Abstract
In the paper, we extend some previous results dealing with the Hermite–Hadamard inequalities with fractal sets and several auxiliary results that vary with local fractional derivatives introduced in the recent literature. We provide new generalizations for the third-order differentiability by employing the local fractional technique for functions whose local fractional derivatives in the absolute values are generalized convex and obtain several bounds and new results applicable to convex functions by using the generalized Hölder and power-mean inequalities.As an application, numerous novel cases can be obtained from our outcomes. To ensure the feasibility of the proposed method, we present two examples to verify the method. It should be pointed out that the investigation of our findings in fractal analysis and inequality theory is vital to our perception of the real world since they are more realistic models of natural and man-made phenomena.
Highlights
Fractals are mathematical developments that present self-similarity over a scope of scales and noninteger measurements
The use of fractal analysis in image processing, machine learning, cryptography, electrochemical processes, physics, diagnostic imagining, neuroscience, image analysis, acoustic, physiology, and Riemann zeta zeros has shown considerable guarantee for estimating forms that have changed as ordinarypartial differential equations [4,5,6,7,8]
The main purpose of the paper is presenting the fundamental basis of fractals and illustrating analysis of fractal sets and related estimations, in particular, establishing the integral inequalities for the functions whose local fractional differentiation in the absolute values are generalized convex
Summary
Fractals are mathematical developments that present self-similarity over a scope of scales and noninteger (fractal) measurements. The fractional operator does provide many potentially helpful tools for numerous problems involving special functions of mathematical science and their extensions and generalizations in one and several variables. The main purpose of the paper is presenting the fundamental basis of fractals and illustrating analysis of fractal sets and related estimations, in particular, establishing the integral inequalities for the functions whose local fractional differentiation in the absolute values are generalized convex. Local fractional inequalities and their fertile applications in pure and applied mathematics have attracted the attention of many researchers [15, 16]. We investigate new concepts of differentiation and integration taking into account the fractal sets and generalized convex functions. Some special cases are correlated with existing results on classical convexity
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