$$\left( p,q\right) $$ p , q -Hermite–Hadamard inequalities and $$\left( p,q\right) $$ p , q -estimates for midpoint type inequalities via convex and quasi-convex functions

Abstract

In this paper, we prove the correct \(\left( p,q\right) \)-Hermite–Hadamard inequality, some new \(\left( p,q\right) \)-Hermite–Hadamard inequalities, and generalized \(\left( p,q\right) \)-Hermite–Hadamard inequality. By using the left hand part of the correct \(\left( p,q\right) \)-Hermite–Hadamard inequality, we have a new equality. Finally using the new equality, we give some \(\left( p,q\right) \)-midpoint type integral inequalities through \(\left( p,q\right) \)-differentiable convex and \(\left( p,q\right) \)-differentiable quasi-convex functions. Many results given in this paper provide extensions of others given in previous works.