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If p(z) = n ∑ j=0 a j j is a polynomial of degree n satisfying p(z) = 0 in |z| < 1 , then for R 1 . Ankeny and Rivlin  proved that M(p,R) R n +1 2 M(p,1) . In this paper we obtain some results in this direction by considering polynomials of degree n 2 , having all its zeros on |z| = k , k 1 which is an improvement of the result recently proved by M. S. Pukhta (2013) [Progress in Applied Mathematics, 6 (2), 50-58]. Mathematics subject classification (2010): 30A10, 30C10, 30D15, 41A17.doi:10.7153/jca-05-09 fatcat:kvuvtzj65na3ne5aviwtisg6pu