INEQUALITIES INVOLVING GAMMA AND PSI FUNCTIONS

Abstract

We prove that certain functions involving the gamma and q-gamma function are monotone. We also prove that (xmψ(x))(m+1) is completely monotonic. We conjecture that -(xmψ(m)(x))(m) is completely monotonic for m ≥ 2; and we prove it, with help from Maple, for 2 ≤ m ≤ 16. We give a very useful Maple procedure to verify this for higher values of m. A stronger result is also formulated where we conjecture that the power series coefficients of a certain function are all positive.