Maximum modulus of derivatives of a polynomial

Abstract

For an arbitrary entire function f(z), let M(f;R) = maxjzj=R jf(z)j<br />and m(f; r) = minjzj=r jf(z)j. If P(z) is a polynomial of degree n having no zeros<br />in jzj < k, k 1, then for 0 r k, it is proved by Aziz et al. that<br />M(P0; ) n<br /> +k f( +k<br />k+r )n[1 􀀀 k(k􀀀 )(nja0j􀀀kja1j)n<br />( 2+k2)nja0j+2k2 ja1j ( 􀀀r<br />k+ )( k+r<br />k+ )n􀀀1]M(P; r)<br />􀀀[ (nja0j +k2ja1j)(r+k)<br />( 2+k2)nja0j+2k2 ja1j [(( +k<br />r+k )n 􀀀 1) 􀀀 n( 􀀀 r)]]m(P; k)g:<br />In this paper, we obtain a re nement of the above inequality. Moreover, we obtain<br />a generalization of above inequality for M(P0;R), where R k.