New bounds for q -midpoint-type inequalities via twice q -differentiable functions on quantum calculus

The aim of this work is to establish new bounds for q -midpoint-type integral inequalities for twice q -differentiable convex functions. We first prove some lemmas which are useful for our main results. Then, we present some midpoint-type inequalities for twice q -differentiable functions. With all these, we showed the results we found in the classical analysis by using q→ 1^-.