A conjecture of Schoenberg
<p/> <p>For an arbitrary polynomial <inline-formula><graphic file="1029-242X-1999-838171-i1.gif"/></inline-formula> with the sum of all zeros equal to zero, <inline-formula><graphic file="1029-242X-1999-838171-i2.gif"/></inline-formula...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
1999-01-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://www.journalofinequalitiesandapplications.com/content/4/838171 |
| Summary: | <p/> <p>For an arbitrary polynomial <inline-formula><graphic file="1029-242X-1999-838171-i1.gif"/></inline-formula> with the sum of all zeros equal to zero, <inline-formula><graphic file="1029-242X-1999-838171-i2.gif"/></inline-formula>, the quadratic mean radius is defined by <inline-formula><graphic file="1029-242X-1999-838171-i3.gif"/></inline-formula> Schoenberg conjectured that the quadratic mean radii of <inline-formula><graphic file="1029-242X-1999-838171-i4.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-1999-838171-i5.gif"/></inline-formula> satisfy <inline-formula><graphic file="1029-242X-1999-838171-i6.gif"/></inline-formula>where equality holds if and only if the zeros all lie on a straight line through the origin in the complex plane (this includes the simple case when all zeros are real) and proved this conjecture for <inline-formula><graphic file="1029-242X-1999-838171-i7.gif"/></inline-formula> and for polynomials of the form <inline-formula><graphic file="1029-242X-1999-838171-i8.gif"/></inline-formula>.</p> <p>It is the purpose of this paper to prove the conjecture for three other classes of polynomials. One of these classes reduces for a special choice of the parameters to a previous extension due to the second and third authors.</p> |
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| ISSN: | 1025-5834 1029-242X |