[Paper Review] Some Refinements of Discrete Jensen's Inequality and Some of Its Applications

Jamal Rooin|ArXiv.org|Oct 24, 2006

Mathematical Inequalities and Applications1 references8 citations

TL;DR

This paper refines discrete Jensen's inequality using weight functions and double stochastic matrices, establishing tighter bounds for convex mappings in abstract spaces. It derives new inequalities in Lp-spaces, measure spaces, and via logarithmic means, extending prior results with applications to functional analysis and harmonic analysis.

ABSTRACT

In this paper, using some aspects of convex functions, we refine discrete Jensen's inequality via weight functions. Then, using these results, we give some applications in different abstract spaces and obtain some new interesting inequalities.

Motivation & Objective

• To refine discrete Jensen's inequality by introducing weight functions that preserve probability measures.
• To extend existing results on Jensen-type inequalities to abstract spaces such as Lp-spaces and measurable function spaces.
• To establish new inequalities involving logarithmic means and arithmetic means through convexity and integral representations.
• To generalize earlier results from [5,6] using convexity and Fubini-type arguments in weighted settings.
• To provide a unified framework for deriving inequalities in diverse functional analytic contexts using convex mappings and weighted averages.

Proposed method

• Introduces a weight function ω(i,j) satisfying row and column normalization with respect to probability measures μ and λ.
• Defines a one-parameter family of functions φω₁,ω₂(t) that interpolate between two weighted convex combinations of points.
• Applies convexity and Jensen's inequality to the function φω₁,ω₂(t), proving it is convex in t and bounded between φ(∑λjxj) and ∑λjφ(xj).
• Uses Fubini's theorem to interchange integrals and apply Fatou’s lemma in Lp and measure-theoretic settings.
• Applies the logarithmic mean L(a,b) = (b−a)/(ln b − ln a) to derive inequalities involving reciprocal norms and integrals.
• Employs Cesàro summability and norm convergence in Lp-spaces to derive limit identities for p-norms and means.

Experimental results

Research questions

• RQ1 How can discrete Jensen’s inequality be refined using two weight functions while preserving convexity and tight bounds?
• RQ2 What are the implications of these refinements in Lp-spaces and measurable function spaces with respect to p-norms?
• RQ3 Can the logarithmic mean be used to derive new inequalities involving integrals of ratios of functions in convex settings?
• RQ4 How do double stochastic matrices and weighted averages improve the classical Jensen inequality in finite-dimensional convex sets?
• RQ5 What are the limit behaviors of the refined inequalities under Cesàro summability and Lp convergence?

Key findings

• The refined inequality φ(∑λjxj) ≤ φω₁,ω₂(t) ≤ ∑λjφ(xj) holds for all t ∈ [0,1], with φω₁,ω₂(t) convex in t.
• The integral ∫₀¹ φω₁,ω₂(t) dt lies between the Jensen lower bound and the upper sum, providing a continuous refinement.
• In Lp-spaces, the limit limₙ→∞ (1/n)∑ᵢ₌₁ⁿ ‖Lₚᵖ(|fᵢ|,|fₙ₊₁₋ᵢ|)‖₁ = ‖f‖ₚᵖ holds under Lp convergence and pointwise a.e. convergence.
• For nonnegative measurable functions f₁,…,fn, the inequality ∑λjφ(fj) ≤ μ(X) − ∑μi‖L⁻¹(1+∑ω₁λjfj, 1+∑ω₂λjfj)‖₁ ≤ φ(∑λjfj) is established, with φ(f) = ∫ f/(1+f) dμ.
• When applied to double stochastic matrices, the inequality reduces to ∑φ(xj)/n ≤ (1/n)∑φ(∑aijxj) ≤ ∑φ(xj)/n, recovering known mean inequalities.
• The identity ∫₀¹ φω₁,ω₂(t) dt = μ(X) − ∑μi‖L⁻¹(1+∑ω₁λjfj, 1+∑ω₂λjfj)‖₁ holds in measure-theoretic settings, linking logarithmic means to convex integrals.