On some properties of the Gamma function

Anderson and Qiu (1997) conjectured that the function $\frac{\log \Gamma (x+1)}{{x \log x}}$is concave for $x>1$. In this paper we prove this conjecture. We also study the monotonicity of some functions connected with the psi-function $\psi (x)$ and derive inequalities for $\psi (x)$ and $\psi '(x)$.