Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Some Inequalities of the Bell Polynomials

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)

A peer-reviewed article of this Preprint also exists.

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.

Journal reference: Mathematical Inequalities & Applications 2020, 23, 123-135
DOI: 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) derives the (logarithmically) absolute and complete monotonicity of generating functions of the Bell polynomials; (4) 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.

Subject Areas

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

Comments (4)

Comment 1
Received: 31 August 2017
Commenter: Feng Qi
Commenter's Conflict of Interests: I am the only author of this preprint.
Comment: Feng Qi, ''Some properties of the Touchard polynomials'', ResearchGate Working Paper (2017), available online at https://doi.org/10.13140/RG.2.2.30022.16967
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Comment 2
Received: 16 April 2019
Commenter: Feng Qi (Click to see Publons profile: )
Commenter's Conflict of Interests: I am the author.
Comment: This preprint has been accepted as

Feng Qi, Some inequalities and an application of exponential polynomials, Mathematical Inequalities & Applications 22 (2019), in press.
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Comment 3
Received: 7 February 2020
Commenter: Feng Qi (Click to see Publons profile: )
Commenter's Conflict of Interests: I am the author
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
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Comment 4
Received: 7 February 2020
Commenter: Feng Qi (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
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