Structural Properties, Complete Monotonicity, Bounds, and Analytic Approximations with Bounded Remainders for a Wallis-Type Gamma Ratio

Abstract In this paper, we investigate several completely monotonic functions involving the Psi function, which is defined as the logarithmic derivative of the Gamma function. By exploiting the analytic and monotonic properties of these functions, we develop three new approximation formulas for a Wallis-type ratio with bounded remainder terms. These approximations lead to a number of new inequalities that are better than some existing bounds in the literature. Finally, an open problem is proposed concerning the monotonic behavior of the remainder functions arising from the new approximations.