scholarly journals On a Conjecture of Alzer, Berg, and Koumandos

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1031
Author(s):  
Ladislav Matejíčka
Keyword(s):  
Gamma Function ◽  
Open Problem ◽  
Logarithmic Derivative ◽  
Completely Monotonic

In this paper, we find a solution of an open problem posed by Alzer, Berg, and Koumandos: determine ( α , m ) ∈ R + × N such that the function x α | ψ ( m ) ( x ) | is completely monotonic on ( 0 , ∞ ) , where ψ ( x ) denotes the logarithmic derivative of Euler’s gamma function.

scholarly journals On Some Complete Monotonicity of Functions Related to Generalized k − Gamma Function

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Hesham Moustafa ◽  
Hanan Almuashi ◽  
Mansour Mahmoud
Keyword(s):  
Lower Bounds ◽  
Gamma Function ◽  
Logarithmic Derivative ◽  
Complete Monotonicity ◽  
Upper And Lower Bounds ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions

In this paper, we presented two completely monotonic functions involving the generalized k − gamma function Γ k x and its logarithmic derivative ψ k x , and established some upper and lower bounds for Γ k x in terms of ψ k x .

scholarly journals Short Remarks on Complete Monotonicity of Some Functions

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 537 ◽  
Author(s):  
Ladislav Matejíčka
Keyword(s):  
Gamma Function ◽  
Logarithmic Derivative ◽  
Complete Monotonicity ◽  
Completely Monotonic ◽  
Β Function

In this paper, we show that the functions x m | β ( m ) ( x ) | are not completely monotonic on ( 0 , ∞ ) for all m ∈ N , where β ( x ) is the Nielsen’s β -function and we prove the functions x m − 1 | β ( m ) ( x ) | and x m − 1 | ψ ( m ) ( x ) | are completely monotonic on ( 0 , ∞ ) for all m ∈ N , m > 2 , where ψ ( x ) denotes the logarithmic derivative of Euler’s gamma function.

scholarly journals A logarithmically completely monotonic function involving the gamma function and originating from the Catalan numbers and function

2015 ◽  
Vol 3 (4) ◽  
pp. 140 ◽  
Author(s):  
Fang-Fang Liu ◽  
Xiao-Ting Shi ◽  
Feng Qi
Keyword(s):  
Gamma Function ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Catalan Numbers ◽  
Monotonic Function ◽  
Completely Monotonic Function ◽  
Completely Monotonic ◽  
Logarithmically Completely Monotonic Function ◽  
And Function

<p><span>In the paper, the authors find necessary conditions and sufficient conditions for a function involving the gamma function and originating from investigation of properties of the Catalan numbers and function in combinatorics to be logarithmically completely monotonic.</span></p>

On the asymptotic expansion of the logarithm of Barnes triple gamma function II

2017 ◽  
Vol 120 (2) ◽  
pp. 291
Author(s):  
Stamatis Koumandos ◽  
Henrik L. Pedersen
Keyword(s):  
Asymptotic Expansion ◽  
Gamma Function ◽  
Positive Order ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions

The remainders in an asymptotic expansion of the logarithm of Barnes triple gamma function give rise to completely monotonic functions of positive order.

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