Abstract. In this paper we establish certain sharp results concerning the maximum modulus of the polar derivative of a polynomial P(z) with restricted zeros. Our results generalize and refine some results of Turán, Malik, Govil and others.Mathematics subject classification (1991): 30A10, 30C10, 30D15.

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Abstract. In this paper we establish some new L q inequalities for polynomials which generalize and refine some results of Szegö, Zygmund, De Bruijn and others.Mathematics subject classification (1991): Primary 26D05, 30D15, Secondary 41A17.

Abstract. In this paper, we present a correct proof of an L p -inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmund's inequality to the polar derivative of a polynomial.

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Certain refinements and generalizations of some well known inequalities concerning the polynomials and their derivatives are obtained.

Abstract. Let P(z) be a polynomial of degree n 1 . In this paper we consider a more general problem of investigating the dependence of maximum ofon the maximum of |P(z)| on |z| = 1 where α , β are arbitrary complex numbers with |α| 1 , |β| 1 and obtain certain sharp compact generalizations of well-known Bernsteintype polynomial inequalities.Mathematics subject classification (2000): 30C10, 30A10.