[Paper Review] Some Hermite-Hadamard Type Inequalities For Harmonically $(s,m)$-convex functions in Second Sense

Imran Abbas Baloch, İmdat Işcan|arXiv (Cornell University)|Apr 24, 2016

Mathematical Inequalities and Applications2 references3 citations

TL;DR

This paper introduces harmonically (s,m)-convex functions in the second sense and establishes new Hermite-Hadamard-type integral inequalities for such functions using Hölder's inequality and integral representations involving hypergeometric functions. The key contribution is a generalized inequality that unifies and extends prior results on harmonically s-convex and (s,m)-convex functions, with special cases recovering known theorems by Iscın, Chen, and Wu.

ABSTRACT

Authors introduce the concept of harmonically $(s,m)$-convex functions in second sense in \cite{II}.In this article, we establish some Hermite-Hadamard type inequalities of this class of functions.

Motivation & Objective

• To introduce and formalize the concept of harmonically (s,m)-convex functions in the second sense.
• To extend existing Hermite-Hadamard-type inequalities to this broader class of functions.
• To unify and generalize prior results on harmonically s-convex and (s,m)-convex functions.
• To derive sharp integral bounds for the difference between the average of function values and the integral mean using higher-order convexity.
• To recover known theorems as special cases by setting specific values of s and m.

Proposed method

• Define harmonically (s,m)-convex functions via the inequality f(xy/(tx + (1-t)y)) ≤ t^s f(y) + m(1-t)^s f(x) for t ∈ [0,1], s ∈ (0,1], m ∈ (0,1].
• Use a lemma expressing the difference between the arithmetic mean and integral mean as a weighted integral involving the derivative.
• Apply Hölder’s inequality to bound the L1-norm of the derivative term using the (s,m)-convexity of |f′|^q.
• Express the resulting integrals in terms of the hypergeometric function _2F_1 and the beta function β.
• Derive two main inequalities: one for q ≥ 1 using the power mean and another for q > 1 using the conjugate exponent p.
• Verify that the results reduce to known theorems (e.g., Iscın, Chen & Wu) when s=1 or m=1.

Experimental results

Research questions

• RQ1 How can the concept of harmonic convexity be generalized to include both s-convexity and m-convexity in the second sense?
• RQ2 What new Hermite-Hadamard-type inequalities emerge when |f′|^q is harmonically (s,m)-convex?
• RQ3 How do the new inequalities relate to and generalize existing results in the literature?
• RQ4 What is the role of the hypergeometric function and beta function in deriving sharp bounds?
• RQ5 Can the new inequalities recover previously established theorems as special cases?

Key findings

• The paper establishes a new Hermite-Hadamard-type inequality for harmonically (s,m)-convex functions in the second sense, valid for q ≥ 1.
• The bound involves the hypergeometric function _2F_1 and beta function, with coefficients ρ₁(s,q;a,b) and ρ₂(s,q;a,b) that depend on s, q, a, and b.
• For m=1, the inequality reduces to Theorem 1.7 by Chen and Wu, confirming consistency with prior work.
• For s=1, the inequality reduces to Theorem 1.3 by Iscın, showing it generalizes his result.
• For s=1 and m=1, the inequality recovers Theorem 1.4 by Iscın, demonstrating full unification of known results.
• The second main inequality for q > 1 uses the conjugate exponent p and involves ν₁(s,q;a,b) and ν₂(s,q;a,b), which are expressed via beta functions and hypergeometric functions.