Some inequalities for the \(q\)-beta and the \(q\)-gamma functions via some \(q\)-integral inequalities (Q2378933)

Some new inequalities for the \(q\)-gamma, the \(q\)-beta and the \(q\)-analogue of the Psi functions are established via some \(q\)-integral inequalities. For instance, it is proved that the mappings \(\Gamma_q^{(2n)}\), \(n=0,1,2,\dots,\) are logarithmically convex on \((0,\infty),\) where \(\Gamma_q\) is the \(q\)-gamma function, and \(\Psi_q=\Gamma'_q/\Gamma_q\) is monotonicly non-decreasing and concave on \((0,\infty)\) \((q\in (0,1)).\)