[Paper Review] Mercer's inequality for h-convex functions

Mohammad W. Alomari|arXiv (Cornell University)|Dec 19, 2017

Mathematical Inequalities and Applications5 references3 citations

TL;DR

This paper generalizes Mercer's inequality to h-convex functions by introducing a weighted inequality involving h-supermultiplicative functions, extending classical Jensen and triangle inequalities. The key result provides an upper bound for the weighted average of h-convex functions, with applications to normed spaces and refined triangle inequalities under specific conditions on h and weights.

ABSTRACT

A generalization of Mercer inequality for h-convex function is presented. As application, a weighted generalization of triangle inequality is given.

Motivation & Objective

• To extend Mercer's inequality from convex functions to h-convex functions, generalizing classical results in convex analysis.
• To establish a weighted inequality involving h-convex functions under conditions on the weight function h.
• To provide a converse refinement of the h-Jensen inequality by introducing a new upper bound involving extreme points of the sequence.
• To demonstrate applications in normed spaces by generalizing the triangle inequality using h-convexity.

Proposed method

• Introduces a generalization of Mercer’s inequality using h-convex functions defined via a non-negative supermultiplicative function h on (0, ∞).
• Applies the h-convexity condition f(αx + βy) ≤ h(α)f(x) + h(β)f(y) for α + β = 1 and α, β ∈ (0,1).
• Derives a weighted inequality involving the weighted average of x_k and the extreme values x₁ and xₙ, under the constraint ∑h(wₖ/Wₙ) ≤ 1.
• Uses the midpoint symmetry between xₖ and yₖ = x₁ + xₙ − xₖ to apply h-convexity and derive the main inequality.
• Reverses the inequality when h is submultiplicative and f is h-concave, under the condition ∑h(wₖ/Wₙ) ≥ 1.
• Applies the result to the absolute value function f(x) = |x| and h(t) = t to refine the triangle inequality in normed spaces.

Experimental results

Research questions

• RQ1 How can Mercer’s inequality be extended to h-convex functions using supermultiplicative weight functions h?
• RQ2 What conditions on h and weights wₖ ensure the validity of the generalized Mercer inequality for h-convex functions?
• RQ3 Can the generalized inequality provide a converse refinement of the h-Jensen inequality?
• RQ4 How does the inequality specialize to yield a refined triangle inequality in normed spaces?
• RQ5 What are the implications of the inequality when f is h-concave and h is submultiplicative?

Key findings

• The main inequality states that f(x₁ + xₙ − (1/Wₙ)∑wₖxₖ) ≤ f(x₁) + f(xₙ) − ∑h(wₖ/Wₙ)f(xₖ) holds for h-supermultiplicative functions and h-convex f when ∑h(wₖ/Wₙ) ≤ 1.
• When h is submultiplicative and f is h-concave, the inequality is reversed under the condition ∑h(wₖ/Wₙ) ≥ 1.
• The inequality refines the h-Jensen inequality by providing an upper bound for ∑h(wₖ/Wₙ)f(xₖ) in terms of f(x₁), f(xₙ), and f(x₁ + xₙ − (1/Wₙ)∑wₖxₖ).
• For f(x) = |x| and h(t) = t, the inequality reduces to ∑wₖ|xₖ| ≤ |x₁| + |xₙ| − |x₁ + xₙ − ∑wₖxₖ|, refining the triangle inequality.
• The result generalizes to norms by setting f(x) = ||x|| for x in a linear space L, yielding a weighted triangle inequality with h-convexity constraints.
• The proof relies on midpoint symmetry and the supermultiplicativity of h to bound f(yₖ) = f(x₁ + xₙ − xₖ) using h-convexity and the assumption on h-sums.