Further inequalities for the gamma function

For $\lambda > 0$ and $k \geqslant 0$ we present a method which permits us to obtain inequalities of the type ${(k + \alpha )^{\lambda - 1}} < \Gamma (k + \lambda )/\Gamma (k + 1) < {(k + \beta )^{\lambda - 1}}$, with the usual notation for the gamma function, where $\alpha$ and $\beta$ are independent of k. Some examples are also given which improve well-known inequalities. Finally, we are also able to show in some cases that the values $\alpha$ and $\beta$ in the inequalities that we obtain cannot be improved.