A NEW IMPROVEMENT OF ZHANG-XU-SITU INEQUALITY ABOUT THE WALLIS RATIO ESTIMATES

Many mathematicians presented inequalities about Wallis ratio and related functions using a various of methods such as mean inequality, Jensen inequality, monotonicity of some sequences and monotonicity or complete monotonicity of some functions. The main aim of this paper is to demonstrate that the natural approach for solving these inequalities is to consider and to exploit the inequalities obtained by truncation of some asymptotic series that are not so easy to use. Such inequalities give estimates of any accuracy n-k , as n approaches infinity. Ultimately, an improvement of an inequality due to X.-M. Zhang, T.-Q. Xu and L.-B. Situ [Geometric convexity of a function involving gamma function and application to inequality theory, J. Inequal. Pure Appl. Math. 8 (1) (2007) Art. 17, 9 pp.] is presented.