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.