Inequalities for Laguerre functions
type: Item
instance of: scholarly article
title: Inequalities for Laguerre functions
zbMATH Open document ID: 0908.33002
DOI: 10.1155/S1025583497000192
published in: Journal of Inequalities and Applications
publication date: 1999-03-08T00:00:00Z
full work available at URL: https://eudml.org/doc/119685
review text: If \(\Phi\) is the confluent hypergeometric function, then the Laguerre functions are defined by \[ L_{\nu}^{\mu}(z)= {\mu +\nu \choose \nu} \Phi (-\nu, 1+\mu;z). \] When \(\nu=n\), a nonnegative integer, \(L_{\nu}^{\mu}\) becomes a polynomial in \(z\) and it has the following nice inequality for \(z=x>0\): \(| L_{n}^{\mu}(x)| \leq {\mu +n \choose \mu} e^{x/2}.\) The author derives several new inequalities both for Laguerre functions \(L_{\nu}^{\mu}(x)\) and for Laguerre polynomials \(L_{n}^{\mu}(x)\). These results do not seem to be available in the literature. Some special inequalities are also observed for confluent hypergeometric functions. In fact, the author obtains an upper bound for the function \(e^{-x}| L_{\nu}^{\mu}(x)| \) whenever \(x>0\), \(\text{Re }\mu >-1/2\) and \(\text{Re }(\mu+\nu +1)>0\). Similarly, a lower bound for the function \(e^{\lambda x}| L_{\nu}^{\mu}(x)| \) is obtained if \(\text{Re }\mu >-1\), \(\text{Re }\nu >0\), \(\mu+\nu +1>0\), \(\lambda \in (0,1)\) and if \(x>0\) is sufficiently large.
Mathematics Subject Classification ID: 33C15
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 26D07
zbMATH DE Number: 1139491
zbMATH Keywords: Laguerre polynomial
zbMATH Keywords: confluent hypergeometric functions and fractional integration
author: E. R. Love
reviewed by: Saminathan Ponnusamy
MaRDI profile type: Publication
OpenAlex ID: W1985152260
description: scientific article; zbMATH DE number 1139491