A class of logarithmically completely monotonic functions and the best bounds in the first Kershawʹs double inequality

In the article, the logarithmically complete monotonicity of a class of functions involving Eulerʹs gamma function are proved, a class of the first Kershaw-type double inequalities are established, and the first Kershawʹs double inequality and Wendelʹs inequality are generalized, refined or extended. Moreover, an open problem is posed.