[Paper Review] Some new inequalities of Hermite-Hadamard's type

Aziz Sağlam, Mehmet Zeki Sarıkaya+1 more|arXiv (Cornell University)|May 4, 2010

Mathematical Inequalities and Applications3 references3 citations

TL;DR

This paper establishes new integral inequalities of Hermite-Hadamard type for differentiable functions whose absolute values of derivatives are convex. Using a double integral identity involving a piecewise-defined kernel function, the authors derive sharp bounds for the difference between the midpoint and integral mean of a function, with applications to special means of real numbers such as arithmetic, logarithmic, and identric means.

ABSTRACT

In this paper, we establish several new inequalities for some differantiable mappings that are connected with the celebrated Hermite-Hadamard integral inequality. Some applications for special means of real numbers are also provided.

Motivation & Objective

• To derive new inequalities related to the Hermite-Hadamard integral inequality for differentiable functions with convex absolute derivatives.
• To extend existing results by introducing a double integral representation involving a piecewise kernel function.
• To provide tighter bounds on the deviation between the midpoint and integral mean of a function.
• To apply the theoretical results to special means of real numbers, including arithmetic, logarithmic, and identric means.
• To generalize previous inequalities by incorporating Lp-norm estimates and convexity assumptions on |f′|q.

Proposed method

• Derives a novel double integral identity (Lemma 2) expressing the difference between f((a+b)/2) and (1/(b−a))∫ₐᵇ f(x)dx using a kernel function m(t) defined piecewise on [0,1/2] and (1/2,1].
• Applies Hölder’s inequality to the double integral representation, leveraging the convexity of |f′|q on [a,b] to bound the integral in terms of |f′(a)| and |f′(b)|.
• Uses the Lp-norm of the kernel function |m(t)−m(s)| to compute the constant factor (2/((p+1)(p+2)))^(1/p) in the final inequality.
• Establishes Theorem 2 and Theorem 3 by combining the kernel identity with Hölder’s inequality under the assumption that |f′|q is convex.
• Applies the general inequalities to specific functions (e.g., f(x)=xⁿ, f(x)=−ln x, f(x)=1/x) to derive concrete bounds for special means.
• Employs integration by parts and symmetry arguments to evaluate the double integrals over the unit square partitioned at t=1/2 and s=1/2.

Experimental results

Research questions

• RQ1 How can new inequalities of Hermite-Hadamard type be derived for differentiable functions when |f′|q is convex?
• RQ2 What is the optimal constant in the bound for |f((a+b)/2) − (1/(b−a))∫ₐᵇ f(x)dx| under convexity of |f′|q?
• RQ3 Can the derived inequalities be applied to estimate the difference between classical means such as A, L, I, and Lp?
• RQ4 How do the bounds depend on the parameter p in the Lp-norm framework?
• RQ5 What are the implications of these inequalities for special functions like f(x)=xⁿ, f(x)=−ln x, and f(x)=1/x in the context of mean inequalities?

Key findings

• The paper establishes a sharp upper bound: |f((a+b)/2) − (1/(b−a))∫ₐᵇ f(x)dx| ≤ (b−a) × (2/((p+1)(p+2)))^(1/p) × (|f′(a)|^q + |f′(b)|^q)^(1/q) / 2^(1/q), under the convexity of |f′|q.
• The double integral identity in Lemma 2 provides a new representation of the Hermite-Hadamard remainder using a piecewise kernel function m(t), enabling precise estimation.
• For f(x) = xⁿ with |n| ≥ 1, the inequality yields |Aⁿ(a,b) − Lₙⁿ(a,b)| ≤ |n|(b−a)/√6 × A(a²⁽ⁿ⁻¹⁾, b²⁽ⁿ⁻¹⁾), linking the result to power means.
• When f(x) = −ln x, the inequality leads to a bound on the logarithmic ratio of the identric and arithmetic means: ln(I/A) ≤ (b−a)/(ab) × (2/((p+1)(p+2)))^(1/p) × A(|b|^q, |a|^q)^(1/q).
• For f(x) = 1/x, the result gives |A⁻¹(a,b) − L⁻¹(a,b)| ≤ (b−a)/(a²b²) × (2/((p+1)(p+2)))^(1/p) × A(|a|²q, |b|²q)^(1/q), relating to harmonic and logarithmic means.
• The constant (2/((p+1)(p+2)))^(1/p) is explicitly derived from the Lp-norm of the kernel function over the unit square, showing tightness in the estimation framework.