Sharp trapezoid inequality for quantum integral operator

Trapezoid inequality estimates the difference of integral mean a function on finite interval [a, b] and arithmetic its values at endpoints b. Quantum calculus is based diference principle or without concept limits. Euler-Jackson q-difference operator q-integral are discretization ordinary derivatives integrals they can be generalized to shifted versions arbitrary domain b]. In this paper we disprove trapezoid for quantum appearing in literature by giving two counterexamples. We point out some differences between definite Riemann explain why mistake made obtain corrected results. also prove sharpness our new bounds estimating value mean. Further derive sharp which case with tightest bounds.