Abstract
The understanding of inequalities in convexity is crucial for studying local fractional calculus efficiency in many applied sciences. In the present work, we propose a new class of harmonically convex functions, namely, generalized harmonically ψ - s -convex functions based on fractal set technique for establishing inequalities of Hermite-Hadamard type and certain related variants with respect to the Raina’s function. With the aid of an auxiliary identity correlated with Raina’s function, by generalized Hölder inequality and generalized power mean, generalized midpoint type, Ostrowski type, and trapezoid type inequalities via local fractional integral for generalized harmonically ψ - s -convex functions are apprehended. The proposed technique provides the results by giving some special values for the parameters or imposing restrictive assumptions and is completely feasible for recapturing the existing results in the relative literature. To determine the computational efficiency of offered scheme, some numerical applications are discussed. The results of the scheme show that the approach is straightforward to apply and computationally very user-friendly and accurate.
Highlights
Fractional calculus is a developing arena in mathematics with profound presentations in all connected areas such as wave liquids, thermodynamics, image processing, virology, biological population models, chaos, and signals processing, see [1,2,3,4,5]
Let ω = 0: Theorem 17 leads to the following local fractional integral inequality of “Ostrowski type”
Let ω = 1: Theorem 17 leads to the following local fractional integral inequality of “Ostrowski type”
Summary
Fractional calculus is a developing arena in mathematics with profound presentations in all connected areas such as wave liquids, thermodynamics, image processing, virology, biological population models, chaos, and signals processing, see [1,2,3,4,5]. Several noted variants via local fractional integral have been investigated by many researchers, for example, Mo et al [10] propounded Hermite-Hadamard inequalities for generalized s-convex functions. 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]. In this regard, integral inequalities have played an important role in describing real world problems such as in elementary research conducted by Niculescu and Persson [30]. We hope that the idea and results obtained will be a catalyst for further investigation
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