Journal of Inequalities and Applications
Volume 1 (1997), Issue 3, Pages 293-299
doi:10.1155/S1025583497000192
Inequalities for Laguerre functions
Department of Mathematics, The University of Melbourne, Parkville 3052, Victoria, Australia
Received 19 June 1996
Copyright © 1997 E. R. Love. 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.
Abstract
The main published inequality for Laguerre functions Lvμ(z) seems to be for Laguerre polynomials Ln0(x) only; it is [2: 10.18(3)]:
|Ln(x)|≤ex/2
for
x>0.This paper presents several inequalities for Laguerre polynomials Lnμ(x) and Laguerre functions Lvμ(x), most of which do not seem to be in the existing literature. The corresponding inequalities for confluent hypergeometric functions are noted.
For our work on expansions in series of Laguerre functions, M.N. Hunter and I needed an inequality for Lvμ(x) when re μ>−12 and re v is large. The only extensions of the above inequality that we could obtain had multiples of ex on the right hand side instead of ex/2. This paper goes on to show that this is inevitable for non-integral v, in that |Lvμ(x)| can exceed a multiple of eλx for every fixed λ<1 if x is sufficiently large.