Monotonicity and Convexity Properties and Some Inequalities Involving a Generalized Form of the Wallis' Cosine Formula

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.