[Paper Review] Generalized s-convex function on fractal sets

Huixia Mo, Xin Sui|arXiv (Cornell University)|May 4, 2014

Mathematical Inequalities and Applications18 references6 citations

TL;DR

This paper introduces two new classes of generalized s-convex functions on real linear fractal sets ℝ^α (0 < α < 1), extending classical s-convexity to local fractional calculus. It establishes properties, relationships, and applications of these functions using local fractional derivatives and integrals, with key results showing that the generalized s-convexity reduces to classical s-convexity when α = 1.

ABSTRACT

We introduce two kinds of generalized $s$-convex functions on real linear fractal sets $\mathbb{R}^α(0

Motivation & Objective

• To extend the concept of s-convex functions to fractal sets ℝ^α (0 < α < 1) using local fractional calculus.
• To define and analyze two distinct classes of generalized s-convex functions: one in the first sense and one in the second sense on fractal sets.
• To investigate the relationship between the two generalized s-convex function classes and their properties under local fractional operations.
• To provide applications of generalized s-convex functions in the context of fractal analysis and optimization.
• To generalize classical results such as Jensen’s and Hermite-Hadamard’s inequalities to the fractal setting.

Proposed method

• Define generalized s-convex functions on ℝ^α using local fractional calculus, with inequalities involving α-powered coefficients: f(λ₁u + λ₂v) ≤ λ₁^{sα}f(u) + λ₂^{sα}f(v).
• Utilize the Gao-Yang-Kang framework for local fractional calculus, including local fractional continuity, derivative, and integral operations.
• Apply the local fractional integral definition: _aI_b^{(α)}f = 1/Γ(1+α) ∫_a^b f(t)(dt)^α, with (Δt_j)^α as the measure.
• Use the Gamma function and fractal arithmetic (e.g., (a+b)^α = a^α + b^α) to maintain consistency in operations on ℝ^α.
• Establish properties via case analysis on domain behavior (e.g., u=0, v>0) and verify inequalities under α-powered weights.
• Construct counterexamples using non-continuous functions on ℝ^α to demonstrate strictness of conditions and distinguish between the two function classes.

Experimental results

Research questions

• RQ1 How can the classical notion of s-convexity be generalized to functions defined on fractal sets ℝ^α with 0 < α < 1?
• RQ2 What are the defining properties and structural differences between generalized s-convex functions in the first and second senses on fractal sets?
• RQ3 How do the two generalized s-convex function classes relate to each other, and under what conditions do they coincide?
• RQ4 Can classical inequalities such as Jensen’s and Hermite-Hadamard’s be extended to the local fractional setting using these generalized functions?
• RQ5 What are the limitations or non-applicability cases for generalized s-convexity, as demonstrated by counterexamples?

Key findings

• The paper successfully defines two distinct classes of generalized s-convex functions on ℝ^α: GK_s^1 and GK_s^2, based on α-powered coefficients and different normalization conditions.
• For f(u) = b^α u^{sα} + c^α, it is proven that f ∈ GK_s^1 under λ₁^{sα} + λ₂^{sα} = 1, but f ∉ GK_s^2 when c^α < 0 and b^α > 0.
• A counterexample function f(u) = u^{(s/(1-s))α} for 0 ≤ u ≤ 1 and k^α u^{(s/(1-s))α} for u > 1 is shown to belong to GK_s^1 but not to GK_s^2, demonstrating the strictness of the second class.
• The generalized s-convex functions reduce to classical s-convex functions when α = 1, confirming consistency with known results.
• The local fractional continuity and differentiability conditions are essential in distinguishing function behavior on fractal sets, as shown by discontinuity-induced violations of convexity conditions.
• The analysis confirms that GK_s^2 requires non-negativity of f on (0, ∞), and violating this leads to contradiction with the definition, thus establishing a necessary condition.