Sharp bounds for the ratio of \(q\)-gamma functions (Q2711597)

The \(q\)-gamma function is defined for \(0<q<1\) by NEWLINE\[NEWLINE\Gamma_q(x) =(1-q)^{1-x} \prod_{n=1}^\infty \frac{1-q^{n+1}}{1-q^{n+x}}NEWLINE\]NEWLINE and for \(q>1\) by NEWLINE\[NEWLINE\Gamma_q(x) = (q-1)^{1-x} q^{x(x-1)/2}\prod_{n=0}^\infty\frac{1-q^{-(n+1)}}{1-q^{-(n+x)}}NEWLINE\]NEWLINE \((x>0).\) The author offers the chain of inequalities NEWLINE\[NEWLINE 1-q^{x+a} < (1-q)[\Gamma_q(x+1)/ \Gamma_q(x+s)]^{1/(1-s)}<1-q^{x+b} \;(1\neq q>0,\;0<s<1;\;x>0), NEWLINE\]NEWLINE where \(b=\frac{\log[1-(1-q)\Gamma_q (s)^{1/(s-1)}]}{\log q}\) and NEWLINE\[NEWLINE a=s/2 (\text{ for } q>1),\quad a=\frac{\log[q(q^{s-1} -1)(1-s)^{-1}(1-q)^{-1}]}{\log q} (\text{ for } 0<q<1) NEWLINE\]NEWLINE are best possible on the right and left side, respectively. The corresponding inequalities for the classical \(\Gamma\) functions are consequences.