Version 1
: Received: 23 August 2017 / Approved: 23 August 2017 / Online: 23 August 2017 (11:35:33 CEST)
Version 2
: Received: 24 August 2017 / Approved: 25 August 2017 / Online: 25 August 2017 (08:41:30 CEST)
Feng Qi, Some inequalities and an application of exponential polynomials, Mathematical Inequalities & Applications, 2020, 1, 123-135; available online at https://doi.org/10.7153/mia-2020-23-10.
Feng Qi, Some inequalities and an application of exponential polynomials, Mathematical Inequalities & Applications, 2020, 1, 123-135; available online at https://doi.org/10.7153/mia-2020-23-10.
Feng Qi, Some inequalities and an application of exponential polynomials, Mathematical Inequalities & Applications, 2020, 1, 123-135; available online at https://doi.org/10.7153/mia-2020-23-10.
Feng Qi, Some inequalities and an application of exponential polynomials, Mathematical Inequalities & Applications, 2020, 1, 123-135; available online at https://doi.org/10.7153/mia-2020-23-10.
Abstract
In the paper, the author (1) presents an explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials, with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds; (2) recovers an explicit formula and its inversion formula for the Bell polynomials in terms of the Stirling numbers of the first and second kinds, with the aid of the above explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials; (3) constructs some determinantal and product inequalities and deduces the logarithmic convexity of the Bell polynomials, with the assistance of the complete monotonicity of generating functions of the Bell polynomials. These inequalities are main results of the paper.
Keywords
Bell polynomial; Bell number; Bell polynomial of the second kind; higher order derivative; generating function; Faa di Bruno formula; inversion theorem; Stirling number of the first kind; Stirling number of the second kind; explicit formula; inversion formula; logarithmically absolute monotonicity; logarithmically complete monotonicity; determinantal inequality; product inequality
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
(Click to see Publons profile: )
Commenter's Conflict of Interests:
I am the author: https://qifeng618.wordpress.com
Comment:
This paper has been formally published as
Feng Qi, Some inequalities and an application of exponential polynomials, Mathematical Inequalities & Applications 23 (2020), no. 1, 123--135; available online at https://doi.org/10.7153/mia-2020-23-10
(Click to see Publons profile: )
Commenter's Conflict of Interests:
I am the author at https://qifeng618.wordpress.com
Comment:
This paper has been formally published as
Feng Qi, Some inequalities and an application of exponential polynomials, Mathematical Inequalities & Applications 23 (2020), no. 1, 123--135; available online at https://doi.org/10.7153/mia-2020-23-10
Commenter: Feng Qi
Commenter's Conflict of Interests: I am the author of this preprint.
Commenter:
Commenter's Conflict of Interests: I am the author.
Feng Qi, Some inequalities and an application of exponential polynomials, Mathematical Inequalities & Applications 22 (2019), in press.
Commenter:
Commenter's Conflict of Interests: I am the author: https://qifeng618.wordpress.com
Feng Qi, Some inequalities and an application of exponential polynomials, Mathematical Inequalities & Applications 23 (2020), no. 1, 123--135; available online at https://doi.org/10.7153/mia-2020-23-10
Commenter:
Commenter's Conflict of Interests: I am the author at https://qifeng618.wordpress.com
Feng Qi, Some inequalities and an application of exponential polynomials, Mathematical Inequalities & Applications 23 (2020), no. 1, 123--135; available online at https://doi.org/10.7153/mia-2020-23-10