Fractal perspective of superquadratic functions with generalized probability estimations

Area of Science:

• Mathematical Analysis
• Fractal Geometry

Background:

• Generalized superquadratic functions are crucial in mathematical analysis.
• Fractal sets present unique challenges for function analysis and inequality development.

Purpose of the Study:

• To introduce and analyze a new class of generalized superquadratic functions on fractal sets.
• To develop novel generalized inequalities based on these functions.
• To explore the practical implications of these inequalities in fractal spaces.

Main Methods:

• Definition and characterization of generalized superquadratic functions on fractal sets.
• Derivation of generalized Jensen's, converse Jensen's, Mercer Jensen's, and Hermite-Hadamard's inequalities.
• Validation through numerical calculations, graphical representations, and comparative analysis with generalized convex functions.

Main Results:

• Introduction of generalized superquadratic functions on fractal sets.
• Establishment of new generalized inequalities with enhanced refinement compared to those from generalized convex functions.
• Demonstration of applicability in probability expectations and special means within fractal space.

Conclusions:

• The newly developed inequalities offer significant improvements and extensions over existing literature.
• Generalized superquadratic functions provide a more refined framework for inequalities on fractal sets.
• The research opens avenues for further exploration in fractal analysis and its applications.