q -Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions

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