MAXIMUM MODULUS OF THE DERIVATIVES OF A POLYNOMIAL

For an arbitrary entire function f(z), let M(f;R) = maxjzj=R jf(z)j and m(f; r) = minjzj=r jf(z)j. If P(z) is a polynomial of degree n having no zeros in jzj < k, k  1, then for 0  r    k, it is proved by Aziz et al. that M(P0; )  n +k f(+k k+r)n[1 􀀀 k(k􀀀)(nja0j􀀀kja1j)n (2+k2)nja0j+2k2ja1j (􀀀r k+)(k+r k+)n􀀀1]M(P; r) 􀀀[(nja0j+k2ja1j)(r+k) (2+k2)nja0j+2k2ja1j  [((+k r+k)n 􀀀 1) 􀀀 n( 􀀀 r)]]m(P; k)g: In this paper, we obtain a re nement of the above inequality. Moreover, we obtain a generalization of above inequality for M(P0;R), where R  k.