Abstract
In this article, we use s-convex and Green functions to obtain a bound for the Jensen gap in discrete form and a bound for the Jensen gap in integral form. We present two numerical examples to verify the main results and to examine the tightness of the bounds. Then, as an application of the discrete result, we derive a converse of the Hölder inequality. Based on the integral result, we obtain a bound for the Hermite-Hadamard gap and present a converse of the Hölder inequality in its integral form. Also, we obtain bounds for the Csiszár and Rényi divergences as applications of the discrete result. Finally, we utilize the bound obtained for the Csiszár divergence to deduce new estimates for some other divergences in information theory.
1. Introduction
Convex functions and their generalizations play a significant role in scientific observation and calculation of various parameters in modern analysis, especially in the theory of optimization. Moreover, convex functions have some nice properties, such as differentiability, monotonicity, and continuity, which are useful in applications [1–5]. Interest in mathematical inequalities for convex and generalized convex functions has been growing exponentially, and research in this respect has had a significant impact on modern analysis [6–20]. Several mathematical inequalities have been established for s-convex functions in particular [21–28], one of the most important being the Jensen inequality. In this paper, we study the Jensen inequality in a more standard framework for s-convex functions.
Definition 1.1 (s-convexity [29]). Fors > 0 and a convex subsetBof a real linear spaceS, a function Γ : B → ℝ is said to bes-convex if the inequality
holds for all ε1, ε2 ∈ Band κ1, κ2 ≥ 0 with κ1 + κ2 = 1.
The function Γ is said to be s-concave if the inequality (1.1) holds in the reverse direction. Obviously, for s = 1 an s-convex function becomes a convex function, which shows that s-convexity of a function is a generalization of ordinary convexity of that function.
Lemma 1.2 ([29]). LetBbe a convex subset of a real linear spaceSand let Γ : B → ℝ be a convex function. Then the following two statements hold:
Γ is s-convex for 0 < s ≤ 1 if Γ is non-negative;
Γ is s-convex for 1 ≤ s < ∞ if Γ is non-positive.
The Green function [30]
defined on [α1, α2] × [α1, α2] and the integral identity
for the function will be used to obtain the main results. Note that G1 is convex and continuous with respect to both variables.
This paper is organized as follows. In section 2 we give a bound for the Jensen gap in discrete form, which pertains to functions for which the absolute value of the second derivative is s-convex. We also derive a bound for the integral version of the Jensen gap. Then we conduct two numerical experiments that provide evidence for the tightness of the bound in the main result. We deduce a converse of the Hölder inequality from the discrete result and a bound for the Hermite-Hadamard gap from the integral result. Moreover, as a consequence of the integral result we obtain a converse of the Hölder inequality in its corresponding integral version. At the beginning of section 3 we present bounds for the Csiszár and Rényi divergences in the discrete case. Finally, we give estimates for the Shannon entropy, Kullback-Leibler divergence, χ2 divergence, Bhattacharyya coefficient, Hellinger distance, and triangular discrimination as applications of the bound obtained for the Csiszár divergence. Conclusions are presented in the final section.
2. Main Results
Using the concept of s-convexity, we derive a bound for the Jensen gap in discrete form, which is presented in the following theorem.
Theorem 2.1. Suppose |Γ|″ is s-convex for a functionand thatzi ∈ [α1, α2] and κi ∈ [0, ∞) fori = 1, …, nwith. Then the following inequality holds:
Proof: Using (1.3), we get
and
Equations (2.5) and (2.6) give
Taking the absolute value of (2.7), we get
By applying a change of variable x = tα1 + (1 − t)α2 for t ∈ [0, 1] and using the convexity of G1(t, x), the inequality (2.8) is transformed to
where The inequality (2.9) leads to the following by using s-convexity of the function |Γ|″:
Now, by using the change of variable x = tα1 + (1 − t)α2 for t ∈ [0, 1], we obtain
Upon replacing zi by in (2.11), we get
Also,
Upon replacing zi by in (2.13), we get
The result (2.4) is then obtained by substituting the values from (2.11)–(2.14) into (2.10).
Remark 2.2. If we use the Green functionG2, G3, orG4instead ofG1in Theorem 2.1, whereG2, G3, andG4are given in [30], we obtain the same result (2.4).
In the following theorem, we give a bound for the Jensen gap in integral form.
Theorem 2.3. Suppose |Γ″| is ans-convex function for, and let ξ1and ξ2be real-valued functions defined on [c1, c2] with ξ1(y) ∈ [α1, α2] for ally ∈ [c1, c2] and such that ξ2, ξ1ξ2, and (Γ ◦ ξ1) ξ 2are all integrable functions on [c1, c2]. Then the inequality
holds provided thatwhen ξ2(y) ∈ [0, ∞) for all y ∈ [c1, c2].
Proof: Using the same procedure as in the proof of Theorem 2.1, (2.15) can be obtained.
Example 1. Let, , and ξ2(y) = 1 for ally ∈ [0, 1]. Thenfor ally ∈ [0, 1]. This shows that Γ is a convex function while |Γ″| is-convex. Also, ξ1(y) ∈ [0, 1] for ally ∈ [0, 1] and we have [α1, α2] = [c1, c2] = [0, 1]. Now, the left-hand side of inequality (2.15) gives, which shows how sharp the Jensen inequality is. The right-hand side of (2.15) gives 0.0274, which is very close to the true discrepancyE1. That is, from inequality (2.15) we have
The difference 0.0274 − 0.0273 = 0.0001 between the two sides of (2.16) shows that the bound for the Jensen gap given by inequality (2.15) is very close to the true value.
Example 2. Let, ξ1(y) = y, and ξ2(y) = 1 for ally ∈ [0, 1]. Thenfor ally ∈ [0, 1], which shows that Γ is a convex function while |Γ″| iss-convex withAlso, ξ1(y) ∈ [0, 1] for ally ∈ [0, 1] and we have [α1, α2] = [c1, c2] = [0, 1]. Therefore, from the left-hand side of inequality (2.15) we obtainwhich shows that the Jensen inequality is quite sharp. The right-hand side of (2.15) gives 0.0387, a value very close to the true discrepancyE2. Finally, from inequality (2.15) we have
The difference 0.0387 − 0.0386 = 0.0001 between the two sides of (2.17) provides further evidence of the tightness of the bound for the Jensen gap given by inequality (2.15).
As an application of Theorem 2.1, we derive a converse of the Hölder inequality, stated in the following proposition.
Proposition 2.4. Letq2 > 1 andq1 ∉ (2, 3) be such that, and lets ∈ (0, 1]. Also, let [α1, α2] be a positive interval and let (d1, …, dn) and (b1, …, bn) be two positiven-tuples such that, withfori = 1, …, n. Then
Proof: Let for x ∈ [α1, α2]; then and which shows that Γ and |Γ″| are convex functions. The function |Γ″| is also non-negative, so by Lemma 1.2 it is also an s-convex function for s ∈ (0, 1]. Thus, using (2.4) with , and we derive
By using the inequality xγ − yγ ≤ (x − y)γ for 0 ≤ y ≤ x and γ ∈ [0, 1] with , , and we obtain
The inequality (2.18) follows from (2.19) and (2.20).
In the following proposition, we provide a converse of the Hölder inequality in integral form as an application of Theorem 2.3.
Proposition 2.5. Letq2 > 1 andq1 ∉ (2, 3) be such thatAlso, letbe two functions such that, , and ζ1(y)ζ2(y) are integrable on [c1, c2] withwhen [α1, α2] ⊂ ℝ. Then the inequality
holds fors ∈ (0, 1].
Proof: Using (2.15) with for , and and following the procedure of Proposition 2.4, we deduce (2.21).
As an application of Theorem 2.3, in the following corollary we establish a bound for the Hermite-Hadamard gap.
Corollary 2.6. Letbe a function such that |ψ″| is s-convex; then
Proof: The inequality (2.22) can be obtained by using (2.15) with ψ = Γ, [α1, α2] = [c1, c2], ξ2(y) = 1, and ξ1(y) = y for y ∈ [c1, c2].
3. Applications to Information Theory
Definition 3.1 (Csiszár f-divergence [31]). Letandwithfor [α1, α2] ⊂ ℝ. For a functionf :[α1, α2] → ℝ, the Csiszárf-divergence functional is defined as
Theorem 3.2. Letbe a function such that |f″| iss-convex. Then forandthe inequality
holds provided thatfor i = 1, …, n.
Proof: The inequality (3.23) can easily be deduced from (2.4) by taking , and
Definition 3.3 (Rényi divergence [31]). For μ ≥ 0 with μ ≠ 1 and two positive probability distributionst = (t1, …, tn) andr = (r1, …, rn), the Rényi divergence is defined as
Corollary 3.4. Let 0 < s ≤ 1 and. Then for positive probability distributionst = (t1, …, tn) andr = (r1, …, rn), the inequality
holds provided thatfor i = 1, …, n with μ > 1.
Proof: Let for x ∈ [α1, α2]. Then and which shows that Γ and |Γ″| are convex functions with |Γ″| ≥ 0; so by Lemma 1.2 the function |Γ″| is s-convex for s ∈ (0, 1]. Therefore, using (2.4) with , and , we derive (3.24).
Definition 3.5 (Shannon entropy [31]). Letr = (r1, …, rn) be a positive probability distribution; then the Shannon entropy is defined as
Corollary 3.6. Let, and letr = (r1, …, rn) be a positive probability distribution such thatfori = 1, …, nwith 0 < s ≤ 1. Then
Proof: Let f(x) = −log x for x ∈ [α1, α2]. Then and , which shows that f and |f″| are convex functions. Also, |f″| is non-negative and so by Lemma 1.2 we conclude that it is s-convex for s ∈ (0, 1]. Therefore, using (3.23) with f(x) = −log x and (t1, …, tn) = (1, …, 1), we get (3.25).
Definition 3.7 (Kullback-Leibler divergence [31]). For two positive probability distributionst = (t1, …, tn) andr = (r1, …, rn), the Kullback-Leibler divergence is defined as
Corollary 3.8. Let 0 < s ≤ 1 and 0 < α1 < α2, and lett = (t1, …, tn) andr = (r1, …, rn) be positive probability distributions such thatfori = 1, …, n. Then
Proof: Let f(x) = x log x for x ∈ [α1, α2]. Then and which shows that f and |f″| are convex functions. Also, |f″| ≥ 0, and so Lemma 1.2 guarantees the s-convexity of |f″| for s ∈ (0, 1]. Therefore, using (3.23) with f(x) = x log x, we get (3.26).
Definition 3.9 (χ2 divergence [31]). The χ2divergencefor two positive probability distributionst = (t1, …, tn) andr = (r1, …, rn) is defined as
Corollary 3.10. Let 0 < s ≤ 1 and 0 < α1 < α2, and let t = (t1, …, tn) and r = (r1, …, rn) be positive probability distributions such thatfori = 1, …, n. Then
Proof: Let f(x) = (x − 1)2 for x ∈ [α1, α2]. Then f″(x) = 2 > 0 and |f″|″(x) = 0, which shows that f and |f″| are convex functions. Also, the function |f″| is non-negative, and so Lemma 1.2 confirms its s-convexity for s ∈ (0, 1]. Therefore, using (3.23) with f(x) = (x − 1)2, we obtain (3.27).
Definition 3.11 (Bhattacharyya coefficient [31]). For two positive probability distributionst = (t1, …, tn) andr = (r1, …, rn), the Bhattacharyya coefficient is defined as
Corollary 3.12. Let 0 < s ≤ 1 and, and lett = (t1, …, tn) andr = (r1, …, rn) be two positive probability distributions such thatfori = 1, …, n.Then
Proof: Let for x ∈ [α1, α2]. Then and which shows that f and |f″| are convex functions. Also, |f″| ≥ 0 implies its s-convexity for s ∈ (0, 1] by Lemma 1.2. Therefore, using (3.23) with we obtain (3.28).
Definition 3.13 (Hellinger distance [31]). The Hellinger distancebetween two positive probability distributionst = (t1, …, tn) andr = (r1, …, rn) is defined as
Corollary 3.14. Let 0 < α1 < α2and 0 < s ≤ 1, and lett = (t1, …, tn) andr = (r1, …, rn) be positive probability distributions such thatfori = 1, …, n. Then
Proof: Let for x ∈ [α1, α2]. Then and which shows that f and |f″| are convex functions. Also, |f″| ≥ 0, and so from Lemma 1.2 we conclude its s-convexity for s ∈ (0, 1]. Therefore, using (3.23) with we deduce (3.29).
Definition 3.15 (Triangular discrimination [31]). For two positive probability distributionst = (t1, …, tn) andr = (r1, …, rn), the triangular discrimination is defined as
Corollary 3.16. Let 0 < s ≤ 1 and 0 < α1 < α2, and lett = (t1, …, tn) andr = (r1, …, rn) be positive probability distributions such thatfori = 1, …, n. Then
Proof: Let for x ∈ [α1, α2]. Then and which shows that f and |f″| are convex functions. Also, |f″| is non-negative, and thus s-convexity of the function |f″| for s ∈ (0, 1] follows from Lemma 1.2. Therefore, using (3.23) with we get (3.30).
Remark 3.17. Analogously, bounds for various divergences in integral form can be derived as applications of Theorem 2.3.
4. Conclusion
The Jensen inequality has numerous applications in engineering, economics, computer science, information theory, and coding; it has been derived for convex and generalized convex functions. This paper presents a novel approach to bounding the Jensen gap. Some bounds are obtained for the Jensen gap via s-convex functions. Numerical experiments not only confirm the sharpness of the Jensen inequality but also provide evidence for the tightness of the bound given in (2.15) for the Jensen gap. These experiments also show that the bound in (2.15) gives very close estimates for the Jensen gap even when the functions are not convex. The bounds are used to obtain new estimates for the Hermite-Hadamard and Hölder inequalities. Furthermore, based on the main results, various divergences are estimated. These estimates for divergences can be applied to signal processing, magnetic resonance image analysis, image segmentation, pattern recognition, and other areas. The ideas in this paper can also be used with other inequalities and for some other classes of convex functions.
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary materials, further inquiries can be directed to the corresponding author/s.
Author contributions
MA gave the main idea. MA and SK worked on Main Results while Y-MC worked on Introduction. All authors checked carefully the whole manuscript and approved.
Acknowledgments
This work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, and 11601485).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Summary
Keywords
Jensen inequality, s-convex function, green function, Csiszár divergence, Hölder inequality
Citation
Adil Khan M, Khan S and Chu Y-M (2020) New Estimates for the Jensen Gap Using s-Convexity With Applications. Front. Phys. 8:313. doi: 10.3389/fphy.2020.00313
Received
26 March 2020
Accepted
09 July 2020
Published
15 October 2020
Volume
8 - 2020
Edited by
Mustafa Inc, Firat University, Turkey
Reviewed by
Praveen Agarwal, Anand International College of Engineering, India; Mustapha Raissouli, Taibah University, Saudi Arabia
Updates
Copyright
© 2020 Adil Khan, Khan and Chu.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Yu-Ming Chu chuyuming2005@126.com
This article was submitted to Mathematical and Statistical Physics, a section of the journal Frontiers in Physics
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