Results 21 to 30 of about 16,133 (297)

Sharp inequalities related to the volume of the unit ball in R n $\mathbb{R}^{n}$

Journal of Inequalities and Applications, 2023
Let Ω n = π n / 2 / Γ ( n 2 + 1 ) $\Omega _{n}=\pi ^{n/2}/\Gamma (\frac{n}{2}+1)$ ( n ∈ N $n \in \mathbb{N}$ ) denote the volume of the unit ball in R n $\mathbb{R}^{n}$ .
Xue-Feng Han, Chao-Ping Chen
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A Sufficient and Necessary Condition for the Power-Exponential Function ${\left(1+\frac{1}{x}\right)}^{\alpha x}$ to Be a Bernstein Function and Related nth Derivatives

Fractal and Fractional, 2023
In the paper, the authors find a sufficient and necessary condition for the power-exponential function 1+1xαx to be a Bernstein function, derive closed-form formulas for the nth derivatives of the power-exponential functions 1+1xαx and (1+x)α/x, and ...
Jian Cao, Bai-Ni Guo, Wei-Shih Du, Feng Qi   +3 more
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Qi’s conjectures on completely monotonic degrees of remainders of asymptotic formulas of di- and trigamma functions

Journal of Inequalities and Applications, 2020
Several conjectures posed by Qi on completely monotonic degrees of remainders for the asymptotic formulas of the digamma and trigamma functions are proved.
Ai-Min Xu, Zhong-Di Cen
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Monotonicity and complete monotonicity of some functions involving the modified Bessel functions of the second kind

Comptes Rendus. Mathématique, 2023
In this paper, we introduce some monotonicity rules for the ratio of integrals. Furthermore, we demonstrate that the function $-T_{\nu ,\alpha ,\beta }(s)$ is completely monotonic in $s$ and absolutely monotonic in $\nu $ if and only if $\beta \ge 1 ...
Mao, Zhong-Xuan, Tian, Jing-Feng
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A complete monotonicity property of the multiple gamma function

Comptes Rendus. Mathématique, 2020
We consider the following functions \[ f_n(x)=1-\ln x+\frac{\ln G_n(x+1)}{x} \text{ and }g_n(x)=\frac{\@root x \of {G_n(x+1)}}{x},\; x\in (0,\infty ),\; n\in \mathbb{N}, \] where $G_n(z)=\left(\Gamma _n(z)\right)^{(-1)^{n-1}}$ and $\Gamma _n$ is the ...
Das, Sourav
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On Some Complete Monotonicity of Functions Related to Generalized k−Gamma Function

Journal of Mathematics, 2021
In this paper, we presented two completely monotonic functions involving the generalized k−gamma function Γkx and its logarithmic derivative ψkx, and established some upper and lower bounds for Γkx in terms of ψkx.
Hesham Moustafa, Hanan Almuashi, Mansour Mahmoud   +2 more
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On rational bounds for the gamma function

Journal of Inequalities and Applications, 2017
In the article, we prove that the double inequality x 2 + p 0 x + p 0 < Γ ( x + 1 ) < x 2 + 9 / 5 x + 9 / 5 $$ \frac{x^{2}+p_{0}}{x+p_{0}}< \Gamma(x+1)< \frac{x^{2}+9/5}{x+9/5} $$ holds for all x ∈ ( 0 , 1 ) $x\in(0, 1)$ , we present the best possible ...
Zhen-Hang Yang, Wei-Mao Qian, Yu-Ming Chu, Wen Zhang   +3 more
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Comparison of visual quantities in untrained neural networks

Cell Reports, 2023
Summary: The ability to compare quantities of visual objects with two distinct measures, proportion and difference, is observed even in newborn animals.
Hyeonsu Lee, Woochul Choi, Dongil Lee, Se-Bum Paik   +3 more
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Completely Monotone Functions: A Digest [PDF]

, 2014
To appear in: K. Alladi, G.V. Milovanovic and M. Th. Rassias (Eds.): "Analytic Number Theory, Approximation Theory and Special Functions", Special Volume dedicated to Professor Hari M.
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Completely Monotonic and Related Functions

Journal of Mathematical Analysis and Applications, 1996
Let \(L^N\) denote the class of functions defined by \[ f\in L^N\Leftrightarrow (-1)^k f^{(k)}(t)\geq 0,\quad \forall t>0,\quad \forall k,\quad 0\leq k\leq N. \] For \(N\to \infty\) we write \(f\in L\); such functions are well known as completely monotonic on \((0,\infty)\). The implication \[ f\in L^N\Rightarrow [\forall \alpha> 1: f^\alpha\in L^N] \]
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