Complete monotonicity results for some functions involving the gamma and 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.

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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 ◽

Necessary And Sufficient

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.

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Sharp Inequalities for Polygamma Functions

Mathematica Slovaca ◽

10.1515/ms-2015-0010 ◽

2015 ◽

Vol 65 (1) ◽

Cited By ~ 11

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.

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Extension of complete monotonicity results involving the digamma function

Moroccan Journal of Pure and Applied Analysis ◽

10.1515/mjpaa-2018-0016 ◽

2018 ◽

Vol 4 (2) ◽

pp. 207-212

Author(s):

Kwara Nantomah

Keyword(s):

Analytical Techniques ◽

Monotonicity Property ◽

Complete Monotonicity ◽

Digamma Function ◽

Special Cases ◽

Monotonicity Results

AbstractBy using some analytical techniques, we prove a complete monotonicity property of a certain function involving the (p, k)-digamma function. Subsequently, we derive some inequalities for the (p, k)- digamma function. As special cases of the established results, we deduce some new results concerning the p-digamma and the k-digamma functions. Our results are extensions of some previous results due to Qiu and Vuorinen, Mortici, and Merovci.

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Completely monotonicity of class functions involving the polygamma and related functions

Asian-European Journal of Mathematics ◽

10.1142/s1793557121500649 ◽

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.

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Complete Monotonicity of Functions Connected with the Exponential Function and Derivatives

Abstract and Applied Analysis ◽

10.1155/2014/851213 ◽

2014 ◽

Vol 2014 ◽

pp. 1-5 ◽

Cited By ~ 3

Author(s):

Chun-Fu Wei ◽

Bai-Ni Guo

Keyword(s):

Exponential Function ◽

Complete Monotonicity ◽

Completely Monotonic ◽

Derivatives Of ◽

Monotonicity Results

Some complete monotonicity results that the functions±1/e±t-1are logarithmically completely monotonic, and that differences between consecutive derivatives of these two functions are completely monotonic, and that the ratios between consecutive derivatives of these two functions are decreasing on0, ∞are discovered. As applications of these newly discovered results, some complete monotonicity results concerning the polylogarithm are found. Finally a conjecture on the complete monotonicity of the above-mentioned ratios is posed.

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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 ◽

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.

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