Generation of new fractional inequalities via n polynomials s-type convexity with applications

Saima Rashid,Dumitru Baleanu,İmdat İşcan,Yu-Ming Chu

doi:10.1186/s13662-020-02720-y

Abstract

The celebrated Hermite–Hadamard and Ostrowski type inequalities have been studied extensively since they have been established. We find novel versions of the Hermite–Hadamard and Ostrowski type inequalities for the n-polynomial s-type convex functions in the frame of fractional calculus. Taking into account the new concept, we derive some generalizations that capture novel results under investigation. We present two different general techniques, for the functions whose first and second derivatives in absolute value at certain powers are n-polynomial s-type convex functions by employing mathcal{K}-fractional integral operators have yielded intriguing results. Applications and motivations of presented results are briefly discussed that generate novel variants related to quadrature rules that will be helpful for in-depth investigation in fractal theory, optimization and machine learning.

Highlights

Introduction and preliminariesA few decades ago, the classical calculus has been revolutionized by tremendous innovations
We propose an novel class of functional variants for convex functions and several other new and effectively applicable generalizations for convexity theory and fractional operators
We demonstrate some essential ideas associated with the fractional integral which is mainly due to Mubeen et al [70]

Summary

Hermite–Hadamard type inequalities for n-polynomial s-type convex function

The aim of this section is to find some inequalities of Hermite–Hadamard type for npolynomial s-type convex functions. Proof It follows from Lemma 3.1 and |P |q is a n-polynomial s-type convex function together with the power mean inequality that 1 q ζ K dζ α ζ z + (1 – ζ )η1 q dζ ζ z + (1 – ζ )η2 q dζ 1 q α+K. Proof Making use of Lemma 3.1 and |P |q is a n-polynomial s-type convex function on I together with the Hölder inequality we get 1 rα. Theorem 4.4 Let α, K > 0, s ∈ [0, 1], q, r > 1 with 1/q + 1/r = 1, n ∈ N, η2 > η1, I = [η1, η2], and P : I → R be a twice differentiable function on I◦ such that P ∈ L1[η1, η2] and |P (z)|q is a n-polynomial s-type convex function on I. Proof Using Lemma 4.1 and the Hölder inequality together with the n-polynomial s-type convexity, we have α r dζ

Applications

Conclusion