\(q\)-extensions of some estimates associated with the digamma function (Q390622)

There are some useful inequalities for the digamma function \(\psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}\) due to Batir and Elezovic et al. which say that NEWLINE\[NEWLINE\log(x+1/2)<\psi(x+1)<\log(x+e^{-\gamma}),NEWLINE\]NEWLINE NEWLINE\[NEWLINE-\gamma-\log(e^{1/x}-1)<\psi(x)<-\log(e^{1/x}-1),NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\psi'(x)e^{\psi(x)}<1NEWLINE\]NEWLINE for all \(x>0\).NEWLINENEWLINEThe author gives the counterparts of these inequalities for the \(q\)-digamma function. Elegant estimates for the (unique) zero of the \(q\)-digamma function are also presented, thanks to the above mentioned estimates.