Sharpening of Bernstein and Turán-type inequalities for polynomials

Abstract

. Let p ( z ) be a polynomial of degree n . The polar derivative of p ( z ) with respect to a real or complex number α is de fi ned by Govil and Mctume [Acta Math. 104, 115–126 (2004)] proved that if p is a of having all its | | 1, then for any complex number α with In this paper, we prove an improvement of the above inequality. Further, we prove an improve- ment of a result due to Govil [Proc. Natl. Acad. Sci., 50, 50–52 (1980)].