M athematical I nequalities & A pplications Volume 4, Number 4 (2001), 491–497 INTEGRAL MEAN ESTIMATES FOR POLYNOMIALIS WITH RESTRICTED ZEROS ABDUL AZIZ AND W ALI MOHAMMAD SHAH Abstract. Let P(z) be a polynomial of degree n which does not vanish in the disk |z| < K . For K = 1 , it is known that for 0 < q < ∞ ,  1 2π  2π 0   P(Re iθ )    q dθ  1/q  B q  1 2π  2π 0   P(e iθ )    q dθ  1/q , where B q =  1 2π  2π 0   1 + R n e inα    q dα  1/q  1 2π  2π 0   1 + e inα    q dα  1/q . In this paper we present a generalization of this result by considering the case K  1 . We shall also prove a similar result for polynomialis having all their zeros in |z|  K , where K  1. Mathematics subject classification (2000): 26D05, 30D15, 41A17. Key words and phrases: Integral mean estimates, admissible operator, complex domain inequalities. REFERENCES [1] A. C. ANKENY AND T. J. RIVLIN, On a theorem of S. Bernstein, Pacific J. Math 5 (1955), 849–852. [2] V. V. ARESTOV, On inequalities for trigonometric polynomials and their derivatives, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), 3–22. [3] A. AZIZ, Integral mean estimates for polynomials with restricted zeros, J. Approx. Theory 55 (1988), 232–239. [4] A. AZIZ AND Q. G. MOHAMMAD, Growth of polynomials with zeros outside a circle, Proc. Amer. Math. Soc. 81 (1981), 549–553. [5] R. P. BOAS,JR. AND Q. I. RAHMAN, L p inequalities for polynomials and entire functions, Arch. Rational Mech. Anal. 11 (1962), 34–39. [6] G. H. HARDY, The mean value of the modulus of na analytic function, Proc. London Math. Soc. 14 (1915), 269–277. [7] M. A. MALIK, An integral mean estimate for polynomials, Proc. Amer. Math. Soc. 91 (1984), 281–284. [8] G. POLYA AND G. SZEG ¨ O, Aufgaben and Lehrs¨ atze aus der Analysis, Springer–Verlag, Berlin, 1925. [9] Q. I. RAHMAN AND G. SCHMEISSER, Les in´ egalit´ es de Markoff et de Bernstein, Presses Univ. Montr´ eal, Montr´ eal, Quebec, 1983. [10] Q. I. RAHMAN AND G. SCHMEISSER, L p inequalities for polynomials, J. Approx. Theory 53 (1998), 26–32. [11] M. RIESZ, ¨ Uber einen satz des Herrn Serge Bernstein, Acta Math. 40 (1916), 337–347. c  , Zagreb Paper MIA-04-43 Mathematical Inequalities & Applications www.ele-math.com mia@ele-math.com