Some extensions of W. Gautschi’s inequalities for the gamma function

Abstract:

It has been shown by W. Gautschi that if $0 < s < 1$, then for $x \geqslant 1$ \[ {x^{1 - s}} < \frac {{\Gamma (x + 1)}}{{\Gamma (x + s)}} < \exp [(1 - s)\psi (x + 1)].\] The following closer bounds are proved: \[ \exp [(1 - s)\psi (x + {s^{1/2}})] < \frac {{\Gamma (x + 1)}}{{\Gamma (x + s)}} < \exp \left [ {(1 - s)\psi \left ( {x + \frac {{s + 1}}{2}} \right )} \right ]\] and \[ {\left [ {x + \frac {s}{2}} \right ]^{1 - s}} < \frac {{\Gamma (x + 1)}}{{\Gamma (x + s)}} < {\left [ {x - \frac {1}{2} + {{\left ( {s + \frac {1}{4}} \right )}^{1/2}}} \right ]^{1 - s}}.\] These are compared with each other and with inequalities given by T. Erber and J. D. Kečkić and P. M. Vasić.