[Paper Review] Hermite-Hadamard type inequalities for GA-s-convex functions

İmdat Işcan|arXiv (Cornell University)|Jun 8, 2013

Mathematical Inequalities and Applications2 references19 citations

TL;DR

This paper introduces GA-s-convex functions in both the first and second senses, extending Hermite-Hadamard-type integral inequalities to this new class. It establishes sharp bounds for the difference between the integral mean and function values using weighted geometric means, leveraging integral identities and power mean estimates under GA-s-convexity assumptions on the derivative's absolute value raised to a power q.

ABSTRACT

In this paper, The author introduces the concepts of the GA-s-convex functions in the first sense and second sense and establishes some integral inequalities of Hermite-Hadamard type related to the GA-s-convex functions.

Motivation & Objective

• To define and investigate GA-s-convex functions in the first and second senses, generalizing classical convexity and GA-convexity.
• To establish new integral inequalities of Hermite-Hadamard type for functions whose derivatives satisfy GA-s-convexity conditions.
• To derive bounds involving logarithmic, geometric, and arithmetic means using integral representations and power mean estimates.
• To apply the derived inequalities to special means such as logarithmic, identric, and p-logarithmic means.
• To generalize prior results on GA-convex functions by incorporating the parameter s ∈ (0,1], thereby enriching the class of functions amenable to Hermite-Hadamard analysis.

Proposed method

• Introduces GA-s-convexity in the first and second senses via functional inequalities involving weighted geometric means and power weights.
• Employs an integral identity involving the logarithmic mean and weighted integrals of the derivative along geometric paths: ∫₀¹ b²ᵗa²⁽¹⁻ᵗ⁾f′(bᵗa¹⁻ᵗ) dt.
• Applies Hölder’s inequality and power mean estimates to bound the difference between the integral mean and function values.
• Uses the condition that |f′|⁰ is GA-s-convex in the first sense to derive bounds involving constants cᵢ(s,q) derived from integrals of the form ∫₀¹ (b/a)^(qt) (1−tˢ)|f′(a)|⁰ dt.
• Applies the results to special means by substituting f(x) = x and f(x) = √x into the derived inequalities.
• Derives explicit bounds using the logarithmic mean L(a,b), geometric mean G(a,b), and arithmetic mean A(a,b) in conjunction with q-norms and power means.

Experimental results

Research questions

• RQ1 How can the concept of s-convexity be adapted to the geometric-arithmetic (GA) setting to define new classes of functions?
• RQ2 What new Hermite-Hadamard-type inequalities emerge when the derivative’s absolute value is GA-s-convex?
• RQ3 How do these inequalities refine or generalize existing bounds for the integral mean of a function over [a,b]?
• RQ4 What are the implications of these inequalities for special means such as L(a,b), A(a,b), and G(a,b)?
• RQ5 Can the parameter s ∈ (0,1] be used to interpolate between classical convexity and GA-convexity, and what are the resulting quantitative improvements?

Key findings

• The paper establishes a new Hermite-Hadamard-type inequality for GA-s-convex functions in the first sense, showing that |f(a)+f(b)|/2 − 1/(ln b − ln a) ∫ₐᵇ f(x)/x dx is bounded by an expression involving a, b, ln(b/a), and integrals of (b/a)^(qt) weighted by (1−tˢ)|f′(a)|⁰ and tˢ|f′(b)|⁰.
• For q > 1, the bound is further refined using Hölder’s inequality, yielding a factor of (q−1)/(2q−1) raised to the power 1−1/q, which improves the estimate as q increases.
• When s = 1, the results reduce to known inequalities for GA-convex functions, confirming consistency with prior work.
• The inequality for the geometric mean f(√(ab)) is bounded by a similar expression involving c₇(1,q/2), c₈(1,q/2), etc., with the same power mean structure.
• Proposition 1 and 2 apply the results to the means A(a,b), L(a,b), and G(a,b), deriving explicit bounds in terms of logarithmic means and power means.
• The derived bounds are sharp in the sense that the constants involved (e.g., (q−1)/(2q−1)) are optimal for the given class of functions.