Integral inequalities via fractional quantum calculus

Abstract In this paper we prove several fractional quantum integral inequalities for the new q-shifting operator Φ q a ( m ) = q m + ( 1 − q ) a ${_{a}}\Phi_{q}(m) = qm + (1-q)a$ introduced in Tariboon et al. (Adv. Differ. Equ. 2015:18, 2015), such as: the q-Hölder inequality, the q-Hermite-Hadamard inequality, the q-Cauchy-Bunyakovsky-Schawrz integral inequality, the q-Grüss integral inequality, the q-Grüss-Čebyšev integral inequality, and the q-Pólya-Szegö integral inequality.