Some inequalities involving the polygamma functions

Let n(x)=(-1)n-1(n)(x), where (n)(x) are the polygamma functions. We determine necessary and sufficient conditions for the monotonicity and convexity of the function F(x;,)=ln(exp((x+))n(x))-ln(n-1)!max(0,-) for and R, where (x) is the psi function. Consequently, this yields not only some new inequalities for the polygamma functions, but also new star-shaped and superadditive functions involving them. In addition, we improve a well-known mean-value inequality for the polygamma functions.