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Let p(z) = a0 +a1z+a2z +a3z + · · ·+anz be a polynomial of degree n . According to a well-known theorem of Rivlin [11], if p(z) is a polynomial of degree n having no zeros inside the unit circle, then for 0 < r 1, max |z|=r |p(z)| ( r +1 2 )n max |z|=1 |p(z)|. In this paper, we generalize and sharpen the above result of Rivlin. Our result also sharpens a recently proved result of Govil and Nwaeze [3]. Also, we present some examples to show that in some cases the improvement obtained by our theorem can be considerably significant. Mathematics subject classification (2010): 15A18, 30C10, 30C15, 30A10.

On sharpening of a Theorem of T. J. Rivlin