相关论文: Two new proofs of the complete monotonicity of a f…

In the paper the authors alternatively prove that the function $x^\alpha\big[\ln\frac{px}{x+p+1}-\psi_p(x)\big]$ is completely monotonic on $(0,\infty)$ if and only if $\alpha \le 1$, where $p\in\mathbb{N}$ and $\psi_p(x)$ is the…

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.

In the present paper, we establish necessary and sufficient conditions for the functions $x^\alpha\bigl\lvert\psi^{(i)}(x+\beta)\bigr\lvert$ and $\alpha\bigl\lvert\psi^{(i)}(x+\beta)\bigr\lvert-x\bigl\lvert\psi^{(i+1)}(x+\beta)\bigr\lvert$…

In this article, a necessary and sufficient condition and a necessary condition are established for a function involving the gamma function to be logarithmically completely monotonic on $(0,\infty)$. As applications of the necessary and…

In the paper the author provides necessary and sufficient conditions on $a$ for the function $\frac{1}{2}\ln(2\pi)-x+\bigl(x-\frac{1}{2}\bigr)\ln x-\ln\Gamma(x)+\frac1{12}{\psi'(x+a)}$ and its negative to be completely monotonic on…

In this paper, we present the necessary and sufficient conditions such that several functions involving $R\left( x\right) =\psi \left( x+1/2\right) -\ln x$ with a parameter are completely monotone on $\left( 0,\infty \right) $, where $\psi…

Let $\psi_q(x)$, $\psi_q'(x)$, and $\psi_q''(x)$ for $q>0$ stand respectively for the $q$-digamma, $q$-trigamma, and $q$-tetragamma functions. In the paper, the author proves along two different approaches that the functions…

The psi function $\psi(x)$ is defined by $\psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}$ and $\psi^{(i)}(x)$ for $i\in\mathbb{N}$ denote polygamma functions, where $\Gamma(x)$ is the gamma function. In this paper, we prove that the function $$…

The psi function $\psi(x)$ is defined by $\psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}$ and $\psi^{(i)}(x)$ for $i\in\mathbb{N}$ denote the polygamma functions, where $\Gamma(x)$ is the gamma function. In this paper we prove that a function…

We simplify the proof of some widely used theoretical theorems, extending their applicability, while correcting some erroneous results. We also generalize key results and present new results that contribute to the development of the theory.…

For $m,n\in\mathbb{N}$, let $f_{m,n}(x)=\bigr[\psi^{(m)}(x)\bigl]^2+\psi^{(n)}(x)$ on $(0,\infty)$. In the present paper, we prove using two methods that, among all $f_{m,n}(x)$ for $m,n\in\mathbb{N}$, only $f_{1,2}(x)$ is nontrivially…

In the article, a notion "logarithmically absolutely monotonic function" is introduced, an inclusion that a logarithmically absolutely monotonic function is also absolutely monotonic is revealed, the logarithmically complete monotonicity…

Let $\psi(x)$ be the di-gamma function, the logarithmic derivative of the classical Euler's gamma function $\Gamma(x)$. In the paper, the author shows that the completely monotonic degree of the function $[\psi'(x)]^2+\psi''(x)$ is $4$,…

In this manuscript, we present the complete monotonicity of functions defined in terms of the poly-double gamma function \begin{align*} \psi_2^{(n)}(x) = (-1)^{n+1} n! \sum_{k=0}^{\infty} \dfrac{(1+k)}{(x+k)^{n+1}}, \quad x > 0, \ n\geq 2.…

Let $ f:(0,\infty)\rightarrow \Bbb{R} $ be a completely monotonic function. In this paper, we present some properties of this functions and several new classes of completely monotonic functions. We also give some special functions such that…

We obtain necessary and sufficient conditions on a function in order that it be the Laplace transform of an absolutely monotonic function. Several closely related results are also given.

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 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 present paper, necessary and sufficient conditions are established for a function involving divided differences of the digamma and trigamma functions to be completely monotonic. Consequently, necessary and sufficient conditions are…

In this note we study the monotonicity of the function $x\mapsto \psi(1 +bx)^a/\psi(1 + ax)^b$. We also give the several inequalities involving the psi function, whic is the logarithmic derivative of the gamma function.