[Paper Review] On Some New Hadamard-Type Inequalities for Coordinated Quasi-convex Functions

M. Emіn Özdemіr, Çetin Yıldız+1 more|arXiv (Cornell University)|Jul 20, 2011

Mathematical Inequalities and Applications6 references21 citations

TL;DR

This paper establishes new Hadamard-type inequalities for co-ordinated quasi-convex functions on a rectangle in ℝ², introducing a novel mapping G(t,s) associated with co-ordinated quasi-convexity and proving its Lipschitz continuity and convexity on coordinates. The key contribution is a sharp integral inequality bounding the average of f over Δ in terms of its values at corner and midpoint points.

ABSTRACT

In this paper, we establish some Hadamard-type inequalities based on coordinated quasi-convexity. Also we define a new mapping associated to coordinated convexity and we prove some properties of this mapping.

Motivation & Objective

• To extend classical Hadamard inequalities to the setting of co-ordinated quasi-convex functions.
• To define and analyze a new integral mapping G(t,s) that generalizes the behavior of f over the rectangle Δ.
• To establish sharp bounds for the double integral of f over Δ using values at the center and corners of Δ.
• To prove that the mapping G is convex on coordinates and satisfies Lipschitz conditions under appropriate assumptions.

Proposed method

• The authors define a new mapping G(t,s) = 1/((b−a)(d−c)) ∫∫ f(tx+(1−t)(a+b)/2, sy+(1−s)(c+d)/2) dxdy over Δ.
• They prove that G is convex on the coordinates in [0,1]×[0,1] using the definition of co-ordinated quasi-convexity.
• The mapping G is shown to be Lipschitz continuous under the assumption that f satisfies an L-Lipschitz condition with respect to its variables.
• The proof uses the triangle inequality and the Lipschitz condition to bound differences |G(t₂,s₂)−G(t₁,s₁)| in terms of |t₂−t₁| and |s₂−s₁|.
• A key step involves integrating a convexity-based inequality over t and s to derive the final double integral bound.
• The final inequality relates the average value of f over Δ to a weighted sum of f at the corners, midpoints of edges, and the center point (a+b)/2, (c+d)/2.

Experimental results

Research questions

• RQ1 Can Hadamard-type inequalities be extended to co-ordinated quasi-convex functions, which are a broader class than co-ordinated convex functions?
• RQ2 What properties does the new integral mapping G(t,s) possess under co-ordinated quasi-convexity?
• RQ3 How can the Lipschitz continuity of G be established when f satisfies an L-Lipschitz condition?
• RQ4 What is the tightest possible bound for the double integral of f over Δ in terms of its values at key points?
• RQ5 Is the resulting inequality sharp, and how does it compare to known inequalities for co-ordinated convex functions?

Key findings

• The mapping G(t,s) is convex on the coordinates in [0,1]×[0,1], generalizing the behavior of the original function f.
• The infimum of G(t,s) occurs at (0,0) and equals f((a+b)/2, (c+d)/2), while the supremum at (1,1) equals the double integral average of f over Δ.
• Under an L-Lipschitz condition, G is shown to be Lipschitz continuous with a bound proportional to (b−a)|t₂−t₁| + (d−c)|s₂−s₁|.
• The paper derives a sharp inequality: the double integral of f is bounded above by a weighted sum of f at the four corners, midpoints of edges, and the center point.
• The final inequality is sharp and reduces to known results when f is co-ordinated convex.
• The bound is expressed as: ∫∫f/(b−a)(d−c) ≤ (1/4)[f(a,c)+f(b,c)+f(a,d)+f(b,d)]/4 + [f((a+b)/2,c)+f((a+b)/2,d)]/2 + f((a+b)/2,(c+d)/2).