Some new fractal Milne-type inequalities for generalized convexity with applications

Abstract

Abstract Fractals are of immense importance across various branches of mathematics, science, and integral inequalities, as their intricate, self-similar structures can model complex natural phenomena and enhance the precision of mathematical descriptions. In this article, we explore generalized Milne-type integral inequalities within the framework of generalized m-convex functions on fractal sets. To achieve this, we introduce a novel fractal integral identity involving differentiable generalized functions. Utilizing this new identity, we derive several contemporary fractal Milne-type integral inequalities and provide specific inequalities for bounded functions. Additionally, we offer illustrative examples and applications, including additional inequalities for generalized special means and various error estimates for the generalized Milne-type quadrature formula.

Keywords

• Milne-type inequality
• Generalized m-convex function
• Local fractional integrals
• Hölder’s inequality
• Power mean inequality
• Special means