scholarly journals Complete monotonicity of a difference defined by four derivatives of a function containing trigamma function

2020 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Gamma Function ◽  
Integral Inequality ◽  
Sufficient Conditions ◽  
Complete Monotonicity ◽  
Logarithmic Convexity ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
Necessary And Sufficient ◽  
Derivatives Of

In the paper, by virtue of convolution theorem for the Laplace transforms, logarithmic convexity of the gamma function, Bernstein's theorem for completely monotonic functions, and other techniques, the author finds necessary and sufficient conditions for a difference defined by four derivatives of a function containing trigamma function to be completely monotonic. Moreover, by virtue of Cebysev integral inequality, the author presents logarithmic convexity of the sequence of polygamma functions.

scholarly journals Monotonicity and complete monotonicity of two functions constituted via three derivatives of a function involving trigamma function

2020 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Sufficient Conditions ◽  
Laplace Transforms ◽  
Necessary And Sufficient Conditions ◽  
Complete Monotonicity ◽  
Convolution Theorem ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
Necessary And Sufficient ◽  
Derivatives Of

In the paper, by convolution theorem of the Laplace transforms, a monotonicity rule for the ratio of two Laplace transforms, Bernstein's theorem for completely monotonic functions, and other analytic techniques, the author (1) presents the decreasing monotonicity of a ratio constituted via three derivatives of a function involving trigamma function; (2) discovers necessary and sufficient conditions for a function constituted via three derivatives of a function involving trigamma function to be completely monotonic. These results conform previous guesses posed by the author.

scholarly journals Complete monotonicity of a difference constituted by four derivatives of a function involving trigamma function

2020 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Sufficient Conditions ◽  
Laplace Transforms ◽  
Necessary And Sufficient Conditions ◽  
Complete Monotonicity ◽  
Convolution Theorem ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
Necessary And Sufficient ◽  
Derivatives Of

In the paper, by virtue of convolution theorem for the Laplace transforms, Bernstein's theorem for completely monotonic functions, and other techniques, the author finds necessary and sufficient conditions for a difference constituted by four derivatives of a function involving trigamma function to be completely monotonic.

scholarly journals Necessary and sufficient conditions for complete monotonicity and monotonicity of two functions defined by two derivatives of a function involving trigamma function

2021 ◽  
pp. 14-14
Author(s):  
Feng Qi
Keyword(s):  
Exponential Function ◽  
Sufficient Conditions ◽  
Laplace Transforms ◽  
Necessary And Sufficient Conditions ◽  
Complete Monotonicity ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
Necessary And Sufficient ◽  
Derivatives Of

In the paper, by virtue of convolution theorem for the Laplace transforms, Bernstein's theorem for completely monotonic functions, some properties of a function involving exponential function, and other analytic techniques, the author finds necessary and sufficient conditions for two functions defined by two derivatives of a function involving trigamma function to be completely monotonic or monotonic. These results generalize corresponding known ones.

scholarly journals A class of completely monotonic functions involving the polygamma functions

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Li-Chun Liang ◽  
Li-Fei Zheng ◽  
Aying Wan
Keyword(s):  
Gamma Function ◽  
Sufficient Conditions ◽  
Psi Function ◽  
Gamma Functions ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Polygamma Functions ◽  
Completely Monotonic Functions ◽  
Alpha Beta ◽  
Necessary And Sufficient

AbstractLet $\Gamma (x)$ Γ ( x ) denote the classical Euler gamma function. We set $\psi _{n}(x)=(-1)^{n-1}\psi ^{(n)}(x)$ ψ n ( x ) = ( − 1 ) n − 1 ψ ( n ) ( x ) ($n\in \mathbb{N}$ n ∈ N ), where $\psi ^{(n)}(x)$ ψ ( n ) ( x ) denotes the nth derivative of the psi function $\psi (x)=\Gamma '(x)/\Gamma (x)$ ψ ( x ) = Γ ′ ( x ) / Γ ( x ) . For λ, α, $\beta \in \mathbb{R}$ β ∈ R and $m,n\in \mathbb{N}$ m , n ∈ N , we establish necessary and sufficient conditions for the functions $$ L(x;\lambda ,\alpha ,\beta )=\psi _{m+n}(x)-\lambda \psi _{m}(x+ \alpha ) \psi _{n}(x+\beta ) $$ L ( x ; λ , α , β ) = ψ m + n ( x ) − λ ψ m ( x + α ) ψ n ( x + β ) and $-L(x;\lambda ,\alpha ,\beta )$ − L ( x ; λ , α , β ) to be completely monotonic on $(-\min (\alpha ,\beta ,0),\infty )$ ( − min ( α , β , 0 ) , ∞ ) .As a result, we generalize and refine some inequalities involving the polygamma functions and also give some inequalities in terms of the ratio of gamma functions.

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 Some logarithmically completely monotonic functions and inequalities for multinomial coefficients and multivariate beta functions

2020 ◽  
pp. 33-33
Author(s):  
Feng Qi ◽  
Da-Wei Niu ◽  
Dongkyu Lim ◽  
Bai-Ni Guo
Keyword(s):  
Gamma Function ◽  
Complete Monotonicity ◽  
Bernoulli Trials ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
Multivariate Beta ◽  
Multinomial Coefficients

In the paper, the authors extend a function arising from the Bernoulli trials in probability and involving the gamma function to its largest ranges, find logarithmically complete monotonicity of these extended functions, and, in light of logarithmically complete monotonicity of these extended functions, derive some inequalities for multinomial coefficients and multivariate beta functions. These results recover, extend, and generalize some known conclusions.

scholarly journals Monotonicity of a ratio involving trigamma and tetragamma functions

2020 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Sufficient Conditions ◽  
Laplace Transforms ◽  
Necessary And Sufficient Conditions ◽  
Exponential Functions ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
The Right ◽  
Necessary And Sufficient ◽  
Logarithmic Concavity

In the paper, by convolution theorem of the Laplace transforms, Bernstein's theorem for completely monotonic functions, and logarithmic concavity of a function involving exponential functions, the author(1) finds necessary and sufficient conditions for a ratio involving trigamma and tetragamma functions to be monotonic on the right real semi-axis;(2) and presents alternative proofs of necessary and sufficient conditions for a function and its negativity involving trigamma and tetragamma functions to be completely monotonic on the positive semi-axis.These results generalizes known conclusions recently obtained by the author.

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.

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