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

Complete Monotonicity ◽

Convolution Theorem ◽

Monotonic Functions ◽

Completely Monotonic ◽

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.

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

10.31219/osf.io/whb2q ◽

2020 ◽

Author(s):

Feng Qi

Keyword(s):

Sufficient Conditions ◽

Laplace Transforms ◽

Complete Monotonicity ◽

Convolution Theorem ◽

Monotonic Functions ◽

Completely Monotonic ◽

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.

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

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm191111014q ◽

2021 ◽

pp. 14-14

Author(s):

Feng Qi

Keyword(s):

Exponential Function ◽

Sufficient Conditions ◽

Laplace Transforms ◽

Complete Monotonicity ◽

Monotonic Functions ◽

Completely Monotonic ◽

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.

Necessary and sufficient conditions for two functions defined by two derivatives of a function involving trigamma function to be completely monotonic or monotonic

10.31219/osf.io/6ar4p ◽

2020 ◽

Author(s):

Feng Qi

Keyword(s):

Exponential Function ◽

Sufficient Conditions ◽

Laplace Transforms ◽

Convolution Theorem ◽

Completely Monotonic ◽

In the paper, by convolution theorem for the Laplace transforms, 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.

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

10.31219/osf.io/56c2s ◽

2020 ◽

Author(s):

Feng Qi

Keyword(s):

Gamma Function ◽

Integral Inequality ◽

Sufficient Conditions ◽

Complete Monotonicity ◽

Logarithmic Convexity ◽

Monotonic Functions ◽

Completely Monotonic ◽

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.

Monotonicity of a ratio involving trigamma and tetragamma functions

10.31219/osf.io/5rfb8 ◽

2020 ◽

Author(s):

Feng Qi

Keyword(s):

Sufficient Conditions ◽

Laplace Transforms ◽

Exponential Functions ◽

Monotonic Functions ◽

Completely Monotonic ◽

The Right ◽

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.

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

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972712001025 ◽

2013 ◽

Vol 88 (2) ◽

pp. 309-319 ◽

Cited By ~ 6

Author(s):

FENG QI ◽

PIETRO CERONE ◽

SEVER S. DRAGOMIR

Keyword(s):

Divided Difference ◽

Sufficient Conditions ◽

Complete Monotonicity ◽

Divided Differences ◽

Psi Function ◽

Gamma Functions ◽

Completely Monotonic ◽

Polygamma Functions ◽

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.

Lower Bound of Sectional Curvature of Manifold of Beta Distributions and Complete Monotonicity of Functions Involving Polygamma Functions

10.20944/preprints202011.0315.v1 ◽

2020 ◽

Author(s):

Feng Qi

Keyword(s):

Lower Bound ◽

Sectional Curvature ◽

Sufficient Conditions ◽

Laplace Transforms ◽

Complete Monotonicity ◽

Convolution Theorem ◽

Open Problems ◽

Polygamma Functions ◽

Beta Distributions ◽

In the paper, by convolution theorem for the Laplace transforms and analytic techniques, the author finds necessary and sufficient conditions for complete monotonicity, monotonicity, and inequalities of several functions involving polygamma functions. By these results, the author derives a lower bound of a function related to the sectional curvature of the manifold of the beta distributions. Finally, the author poses several guesses and open problems related to monotonicity, complete monotonicity, and inequalities of several functions involving polygamma functions.

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

Georgian Mathematical Journal ◽

10.1515/gmj-2016-0004 ◽

2016 ◽

Vol 23 (2) ◽

Cited By ~ 5

Author(s):

Feng Qi ◽

Bai-Ni Guo

Keyword(s):

Sufficient Conditions ◽

Complete Monotonicity ◽

Divided Differences ◽

Gamma Functions ◽

Completely Monotonic ◽

Polygamma Functions ◽

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.

Necessary and sufficient conditions for a difference constituted by four derivatives of a function involving trigamma function to be completely monotonic

Mathematical Inequalities & Applications ◽

10.7153/mia-2021-24-58 ◽

2021 ◽

pp. 845-855

Author(s):

Feng Qi

Keyword(s):

Sufficient Conditions ◽

Completely Monotonic ◽

A class of completely monotonic functions involving the polygamma functions

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02744-2 ◽

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 ◽

Alpha Beta ◽

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.