General Inequalities

Log-convexity and log-concavity for series in gamma ratios and applications

“…its logarithm is a convex function). A sum of log-convex functions can be shown to be log-convex using Hölder inequality or a theorem of Montel [21,Theorem 1.4.5.2]. Additivity implies then that the (finite or infinite) sum f (µ; x) := f k Γ(µ + k)x k is logarithmically convex function of µ for fixed x ≥ 0 once the coefficients f k are assumed to be nonnegative.…”