Monotonic and Logarithmically Convex Properties of a Function Involving Gamma Functions.

Using the series-expansion of digamma functions and other techniques, some monotonicity and logarithmical concavity involving the ratio of gamma function are obtained, which is to give a partially affirmative answer to an open problem posed by B.-N. Guo and F. Qi. Several inequalities for the geometric means of natural numbers are established. [ABSTRACT FROM AUTHOR]