Integral representations and properties of some functions involving the logarithmic function

摘要

By using Cauchy integral formula in the theory of complex functions, theauthors establish some integral representations for the principal branches ofseveral complex functions involving the logarithmic function, find someproperties, such as being operator monotone function, being complete Bernsteinfunction, and being Stieltjes function, for these functions, and verify aconjecture on complete monotonicity of a function involving the logarithmicfunction.