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Some New Inequalities for the Gamma and Polygamma Functions

Authors

Waad Al Sayed, H.M. Moustafa
Type: Article
Publication Date: 2025-04-14
Citations: 0
DOI: https://doi.org/10.3390/sym17040595

Abstract

In this paper, we present some new symmetric bounds for the gamma and polygamma functions. For this goal, we present two functions involving gamma and polygamma functions and we investigate their complete monotonicity. Also, we investigate their completely monotonic degrees. This concept gives more accuracy in measuring the complete monotonicity property. These new bounds are better than some of the recently published results.

Topics

Locations

SOME COMPLETELY MONOTONIC FUNCTIONS INVOLVING THE GAMMA AND POLYGAMMA FUNCTIONS

2008-01-31
Ai‐Jun Li , Chao-Ping Chen
In this paper, some logarithmically completely monotonic, strongly completely monotonic and completely monotonic functions related to the gamma, digamma and polygamma functions are established. Several inequalities, whose bounds are best … In this paper, some logarithmically completely monotonic, strongly completely monotonic and completely monotonic functions related to the gamma, digamma and polygamma functions are established. Several inequalities, whose bounds are best possible, are obtained.
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Some Inequalities for the $q$-Gamma and the $q$-Polygamma Functions

2018-12-09
Ahmed Salem
In this paper, the complete monotonicity property for functions related to the $q$-gamma and the $q$-polygamma functions, where $q$ is a positive real number, is proved and exploited to establish … In this paper, the complete monotonicity property for functions related to the $q$-gamma and the $q$-polygamma functions, where $q$ is a positive real number, is proved and exploited to establish some inequalities for the $q$-gamma and the $q$-polygamma functions.

Complete monotonicity results for some functions involving the gamma and polygamma functions

2011-01-07
Hamdullah Şevli̇ , Necdet Batır

On some properties of the gamma function

2007-10-05
Necdet Batır

Some New Inequalities for Gamma and Polygamma Functions

2004-01-01
Necdet Batır
In this paper we derive some new inequalities involving the gamma function Γ, polygamma functions ψ = Γ'/Γ and ψ'. We also obtained two new sequences converging to Euler-Mascheroni constant … In this paper we derive some new inequalities involving the gamma function Γ, polygamma functions ψ = Γ'/Γ and ψ'. We also obtained two new sequences converging to Euler-Mascheroni constant γ very quickly. The method we use is new and can be applied to other similar problems.

Complete monotonicity and inequalities related to generalized k-gamma and k-polygamma functions

2020-01-30
Jumei Zhang , Li Yin , Honglian You
Abstract In this paper, we prove new complete monotonicity properties of some functions related to generalized k -gamma and k -polygamma functions. Applications of the results yield various new inequalities. … Abstract In this paper, we prove new complete monotonicity properties of some functions related to generalized k -gamma and k -polygamma functions. Applications of the results yield various new inequalities. In the end, double inequalities are constructed involving the k -generalized digamma and polygamma functions.
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Complete monotonicity related to the k-polygamma functions with applications

2019-08-28
Li Yin , Jumei Zhang , Xiuli Lin
In this paper, we prove complete monotonicity of some functions involving k-polygamma functions. As an application of the main result, we also give new upper and lower bounds of the … In this paper, we prove complete monotonicity of some functions involving k-polygamma functions. As an application of the main result, we also give new upper and lower bounds of the k-digamma function.
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Complete Monotonicity Properties of a Function Involving the Polygamma Function

2018-10-30
Kwara Nantomah
In this paper, we study completete monotonicity properties of certain functions associated with the polygamma functions.Subsequently, we deduce some inequalities involving difference of polygamma functions. In this paper, we study completete monotonicity properties of certain functions associated with the polygamma functions.Subsequently, we deduce some inequalities involving difference of polygamma functions.
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Some monotonicity properties and inequalities for the generalized digamma and polygamma functions

2018-09-20
Li Yin , Li-Guo Huang , Zhi-Min Song +1 more
Several monotonicity and concavity results related to the generalized digamma and polygamma functions are presented. This extends and generalizes the main results of Qi and Guo and others. Several monotonicity and concavity results related to the generalized digamma and polygamma functions are presented. This extends and generalizes the main results of Qi and Guo and others.
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Completely monotonicity of class functions involving the polygamma and related functions

2020-07-17
B. Ravi , A. Venakata Lakshmi
In this paper, the authors prove some inequalities and completely monotonic properties of polygamma functions. As an application, we give lower bound for the zeta function on natural numbers. Partially, … In this paper, the authors prove some inequalities and completely monotonic properties of polygamma functions. As an application, we give lower bound for the zeta function on natural numbers. Partially, we answer the fifth and sixth open problems listed in [F. Qi and R. P. Agarwal, On complete monotonicity for several classes of functions related to ratios of gamma functions, J. Inequalities Appl. 2019(36) (2019) 42]. We propose two open problems on completely monotonic functions related to polygamma functions.

Some classes of completely monotonic functions related to q-gamma and q-digamma functions

2016-01-01
Ahmed Salem
In the paper, some classes of completely monotonic functions involving the q -gamma and q -digamma functions are derived.The monotonicity properties of these functions are exploited to establish a double … In the paper, some classes of completely monotonic functions involving the q -gamma and q -digamma functions are derived.The monotonicity properties of these functions are exploited to establish a double inequality for the ratio of the q -gamma function and a double inequality for the q -digamma function.Moreover, a class of inequalities for the q -polygamma functions is presented.
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Monotonicity properties and inequalities of gamma and psi functions

2009-01-01
Chen Chan-ping
A class of logarithmically completely monotonic function involving the gamma function is presented,and some inequalities involving the psi and polygamma functions are established.Theorems are generalized and some known results are … A class of logarithmically completely monotonic function involving the gamma function is presented,and some inequalities involving the psi and polygamma functions are established.Theorems are generalized and some known results are improved.Finaly,a new proof of the increasingness for the function x2[ψ(x+1/2)-ψ(x)-1/2x]is given.

Improvements of bounds for the q-gamma and the $q$-polygamma functions

2017-01-01
Ahmed Hamed Salem , Faris Alzahrani
In this paper, the complete monotonicity property of functions involving the q -gamma function is proven and used to establish sharp inequalities for the q -gamma and the q -polygamma … In this paper, the complete monotonicity property of functions involving the q -gamma function is proven and used to establish sharp inequalities for the q -gamma and the q -polygamma functions for all q > 0 .These bounds for the q -gamma and the q -polygamma functions refine those given by Salem [17].
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Improvements of bounds for the q-gamma and the $q$-polygamma functions

2017-01-01
Ahmed Salem , Faris Alzahrani
In this paper, the complete monotonicity property of functions involving the q -gamma function is proven and used to establish sharp inequalities for the q -gamma and the q -polygamma … In this paper, the complete monotonicity property of functions involving the q -gamma function is proven and used to establish sharp inequalities for the q -gamma and the q -polygamma functions for all q > 0 .These bounds for the q -gamma and the q -polygamma functions refine those given by Salem [17].
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Asymptotic expansions of the logarithm of the gamma function in the terms of the polygamma functions

2014-01-01
Chao-Ping Chen
We present new asymptotic expansions of the logarithm of the gamma function in terms of the polygamma functions.Based on these expansions, we prove new complete monotonicity properties of some functions … We present new asymptotic expansions of the logarithm of the gamma function in terms of the polygamma functions.Based on these expansions, we prove new complete monotonicity properties of some functions involving the gamma and polygamma functions.As consequences of them we establish new upper and lower bounds for the gamma function in terms of the polygamma functions.
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Complete monotonicity properties of a function involving the polygamma function

2018-01-01
Kwara Nantomah
In this paper, we study completete monotonicity properties of the function $f_{a,k}(x)=\psi^{(k)}(x+a) - \psi^{(k)}(x) - \frac{ak!}{x^{k+1}}$, where $a\in(0,1)$ and $k\in \mathbb{N}_0$. Specifically, we consider the cases for $k\in \{ 2n: … In this paper, we study completete monotonicity properties of the function $f_{a,k}(x)=\psi^{(k)}(x+a) - \psi^{(k)}(x) - \frac{ak!}{x^{k+1}}$, where $a\in(0,1)$ and $k\in \mathbb{N}_0$. Specifically, we consider the cases for $k\in \{ 2n: n\in \mathbb{N}_0 \}$ and $k\in \{ 2n+1: n\in \mathbb{N}_0 \}$. Subsequently, we deduce some inequalities involving the polygamma functions.
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Complete monotonicity properties of a function involving the polygamma function

2018-06-27
Kwara Nantomah
In this paper, we study completete monotonicity properties of the function $f_{a,k}(x)=\psi^{(k)}(x+a) - \psi^{(k)}(x) - \frac{ak!}{x^{k+1}}$, where $a\in(0,1)$ and $k\in \mathbb{N}_0$. Specifically, we consider the cases for $k\in \{ 2n: … In this paper, we study completete monotonicity properties of the function $f_{a,k}(x)=\psi^{(k)}(x+a) - \psi^{(k)}(x) - \frac{ak!}{x^{k+1}}$, where $a\in(0,1)$ and $k\in \mathbb{N}_0$. Specifically, we consider the cases for $k\in \{ 2n: n\in \mathbb{N}_0 \}$ and $k\in \{ 2n+1: n\in \mathbb{N}_0 \}$. Subsequently, we deduce some inequalities involving the polygamma functions.

Complete monotonicity of a family of functions involving the tri- and tetra-gamma functions

2013-01-01
Feng Qi
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 $$ … The psi function $\psi(x)$ is defined by $\psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}$ and $\psi^{(i)}(x)$ for $i\in\mathbb{N}$ denote polygamma functions, where $\Gamma(x)$ is the gamma function. In this paper, we prove that the function $$ [\psi'(x)]^2+\psi"(x)-\frac{x^2+\lambda x+12}{12x^4(x+1)^2} $$ is completely monotonic on $(0,\infty)$ if and only if $\lambda\le0$, and so is its negative if and only if $\lambda\ge4$. From this, some inequalities are refined and sharpened.
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Complete monotonicity properties and asymptotic expansions of the logarithm of the gamma function

2014-11-04
Chao-Ping Chen , Jingyun Liu
We prove two conjectures of Chen concerning the complete monotonicity properties of some functions involving the gamma and polygamma functions.We prove asymptotic expansions of the logarithm of the gamma function … We prove two conjectures of Chen concerning the complete monotonicity properties of some functions involving the gamma and polygamma functions.We prove asymptotic expansions of the logarithm of the gamma function in terms of the polygamma functions, and provide recurrence relations to calculate the coefficients of the asymptotic expansions.By using the results obtained, we derive recursive relations of the Bernoulli numbers.
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Some completely monotonic functions involving the polygamma functions

2010-01-01
Peng Gao
Motivated by existing results, we present some completely monotonic functions involving the polygamma functions. Motivated by existing results, we present some completely monotonic functions involving the polygamma functions.
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