Inequalities involving the gamma and digamma functions

Abstract: We improve the upper bounds of the following inequalities proved in [H. Alzer and N. Batir, Monotonicity properties of the gamma function, Appl. Math. Letters, 20(2007), 778-781]. \begin{equation*} exp\left(-\frac{1}{2}\psi\left(x+\frac{1}{3}\right)\right)<\frac{\Gamma(x)}{\sqrt{2\pi}x^xe^{-x}}<exp\left(-\frac{1}{2}\psi(x)\right), \end{equation*} and $$ \frac{1}{2}\psi'(x+1/3))<\log x-\psi(x)<\frac{1}{2}\psi'(x).$$ Here $\Gamma$ is the classical gamma function and $\psi$ is the digamma function.