scholarly journals Lower Bound of Sectional Curvature of Fisher–Rao Manifold of Beta Distributions and Complete Monotonicity of Functions Involving Polygamma Functions

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Feng Qi
Keyword(s):  
Lower Bound ◽  
Sectional Curvature ◽  
Complete Monotonicity ◽  
Polygamma Functions ◽  
Beta Distributions

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

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.

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 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.

Curvature Bounds for the Spectrum of Closed Einstein Spaces

1978 ◽  
Vol 30 (5) ◽  
pp. 1087-1091 ◽  
Author(s):  
Udo Simon
Keyword(s):  
Lower Bound ◽  
Scalar Curvature ◽  
Sectional Curvature ◽  
Constant Scalar Curvature ◽  
Einstein Space ◽  
Curvature Bounds ◽  
Nonzero Eigenvalue ◽  
Einstein Spaces ◽  
Image Position

The following is our main result.(A) THEOREM. Let (M, g) be a closed connected Einstein space, n = dim M ≧ 2 (with constant scalar curvature R). Let K0 be the lower bound of the sectional curvature. Then either (M, g) is isometrically diffeomorphic to a sphere and the first nonzero eigenvalue ƛ1of the Laplacian fulfils

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

2016 ◽  
Vol 23 (2) ◽  
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.

scholarly journals Lower Bound for the First Steklov Eigenvalue

2018 ◽  
Vol 18 (2) ◽  
pp. 8
Author(s):  
SISTEMAS REVISTAS ◽  
Óscar Andrés Montaño
Keyword(s):  
Lower Bound ◽  
Sectional Curvature ◽  
Lower Bounds ◽  
Positive Sectional Curvature ◽  
Steklov Eigenvalue

In this paper we find lower bounds for the first Steklov eigenvalue in Riemannian n-manifolds, n = 2, 3, with non-positive sectional curvature.

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