Completely monotonic functions involving Bateman’s G−function

Abstract

n this paper, we prove the complete monotonicity of some functions involving Bateman’s G−function and show that where α = 1 and β = 0 are the best possible constants, which is a refinement of a recent result. Then, we give a new proof of Slavi´c inequality about Wallis ratio Wm and provide a new inequality for Wm. Our new inequality improves some recent related works. We also present two inequalities for the hyperbolic tangent function.