Markov Inequalities for Polynomials with Restricted Coefficients

• Main Authors: Lin Shaobo, Cao Feilong
• Format: Article
• Language: English
• Published: SpringerOpen 2009-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2009/808720

<p/> <p>Essentially sharp Markov-type inequalities are known for various classes of polynomials with constraints including constraints of the coefficients of the polynomials. For <inline-formula> <graphic file="1029-242X-2009-808720-i1.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2009-808720-i2.gif"/></inline-formula> we introduce the class <inline-formula> <graphic file="1029-242X-2009-808720-i3.gif"/></inline-formula> as the collection of all polynomials of the form <inline-formula> <graphic file="1029-242X-2009-808720-i4.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2009-808720-i5.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2009-808720-i6.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2009-808720-i7.gif"/></inline-formula>. In this paper, we prove essentially sharp Markov-type inequalities for polynomials from the classes <inline-formula> <graphic file="1029-242X-2009-808720-i8.gif"/></inline-formula> on <inline-formula> <graphic file="1029-242X-2009-808720-i9.gif"/></inline-formula>. Our main result shows that the Markov factor <inline-formula> <graphic file="1029-242X-2009-808720-i10.gif"/></inline-formula> valid for all polynomials of degree at most <inline-formula> <graphic file="1029-242X-2009-808720-i11.gif"/></inline-formula> on <inline-formula> <graphic file="1029-242X-2009-808720-i12.gif"/></inline-formula> improves to <inline-formula> <graphic file="1029-242X-2009-808720-i13.gif"/></inline-formula> for polynomials in the classes <inline-formula> <graphic file="1029-242X-2009-808720-i14.gif"/></inline-formula> on <inline-formula> <graphic file="1029-242X-2009-808720-i15.gif"/></inline-formula>.</p>