growth estimate for rational functions with prescribed poles and restricted zeros

Let r(z) = f(z)/w(z) where f(z) be a polynomial of degree at most n and w(z) = ∏n j=1(z − aj), |aj | > 1 for 1 ≤ j ≤ n. if the rational function r(z) ̸= 0 in |z| < k, then for k = 1, it is known that [a.aziz and n.a.rather, journal mathematical inequalities and applications, vol.2, no.2(1999), 165 - 173] |r(rz)| ≤ (|b(rz)| + 1/2) sup |z|=1 |r(z)| for |z| = 1 where b(z) = ∏n j=1 {(1 − a¯jz)/(z − aj)}. in this paper, we consider the case k ≥ 1 and obtain cer tain results concerning the growth of the maximum modulus of the rational functions with prescribed poles and restricted zeros in the chebyshev norm on the unit circle in the complex plane.