Generalized trapezium-type inequalities in the settings of fractal sets for functions having generalized convexity property

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.

Inverse fractional Sturm‐Liouville problem with eigenparameter in the boundary conditions

“…The concept of fractional differential equation is an extension of the classical differential equation which gives the chance of taking the $$$ \alpha $$$‐order derivative of a function, where $$$ 0<\alpha \le 1 $$$. This extension has been proved to be very effective as it gives many chances of calculating and computing on some phenomenon that can't be done by the classical differentiation and integration as described in many works 13–43 . Rashid et al 36 derived some generalizations that captured novel results under investigation by fractional operators.…”

Local Fractional Integral Hölder-Type Inequalities and Some Related Results

“…In particular, the local fractional calculus is utilized to establish some new inequalities which are extensions of classical real inequalities on certain fractal spaces. For example, Mo et al [32] derived generalized Jensen's inequality and generalized Hermite-Hadamard's inequality on fractal space, Khan et al [33] obtained generalized trapezium-type inequalities in the settings of fractal sets, Sun [29] given generalization of some inequalities for generalized harmonically convex functions via local fractional integrals. More recent results in this direction can be found in [13][14][15][16][17]27,29,30,32,33].…”

Some Inequalities for a New Class of Convex Functions with Applications via Local Fractional Integral

“…It is well known that convexity theory has potential applications in many intriguing and captivating fields of research and furthermore played a remarkable role in numerous areas, such as coding theory, optimization, physics, information theory, engineering, and inequality theory. Several new classes of classical convexity have been proposed in the literature (see [16][17][18][19][20][21][22][23][24][25][26]). Many researchers endeavored, attempted, and maintain their work on the concept of convex functions, generalize its variant forms in different ways using innovative ideas and fruitful techniques [27][28][29].…”