Related papers: A completely monotonic function involving the tri-…
In this paper the incomplete gamma function $\gamma(\alpha,x)$ and its derivative is considered for negative values of $\alpha $ and the incomplete gamma type function $\gamma_*(\alpha,x_-)$ is introduced. Further the polygamma functions…
In this article, logarithmically complete monotonicity properties of some functions such as $\frac1{[\Gamma(x+1)]^{1/x}}$, $\frac{[{\Gamma(x+\alpha+1)}]^{1/(x+\alpha)}}{[{\Gamma(x+1)}]^{1/x}}$, $\frac{[\Gamma(x+1)]^{1/x}}{(x+1)^\alpha}$ and…
In this paper, the authors establish some inequalities involving the Psi and $k$-Gamma functions. The procedure utilizes some monotonicity properties of some functions associated with the Psi and $k$-Gamma functions.
Motivated by several conjectures posed in the paper " Completely monotonic degrees for a difference between the logarithmic and psi functions",we confirm in this work some conjectures on completely monotonic degrees of remainders of the…
In the paper, the authors concisely survey and review some functions involving the gamma function and its various ratios, simply state their logarithmically complete monotonicity and related results, and find necessary and sufficient…
In the article we present necessary and sufficient conditions for a function involving the logarithm of the gamma function to be completely monotonic and apply these results to bound the gamma function $\Gamma(x)$, the $n$-th harmonic…
In this paper, we investigate the complete monotonicity of some functions involving gamma function. Using the monotonic properties of these functions, we derived some inequalities involving gamma and beta functions. Such inequalities…
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…
Motivated by existing results, we present some completely monotonic functions involving the polygamma functions.
In this paper we define the polygamma functions $\psi^{(n)}(x)$ for negative integers by using neutrix calculus.
Motivated by open questions in the papers " Refinements and sharpenings of some double inequalities for bounding the gamma function" and "Complete monotonicity and monotonicity of two functions defined by two derivatives of a function…
In the paper, necessary and sufficient conditions are presented for a function involving a ratio of gamma functions to be logarithmically completely monotonic. This extends and generalizes the main result in [\emph{Inequalities and…
In the paper, by convolution theorem of the Laplace transforms, a monotonicity rule for the ratio of two Laplace transforms, Bernstein's theorem for completely monotonic functions, and other analytic techniques, the authors verify…
In 1974, Gautschi proved an intriguing inequality involving the gamma function $\Gamma$. Precisely, he proved that, for $z>0$, the harmonic mean of $\Gamma(z)$ and $\Gamma(1/z)$ can never be less than 1. In 2017, Alzer and Jameson extended…
We present some completely monotonic functions involving the $q$-gamma function that are inspired by their analogues involving the gamma function.
We prove some properties of completely monotonic functions and apply them to obtain results on gamma and $q$-gamma functions.
In the note, two alternative proofs are provided for a monotonicity result that the function $\psi(x)+\ln\bigl(e^{1/x}-1\bigr)$ is strictly increasing on $(0,\infty)$, where $\psi(x)$ is the psi function.
We present some completely monotonic functions involving the$q$-polygamma functions, our result generalizes some known results.
In this expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some inequalities, the complete monotonicity of several functions involving ratios of two gamma or $q$-gamma…
In the paper, the authors verify the complete monotonicity of the difference $e^{1/t}-\psi'(t)$ on $(0,\infty)$, compute the completely monotonic degree and establish integral representations of the remainder of the Laurent series expansion…