MAXIMUM MODULUS OF THE DERIVATIVES OF A POLYNOMIAL

Abstract. For an arbitrary entire function f(z), let M(f,R) = max|z|=R |f(z)| and m(f, r) = min|z|=r |f(z)|. If P (z) is a polynomial of degree n having no zeros in |z | < k, k ≥ 1, then for 0 ≤ r ≤ ρ ≤ k, it is proved by Aziz et al. that M(P ′, ρ) ≤ nρ+k{(ρ+kk+r)n[1 − k(k−ρ)(n|a0|−k|a1|)n(ρ2+k2)n|a0|+2k2ρ|a1 | ( ρ−r k+ρ)( k+r k+ρ) n−1]M(P, r) − [ (n|a0|ρ+k2|a1|)(r+k)(ρ2+k2)n|a0|+2k2ρ|a1 | × [(( ρ+k r+k) n − 1) − n(ρ − r)]]m(P, k)}. In this paper, we obtain a refinement of the above inequality. Moreover, we obtain a generalization of above inequality for M(P ′, R), where R ≥ k. 1. Introduction an