Maximum modulus of derivatives of a polynomial


Department of Mathematics, Shahrood University of Technology, Shahrood, Iran.


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.