Markov and Bernstein type inequalities for polynomials

• Main Authors: Mohapatra RN, Govil NK
• Format: Article
• Language: English
• Published: SpringerOpen 1999-01-01
• Series: Journal of Inequalities and Applications
• Subjects: Inequalities in the complex domain; Inequalities for trigonometric functions and polynomials; Rational functions
• Online Access: http://www.journalofinequalitiesandapplications.com/content/3/156027
• ISSN: 1025-5834; 1029-242X

<p/> <p>In an answer to a question raised by chemist Mendeleev, A. Markov proved that if <inline-formula><graphic file="1029-242X-1999-156027-i1.gif"/></inline-formula> is a real polynomial of degree <inline-formula><graphic file="1029-242X-1999-156027-i2.gif"/></inline-formula>, then <inline-formula><graphic file="1029-242X-1999-156027-i3.gif"/></inline-formula> The above inequality which is known as Markov's Inequality is best possible and becomes equality for the Chebyshev polynomial <inline-formula><graphic file="1029-242X-1999-156027-i4.gif"/></inline-formula>.</p> <p>Few years later, Serge Bernstein needed the analogue of this result for the unit disk in the complex plane instead of the interval <inline-formula><graphic file="1029-242X-1999-156027-i5.gif"/></inline-formula> and the following is known as Bernstein's Inequality.</p> <p>If <inline-formula><graphic file="1029-242X-1999-156027-i6.gif"/></inline-formula> is a polynomial of degree <inline-formula><graphic file="1029-242X-1999-156027-i7.gif"/></inline-formula> then <inline-formula><graphic file="1029-242X-1999-156027-i8.gif"/></inline-formula> This inequality is also best possible and is attained for <inline-formula><graphic file="1029-242X-1999-156027-i9.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-1999-156027-i10.gif"/></inline-formula> being a complex number.</p> <p>The above two inequalities have been the starting point of a considerable literature in Mathematics and in this article we discuss some of the research centered around these inequalities.</p>