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>

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 Sharp Inequalities for Polygamma Functions

2015 ◽  
Vol 65 (1) ◽  
Author(s):  
Bai-Ni Guo ◽  
Feng Qi ◽  
Jiao-Lian Zhao ◽  
Qiu-Ming Luo
Keyword(s):  
Rational Functions ◽  
Complete Monotonicity ◽  
Double Inequality ◽  
Polygamma Function ◽  
Polygamma Functions

AbstractIn the paper, the authors review some inequalities and the (logarithmically) complete monotonicity concerning the gamma and polygamma functions and, more importantly, present a sharp double inequality for bounding the polygamma function by rational functions.

scholarly journals Sharp bounds for a ratio of the q-gamma function in terms of the q-digamma function

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Faris Alzahrani ◽  
Ahmed Salem ◽  
Moustafa El-Shahed
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Gamma Function ◽  
Open Problem ◽  
Upper And Lower Bounds ◽  
Digamma Function ◽  
Sharp Bounds ◽  
Gamma Functions ◽  
Numer Math

AbstractIn the present paper, we introduce sharp upper and lower bounds to the ratio of two q-gamma functions ${\Gamma }_{q}(x+1)/{\Gamma }_{q}(x+s)$ Γ q ( x + 1 ) / Γ q ( x + s ) for all real number s and $0< q\neq1$ 0 < q ≠ 1 in terms of the q-digamma function. Our results refine the results of Ismail and Muldoon (Internat. Ser. Numer. Math., vol. 119, pp. 309–323, 1994) and give the answer to the open problem posed by Alzer (Math. Nachr. 222(1):5–14, 2001). Also, for the classical gamma function, our results give a Kershaw inequality for all $0< s<1$ 0 < s < 1 when letting $q\to 1$ q → 1 and a new inequality for all $s>1$ s > 1 .

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 On Alzer and Qiu's Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function

2011 ◽  
Vol 2011 ◽  
pp. 1-7 ◽  
Author(s):  
Yu-Ming Chu ◽  
Miao-Kun Wang ◽  
Ye-Fang Qiu
Keyword(s):  
Open Problem ◽  
Elliptic Integral ◽  
Elliptic Integrals ◽  
Complete Elliptic Integral ◽  
Hyperbolic Tangent ◽  
Hyperbolic Tangent Function ◽  
Double Inequality ◽  
Tangent Function ◽  
Complete Elliptic Integrals

We prove that the double inequality(π/2)(arthr/r)3/4+α*r<K(r)<(π/2)(arthr/r)3/4+β*rholds for allr∈(0,1)with the best possible constantsα*=0andβ*=1/4, which answer to an open problem proposed by Alzer and Qiu. Here,K(r)is the complete elliptic integrals of the first kind, and arth is the inverse hyperbolic tangent function.

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