Sharp Trapezoid Inequality for Quantum Integral Operator

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