An inequality for section of certain power series
type: Item
instance of: scholarly article
title: An inequality for section of certain power series
zbMATH Open document ID: 0784.26013
DOI: 10.1007/BF01207189
author: Karl Dichler
published in: Archiv der Mathematik
publication date: 1993-11-15T00:00:00Z
review text: The following result is proved: Suppose that the power series \(f(x)=\sum^ \infty_{j=0}a_ jx^ j\) with positive coefficients is convergent in the interval \((-R,R)\), \(0<R \leq \infty\), and define \[ I_ n(x)=f(x)-\sum^ n_{j=0}a_ jx^ j= \sum^ \infty_{j=n+1} a_ jx^ j. \] Further, suppose there is a positive integer \(N\) such that for all \(n \geq N\) we have \[ {a_{n+1} \over a_ n}-{a_{n+2} \over a_{n+1}} \geq 0 \quad \text{ and } \quad {a_{n+1} \over a_ n}-2{a_{n+2} \over a_{n+1}}+{a_{n+3} \over a_{n+2}} \geq 0. \tag{*} \] Then for all \(n \geq N\) and \(x \in (0,R)\) we have \[ I_{n-1}(x)I_{n+1}(x) \geq {a_ na_{n+2} \over (a_{n+1})^ 2} \bigl( I_ n(x) \bigr)^ 2, \tag{**} \] and the coefficient \(a_ na_{n+2}/a^ 2_{n+1}\) is best possible in the sense that it cannot be replaced by a larger number. The inequality \((**)\) is strict if the inequalities \((*)\) are all strict.
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 33B99
zbMATH DE Number: 446308
zbMATH Keywords: Mittag-Leffler functions
zbMATH Keywords: power series
zbMATH Keywords: positive coefficients
zbMATH Keywords: inequality
reviewed by: J. Pečarić
MaRDI profile type: Publication
cites work: Q5558293
cites work: An inequality for the exponential function
cites work: Zeros of sections of power series
cites work: Q5847356
cites work: Q3270032
description: scientific article; zbMATH DE number 446308