Preprint Article Version 1 This version is not peer-reviewed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)

How to cite: Qi, F. Some Inequalities of the Bell Polynomials. Preprints 2017, 2017080079 (doi: 10.20944/preprints201708.0079.v1). Qi, F. Some Inequalities of the Bell Polynomials. Preprints 2017, 2017080079 (doi: 10.20944/preprints201708.0079.v1).

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

Readers' Comments and Ratings (1)

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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