⋯,γn and m2=ζ1,ζ2,⋯,ζn, the Rényi divergence is defined by
(37)Drem1,m2=1δ−1log∑i=1nγiδζi1−δ.Corollary 30.
Assume thatm1=γ1,γ2,⋯,γn and m2=ζ1,ζ2,⋯,ζn are two positive probability distributions and δ>1, then
(38)Drem1,m2−1δ−1∑i=1nγiδlogγiζiδ−1≤16∑i=1nγi∑i=1nγiδζi1−δ−γiζiδ−12×2δ−1∑i=1nγiδζi1−δ−2+1δ−1γiζi2−2δ.Proof.
LetΦx=1/δ−1logx,x>0. Then, Φ″x=1/δ−1x2andΦ″″x=6/δ−1x4. Clearly, Φ″x>0andΦ″″x>0 for all x∈0,∞. Thus, this verifies that the function Φx=1/δ−1logx is convex as well as 4-convex on (0,∞). Therefore, using (2) for Φx=1/δ−1logx,pi=γi,andxi=γi/ζiδ−1, we get (38).Corollary 31.
Let the hypotheses of Corollary30 hold. Then
(39)Drem1,m2−1δ−1∑i=1nγiδlogγiζiδ−1≥12δ−1∑i=1nγi∑i=1nγiδζi1−δ−γiζiδ−12×32∑i=1nγiδζi1−δ+γi/ζiδ−12.Proof.
UsingΦx=1/δ−1logx,pi=γi and xi=γi/ζiδ−1 in (7), we get (39).Definition 32.
Letm2=ζ1,ζ2,⋯,ζn be a probability distribution with positive entries. Then the Shannon entropy is defined by
(40)Esm2=−∑i=1nζilogζi.Corollary 33.
Suppose thatm2=γ1,γ2,⋯,γn is a probability distribution such that γi>0 for all i∈1,2,⋯,n, then
(41)logn−Esm2≤16∑i=1nγin−1γi22γi2+γi2.Proof.
Since the functionΦx=−logx is both convex and 4-convex on (0,∞) because Φx=1/x2 and Φ″″x=6/x4 are positive for x>0, therefore (41) can easily be obtained by putting Φx=−logx and ζi=1i=1,2,⋯,n in (35).Corollary 34.
Assume that the conditions of Corollary33 are fulfilled, then
(42)logn−Esm2≥12∑i=1nγin−1γi23γi2nγi+12.Proof.
Utilizing the functionΦx=−logx and ζi=11,2,⋯,n in (36), we obtain (42).Definition 35.
Letm1=γ1,γ2,⋯,γn and m2=ζ1,ζ2,⋯,ζn be arbitrary probability distributions such that γi,ζi>0 for all i∈1,2,⋯,n. Then, the Kullback–Liebler divergence is defined by
(43)Dklm1,m2=∑i=1nζilogζiγi.Corollary 36.
Assume thatm1=γ1,γ2,⋯,γn and m2=ζ1,ζ2,⋯,ζn are positive probability distributions, then
(44)Dklm1,m2≤16∑i=1nγi1−ζiγi22+γiζi.Proof.
The functionΦx=xlogx is convex and 4-convex on (0,∞) because Φ″x=1/x>0 and Φ″″x=2/x3>0 for all x∈0,∞. Thus, using (35) by taking Φx=xlogx, we get (44).Corollary 37.
Let the postulates of Corollary36 be true. Then
(45)Dklm1,m2≥32∑i=1nγi−ζi22γi−ζi.Proof.
SubstitutingΦx=xlogx, x>0 in (36), we acquire (45).Definition 38.
For any positive probability distributionsm1=γ1,γ2,⋯,γn and m2=ζ1,ζ2,⋯,ζn, the Bhattacharyya coefficient is defined as
(46)Cbm1,m2=∑i=1nγiζi.Corollary 39.
Assume thatm1=γ1,γ2,⋯,γn and m2=ζ1,ζ2,⋯,ζn are positive probability distributions; then
(47)1−Cbm1−m2≤124∑i=1nγi1−ζiγi22+γiζi3/2.Proof.
IfΦx=−x,x>0, then Φ″x=1/4x−3/2 and Φ″″x=15/16x−7/2. Thus, this shows that both Φ″ and Φ″″ are positive on (0,∞). Hence, this confirms the convexity as well as 4-convexity of the function Φx=−x. Therefore, using (35) by choosing Φx=−x, we obtain (47).Corollary 40.
Suppose that the assumptions of Corollary39 hold, then
(48)1−Cbm1,m2≥12∑i=1nγi−ζi2γi3γi2γi+ζi3/2.Proof.
UsingΦx=−x,x>0, in (36), we obtain (48).Remark 41.
The analogous form of above discrete forms for different divergences, Shannon entropy and Bhattacharyya coefficient, can easily be obtained by utilizing Theorems3 and 5.


## 6. Conclusion

There are extensive literature devoted to the Jensen inequality concerning different refinements, extensions, and improvements. Also, there are many bounds obtained for the Jensen gap which provides many interesting and valuable estimates for the Jensen gap. In this note, we proposed a novel technique of obtaining of some significant estimates for Jensen’s gap while utilizing a 4-convex function. We obtained the required estimates for the Jensen gap by utilizing the definition of convex function and the famous Jensen inequality. For the support of our main results, we provided some examples for taking some particular convex functions. We presented some consequences of the main results in which some new important improvements for the Hölder and Hermite–Hadamard inequalities are acquired. Furthermore, for some more consequences of the main results, we obtain several relations for power and quasiarithmetic means. Applications of the main results are discussed in the information theory. These applications give many interesting estimates for several divergences, Bhattacharyya coefficient and Shannon entropy.



---

*Source: 2157375-2022-01-17.xml*