Integral representations and properties of some functions involving the logarithmic function

Abstract

By using Cauchy integral formula in the theory of complex functions, the authors establish some integral representations for the principal branches of several complex functions involving the logarithmic function, find some properties, such as being operator monotone function, being complete Bernstein function, and being Stieltjes function, for these functions, and verify a conjecture on complete monotonicity of a function involving the logarithmic function.

15 pages

References in corpus (4)

• Limit formulas for ratios of derivatives of the gamma and digamma functions at their singularities
• Limit formulas for ratios of polygamma functions at their singularities
• Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions
• Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions

Cited by in corpus (3)

• Completely monotonic degree of a function involving the tri- and tetra-gamma functions
• A ratio of many gamma functions and its properties with applications
• A double inequality for completely monotonic degree of a remainder for an asymptotic expansion of the trigamma function