Abstract. In this paper we consider a class of polynomials P(z) = a 0 + n j=μ a j z j , 1 μ n , not vanishing in the disk |z| < K . For K 1 , we investigate the dependence of max |z|=1 |P(Rz) − P(z)| on max |z|=1 |P(z)| and for K > 0 we estimate max |z|=R |P (z)| in terms of max |z|=r |P(z)| , 0 r R K . Our results not only generalize some known polynomial inequalities, but olso a variety of interesting results can be deduced from these by a fairly uniform procedure. We also obtain a generalization of a Theorem of Paul Turan. (2000): 30A10, 30C10, 30D15. Mathematics subject classification

Let P n be the class of polynomials of degree at most n. In 1969, Rahman introduced a class B n of operators B that map P n into itself and proved thatIn this paper, we show that this inequality holds analogously for the norm • q with q 1 and for some of its refinements as well. Mathematics subject classification (2010): 30A06, 30A64.

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In this paper, we prove some extensions and generalizations of the classical Eneström-Kakeya theorem