Monotonicity and Convexity Properties and Some Inequalities Involving a Generalized Form of the Wallis' Cosine Formula
Kwara Nantomah *
Department of Mathematics, Faculty of Mathematical Sciences, University for Development Studies, Navrongo Campus, P. O. Box 24, Navrongo, UE/R, Ghana.
*Author to whom correspondence should be addressed.
Abstract
This study is focused on monotonicity and convexity properties of a generalized form of the Wallis’ cosine formula. Specifically, by using the integral form of the Nielsen’s β-function, we prove that the generalized Wallis’ cosine formula is logarithmically completely monotonic, logarithmically convex and decreasing. Furthermore, by using the classical Wendel’s, Hölder’s and Young’s inequalities, among other analytical techniques, we establish some new inequalities involving the generalized function.
Keywords: Nielsen's β-function, Wendel's inequality, Hölder's inequality, Young's inequality, logarithmically completely monotonic function, Wallis' cosine formula.