Inequalities for the Gamma function via convexity

Abstract. We review techniques based on convexity, logarithmic convexity and Schur-convexity, for producing inequalities and asymptotic expansions for ratios of Gamma functions. As an illustration, results for the Gautschi’s and Gurland’s ratio are pre-sented, as well as asymptotic expansions for the Gamma function, along the lines of W. Krull’s work. We argue that convexity-based techniques are advantageous over other methods, because they enable a comparision of inequalities, provide two transforma-tions for their sharpening, and also yield two sided asymptotic expansions. The Gamma function... is simple enough for juniors in college to meet, but deep enough to have called forth contributions from the finest mathematicians. Philip Davis [7]