Some properties of the k-Gamma function

We give completely monotonicity properties and inequalities for functions involving the Γ_k functions and their logarithmic derivatives ψ_k functions. We introduce a k-analogue of the Riemann Zeta function ζ_k as an integral and using Schwarz’s and Holder’s inequalities we obtain some inequalities relating ζ_k and Γ_k functions. The obtained results are the k-anologues of known results concerning functions involving the Gamma and psi functions

A class of completely monotonic functions involving divided differences of the psi and polygamma functions and some applications

A class of functions involving the divided differences of the psi function and the polygamma functions and originating from Kershaw's double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving ratio of two gamma functions and originating from establishment of the best upper and lower bounds in Kershaw's double inequality are derived, two sharp double inequalities involving ratios of double factorials are recovered, the probability integral or error function is estimated, a double inequality for ratio of the volumes of the unit balls in $\mathbb{R}^{n-1}$ and $\mathbb{R}^n$ respectively is deduced, and a symmetrical upper and lower bounds for the gamma function in terms of the psi function is generalized.Comment: 11 page

The monotonicity and convexity of a function involving digamma one and their applications

Let $\mathcal{L}(x,a)$ be defined on $\left( -1,\infty \right) \times \left( 4/15,\infty \right)$ or $\left( 0,\infty \right) \times \left( 1/15,\infty \right)$ by the formula% \begin{equation*} \mathcal{L}(x,a)=\tfrac{1}{90a^{2}+2}\ln \left( x^{2}+x+\tfrac{3a+1}{3}% \right) +\tfrac{45a^{2}}{90a^{2}+2}\ln \left( x^{2}+x+\allowbreak \tfrac{% 15a-1}{45a}\right) . \end{equation*} We investigate the monotonicity and convexity of the function $x\rightarrow F_{a}\left( x\right) =\psi \left( x+1\right) -\mathcal{L}(x,a)$, where $\psi$ denotes the Psi function. And, we determine the best parameter $a$ such that the inequality \psi \left( x+1\right) \right) \mathcal{L}% (x,a) holds for $x\in \left( -1,\infty \right)$ or $\left( 0,\infty \right)$, and then, some new and very high accurate sharp bounds for pis function and harmonic numbers are presented. As applications, we construct a sequence $\left( l_{n}\left( a\right) \right)$ defined by $l_{n}\left( a\right) =H_{n}-\mathcal{L}\left( n,a\right)$, which gives extremely accurate values for $\gamma$.Comment: 20 page