Inequalities for Laguerre functions
<p/> <p>The main published inequality for Laguerre functions <inline-formula><graphic file="1029-242X-1997-936095-i1.gif"/></inline-formula> seems to be for Laguerre polynomials <inline-formula><graphic file="1029-242X-1997-936095-i2.gif"/&g...
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
1997-01-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://www.journalofinequalitiesandapplications.com/content/1/936095 |
| Summary: | <p/> <p>The main published inequality for Laguerre functions <inline-formula><graphic file="1029-242X-1997-936095-i1.gif"/></inline-formula> seems to be for Laguerre polynomials <inline-formula><graphic file="1029-242X-1997-936095-i2.gif"/></inline-formula> only; it is [2: 10.18(3)]: <inline-formula><graphic file="1029-242X-1997-936095-i3.gif"/></inline-formula>This paper presents several inequalities for Laguerre polynomials <inline-formula><graphic file="1029-242X-1997-936095-i4.gif"/></inline-formula> and Laguerre functions <inline-formula><graphic file="1029-242X-1997-936095-i5.gif"/></inline-formula>, most of which do not seem to be in the existing literature. The corresponding inequalities for confluent hypergeometric functions are noted.</p> <p>For our work on expansions in series of Laguerre functions, M.N. Hunter and I needed an inequality for <inline-formula><graphic file="1029-242X-1997-936095-i6.gif"/></inline-formula> when re <inline-formula><graphic file="1029-242X-1997-936095-i7.gif"/></inline-formula> and re <inline-formula><graphic file="1029-242X-1997-936095-i8.gif"/></inline-formula> is large. The only extensions of the above inequality that we could obtain had multiples of <inline-formula><graphic file="1029-242X-1997-936095-i9.gif"/></inline-formula> on the right hand side instead of <inline-formula><graphic file="1029-242X-1997-936095-i10.gif"/></inline-formula>. This paper goes on to show that this is inevitable for non-integral <inline-formula><graphic file="1029-242X-1997-936095-i11.gif"/></inline-formula>, in that <inline-formula><graphic file="1029-242X-1997-936095-i12.gif"/></inline-formula> can exceed a multiple of <inline-formula><graphic file="1029-242X-1997-936095-i13.gif"/></inline-formula> for every fixed <inline-formula><graphic file="1029-242X-1997-936095-i14.gif"/></inline-formula> if <inline-formula><graphic file="1029-242X-1997-936095-i15.gif"/></inline-formula> is sufficiently large.</p> |
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| ISSN: | 1025-5834 1029-242X |