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

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.

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 that is an interval of real numbers and is a convex function on . If and with , then Inequality (1) will be true in the reverse direction, if the function is concave on .

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 that is an interval in with and . If is a twice differentiable function such that is 4-convex on , then Inequality (2) will be true in the opposite direction, if the function is 4-concave.

Proof. Without misfortune of sweeping statement, assume that . Utilizing integration by parts, we have which implies that Since, the function is 4-convex on . Therefore, using the definition of convex function on the right hand side of (4), we receive which is equivalent to (2).

The integral version of (2) is stated in the following theorem.

Theorem 3. Assume that is an interval in that is a twice differentiable function such that is 4-convex and are integrable functions with . Also, assume that is an integrable function, . Then Inequality (6) will be true in the reverse sense, if   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 Theorem 2 are valid, then Inequality (7) will become true in the reverse direction, if the function is 4-concave.

Proof. Since, the function Ψ is 4-convex on I. Therefore, applying integral Jensen’s inequality on the right of (4), we get which is the required inequality.

The analogous inequality of (7) is given in the following theorem.

Theorem 5. Suppose that all the hypotheses of Theorem 3 are true. Then If the function 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 functions and . Then, . This verifies that the function is convex as well as 4-convex. Now, utilizing (6) for , , , , and , we get Using above functions with the given interval in inequality (4) in [17], we acquire 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 functions , , and for all . Then, . Clearly, both are nonnegative on [0,1]. This shows that the function is convex as well as 4-convex. Utilizing and in (6), we obtain Now, using the chosen functions in the inequality (4) in [17], we acquire 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 functions , , and are defined on [0,1]. Then, and . Obviously, both the functions and are nonnegative on . This confirms the convexity and 4-convexity of the function . Choosing in (6), we obtain Now, using the given functions in the inequality (4) in [17], we acquire 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 Theorem 2.

Proposition 10. Let and be two positive -tuples and , such that . If , then

Proof. Since the function is convex as well as 4-convex on (0,∞) for all . Therefore, utilizing (2) by choosing and for all and then taking power , we get As the inequality holds, for all and , thus using (18) for , we obtain Now, comparing (17) and (19), we acquire (16).

Another consequence of Theorem 2 is given in the following corollary, in which we provide a relation for the Hölder inequality.

Corollary 11. Let and be two positive -tuples, and such that . If , then

Proof. For such that , the function is convex as well as 4-convex on [0,∞]. Therefore, utilizing (2) by choosing , we get (20).

As a consequence of Theorem 4, we acquire another improvement for the Hölder inequality which is stated in the following corollary.

Corollary 12. Assume that and are two -tuples such that for all . If and such that , then

Proof. The function is both convex and 4-convex for with such that . Therefore, inequality (21) can easily be acquired by putting in (7).

Now, we recall the definition of power mean.

Definition 13. Let and be two -tuples such that for all with . Then, the power mean of order is defined by

As a consequence of Theorem 2, in the following corollary, we give bound for the power mean.

Corollary 14. Let and be two positive -tuples such that and and be nonzero real numbers. Then the following statements are true: (i)If with or or , then(ii)If with or or , then (23) holds.(iii)If with or with , then (23) holds in the reverse direction.

Proof. (i)For , the function is 4-convex with the given conditions.Therefore, using (2) by taking , we obtain (23) (ii)If the given conditions are hold, then the function will be 4-convex on (0,∞). Thus, (23) can easily be obtained by adopting the procedure of (i)(iii)The function is 4-concave on (0,∞) for the given values of and . Therefore, we can get the reverse inequality in (23) by adopting the procedure of (i).

In the following result, we present an application of Theorem 4.

Corollary 15. Let , , and be the same as that of Corollary 14 and . Then (A)If the conditions given in (i) and (ii) are satisfied, then (B)If the conditions in (iii) are fulfilled, then (24) holds in the reverse direction.

Proof. (A)Since, the function is 4-convex on (0,∞) for the conditions given in (i) and (ii) of Corollary 14. Therefore, using (7) for , we obtain (24)(B)If the condition on and mentioned in (iii) of Corollary 14 is true, then the function will be 4-concave for . Thus, utilizing (7) while choosing , we obtain the reverse inequality in (24).

In the following corollary, we obtain an interesting relation for different means as a consequence of Theorem 2.

Corollary 16. Let and be positive -tuples with . Then

Proof. Let , . Then, . Clearly, both are positive for all . This confirms that the function is convex as well as 4-convex on (0,∞). Therefore, putting in (2), we acquire (25).

In the following corollary, a relation for distinct means is obtain with the help of Theorem 2.

Corollary 17. Let hypotheses of Corollary 16 hold. Then

Proof. Consider function . Then clearly, . This shows that the given function is convex as well as 4-convex. Thus, applying (2) by choosing , , and , we get (26).

An application of Theorem 4 is acquired in the below corollary.

Corollary 18. Suppose that all the assumptions of Corollary 16 are true, then

Proof. Put , , and in (7), we get (27).

The following is another relation for distinct means which is the consequence of Theorem 4.

Corollary 19. Let the hypotheses of Corollary 16 be fulfilled. Then

Proof. Utilizing (7) for , , and , we get (28).
Now, we give the definition of quasiarithmetic mean.

Definition 20. Let and be positive -tuples with be strictly monotonic continuous function. Then the quasi-arithmetic mean is defined as

In the following corollary, we obtain a relation for the quasi-arithmetic mean with the help of Theorem 2.

Corollary 21. Let and be positive -tuples with . Also, let be strictly monotonic continuous function and be 4-convex on (0,∞). Then

Proof. Since, the function is 4-convex on (0,∞). Therefore, choosing , , and in (2), we obtain (30).

As an application of Theorem 4, in the following corollary, we present a relation for the quasi-arithmetic mean.

Corollary 22. Let the hypotheses of Corollary 21 hold. Then

Proof. Using (7) for , , and , 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. Let be a 4-convex function. Then

If the function is 4-concave, then the inequality (32) holds in the reverse direction.

Proof. Since, the function is 4-convex on . Therefore, using (6) for and , we get (32).

Corollary 24. Assume that the function is 4-convex, then Inequality (33) will be true in the opposite sense, if the function is 4-concave.

Proof. Inequality (33) can easily be deduced by choosing and 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 Theorems 3 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. Let be a real valued function defined on and and be two -tuples such that and for all . Then the Csisźar divergence is defined by

Theorem 27. Assume that the function is 4-convex and and are two -tuples such that and for all , then

Proof. All the hypotheses of this theorem are same as that of Theorem 2.
Thus, using (2) by taking , we obtain (35).

Theorem 28. Suppose that all the assumptions of Theorem 27 are satisfied, then

Proof. Since, the function is 4-convex. Therefore, utilizing (7) for , we get (36).

Definition 29. For any such that and arbitrary positive probability distributions and , the Rényi divergence is defined by

Corollary 30. Assume that and are two positive probability distributions and , then

Proof. Let . Then, . Clearly, for all . Thus, this verifies that the function is convex as well as 4-convex on (0,∞). Therefore, using (2) for , we get (38).

Corollary 31. Let the hypotheses of Corollary 30 hold. Then

Proof. Using and in (7), we get (39).

Definition 32. Let be a probability distribution with positive entries. Then the Shannon entropy is defined by

Corollary 33. Suppose that is a probability distribution such that for all , then

Proof. Since the function is both convex and 4-convex on (0,∞) because and are positive for , therefore (41) can easily be obtained by putting and in (35).

Corollary 34. Assume that the conditions of Corollary 33 are fulfilled, then

Proof. Utilizing the function and in (36), we obtain (42).

Definition 35. Let and be arbitrary probability distributions such that for all . Then, the Kullback–Liebler divergence is defined by

Corollary 36. Assume that and are positive probability distributions, then

Proof. The function is convex and 4-convex on (0,∞) because and for all . Thus, using (35) by taking , we get (44).

Corollary 37. Let the postulates of Corollary 36 be true. Then

Proof. Substituting , in (36), we acquire (45).

Definition 38. For any positive probability distributions and , the Bhattacharyya coefficient is defined as

Corollary 39. Assume that and are positive probability distributions; then

Proof. If , then and . Thus, this shows that both and are positive on (0,∞). Hence, this confirms the convexity as well as 4-convexity of the function . Therefore, using (35) by choosing , we obtain (47).

Corollary 40. Suppose that the assumptions of Corollary 39 hold, then

Proof. Using , 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 Theorems 3 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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally to writing of this paper. All authors read and approved the final manuscript.