New Perspectives of Symmetry Conferred by q-Hermite-Hadamard Type Integral Inequalities

Abstract

The main goal of this work is to provide quantum parametrized Hermite-Hadamard like type integral inequalities for functions whose second quantum derivatives in absolute values follow different type of convexities. A new quantum integral identity is derived for twice quantum differentiable functions, which is used as a key element in our demonstrations along with several basic inequalities such as: power mean inequality, and Holder’s inequality. The symmetry of the Hermite-Hadamard type inequalities is stressed by the different types of convexities. Several special cases of the parameter are chosen to illustrate the investigated results. Four examples are presented.

On q-Hermite–Hadamard Type Inequalities via s-Convexity and (α,m)-Convexity

“…This second result is the main result for the last theorem of this subsection, and is a refinement of the parametric identity given in Lemma 2 from [28] when we consider a second parameter m. Lemma 2. Let k : [cm, d] → R be a twice q-differentiable function on (cm, d) and 0 < c < d, c d < m ≤ 1.…”

“…This second result is the main result for the last theorem of this subsec a refinement of the parametric identity given in Lemma 2 from [28] when w second parameter m.…”

“…The advantage of using one integral instead of two integrals from 0 to 1 in Lemma 1 in the right member of the identity is that in the proofs of the next inequalities in which it is used, we do not have a sum of two modulus; we have only a modulus, which is better for our majorizations. For the second goal, a new lemma has been demonstrated, which is a generalization of the corresponding identity from [28] when the parameter m = 1.…”