Some Extensions of W. Gautschi's Inequalities for the Gamma Function

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: $$\exp[(1 - s)\psi(x + s^{1/2})] < \frac{\Gamma(x + 1)}{\Gamma(x + s)} < \exp\bigg[(1 - s)\psi\bigg(x + \frac{s + 1}{2}\bigg)\bigg]$$ and $$\bigg[x + \frac{s}{2}\bigg]^{1 - s} < \frac{\Gamma(x + 1)}{\Gamma(x + s)} < \bigg[x - \frac{1}{2} + \bigg(s + \frac{1}{4}\bigg)^{1/2}\bigg]^{1 - s}.$$ These are compared with each other and with inequalities given by T. Erber and J. D. Kečkić and P. M. Vasić.