Complete Monotonicity of Functions Defined by λ Generalized Psi Function and Its Derivatives
Emrah Yildirim *
Department of Mathematics, Aydin Adnan Menderes University, Aydin, Turkey.
Ahmet Ates
Graduate School of Natural and Applied Sciences, Aydin Adnan Menderes University, Aydin, Turkey.
*Author to whom correspondence should be addressed.
Abstract
In this work, we firstly give integral representations of λ-psi (or λ-digamma) and λ-zeta functions and then obtain λ-generalization of Binet’s first formula for the logarithms of λ-gamma function ln Γλ(x) as
$$\ln \Gamma_\lambda(x)=\left(x-\frac{1}{2}\right) \ln x-x \ln \lambda+\frac{1}{2} \ln (2\pi)+\int_0^{\infty}\left[\frac{1}{2}-\frac{1}{t}+\frac{1}{e^t-1}\right] \frac{e^{-t x}}{t} d t$$
for all positive real values of x and λ. As immediate consequences, we get some completely monotonicity properties on functions related to λ-psi function and its derivatives defined by $$f1(x) = ψλ(x) + ln λ − ln x +
\frac{1}{2 x}+\frac{1}{12 x^2}, f_2(x) $$$$ =\ln x-\frac{1}{2 x}-\ln \lambda-\psi_\lambda(x), f_3(x)$$$$=\psi_\lambda^{\prime}(x)-\frac{1}{x}-\frac{1}{2 x^2}-\frac{1}{6 x^3}+\frac{1}{30 x^5}, f_4(x)$$$$=\frac{1}{x}+\frac{1}{2 x^2}+\frac{1}{6 x^3}-\psi_\lambda^{\prime}(x)$$ for all x, λ > 0. At last, we obtain some mean inequalities on λ-psi function.
Keywords: λ-polygamma function, λ-psi function, inequality, Binet’s first formula for ln Γλ(x)