q-Non uniform difference calculus and classical integral inequalities

We first establish q -non uniform difference versions of the integral inequalities of Hölder, Cauchy–Schwarz, and Minkowski of classical mathematical analysis and then integral inequalities of Grönwall and Bernoulli based on the Lagrange method of linear q -non uniform difference equations of first order. Finally, we prove the Lyapunov inequality for the solutions of the q -non uniform Sturm–Liouville equation.