scholarly journals On the complete monotonicity of quotient of gamma functions

2012 ◽  
pp. 395-402 ◽  
Author(s):  
Yi-Chao Chen ◽  
Toufik Mansour ◽  
Qian Zou
Keyword(s):  
Complete Monotonicity ◽  
Gamma Functions

scholarly journals COMPLETE MONOTONICITY OF A FUNCTION INVOLVING THE DIVIDED DIFFERENCE OF PSI FUNCTIONS

2013 ◽  
Vol 88 (2) ◽  
pp. 309-319 ◽  
Author(s):  
FENG QI ◽  
PIETRO CERONE ◽  
SEVER S. DRAGOMIR
Keyword(s):  
Divided Difference ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complete Monotonicity ◽  
Divided Differences ◽  
Psi Function ◽  
Gamma Functions ◽  
Completely Monotonic ◽  
Polygamma Functions ◽  
Necessary And Sufficient

AbstractNecessary and sufficient conditions are presented for a function involving the divided difference of the psi function to be completely monotonic and for a function involving the ratio of two gamma functions to be logarithmically completely monotonic. From these, some double inequalities are derived for bounding polygamma functions, divided differences of polygamma functions, and the ratio of two gamma functions.

scholarly journals A Solution to Qi’s Conjecture on a Double Inequality for a Function Involving the Tri- and Tetra-Gamma Functions

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1098 ◽  
Author(s):  
Ladislav Matejíčka
Keyword(s):  
Open Problem ◽  
Complete Monotonicity ◽  
Double Inequality ◽  
Gamma Functions

In the paper, the author gives a solution to a conjecture on a double inequality for a function involving the tri- and tetra-gamma functions, which was first posed in Remark 6 of the paper “Complete monotonicity of a function involving the tri- and tetragamma functions” (2015) and repeated in the seventh open problem of the paper “On complete monotonicity for several classes of functions related to ratios of gamma functions” (2019).

An integral representation of the Catalan numbers

2015 ◽  
Vol 3 (3) ◽  
pp. 130 ◽  
Author(s):  
Xiao-Ting Shi ◽  
Fang-Fang Liu ◽  
Feng Qi
Keyword(s):  
Integral Representation ◽  
Open Problem ◽  
Catalan Numbers ◽  
Complete Monotonicity ◽  
Gamma Functions

<p><span>In the paper, the authors establish an integral representation of the Catalan numbers, connect the Catalan numbers with the (logarithmically) complete monotonicity, and pose an open problem on the logarithmically complete monotonicity of a function involving ratio of gamma functions.</span></p>

Completely monotonicity of class functions involving the polygamma and related functions

2020 ◽  
pp. 2150064
Author(s):  
B. Ravi ◽  
A. Venakata Lakshmi
Keyword(s):  
Lower Bound ◽  
Zeta Function ◽  
Complete Monotonicity ◽  
Open Problems ◽  
Gamma Functions ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Polygamma Functions ◽  
Completely Monotonic Functions ◽  
Monotonic Properties

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.

scholarly journals Complete monotonicity of divided differences of the di- and tri-gamma functions with applications

2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complete Monotonicity ◽  
Divided Differences ◽  
Gamma Functions ◽  
Completely Monotonic ◽  
Polygamma Functions ◽  
Necessary And Sufficient

AbstractIn the paper, we establish necessary and sufficient conditions for two families of functions involving divided differences of the di- and tri-gamma functions to be completely monotonic. Consequently, we derive necessary and sufficient conditions for two families of functions involving the ratio of two gamma functions to be logarithmically completely monotonic. Furthermore, we deduce some inequalities for bounding the ratio of two gamma functions and divided differences of polygamma functions.

scholarly journals Logarithmically Complete Monotonicity of a Function Involving the Gamma Functions

2020 ◽  
Vol 8 (1) ◽  
pp. 17
Author(s):  
Mohammad Soueycatt ◽  
Abedalbaset Yonsoo ◽  
Ahmad Bekdash ◽  
Nabil Khuder Salman
Keyword(s):  
Complete Monotonicity ◽  
Gamma Functions
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