Refinement of an Inequality Of

We prove: Let P(z) = P n k=0 a k z k be a complex polynomial with n 1 and a 0 an 6 = 0. If z is a zero of P, then we have for all real numbers t > 0: (*) jzj > ja 0 jt ja 0 j + Kn(t) with Kn(t) = 1 1 ? n(t) n min 1mn h (n(t) m ? n(t) n) max mpn Ap(t) + (1 ? n(t) m) max 1pn Ap(t) i ; n(t) = ja 0 j ja 0 j + max 1pn Ap(t) ; Ap(t) = 1 p p X k=1 ja k jt k : Inequality (*) sharpens a result of E. Landau. In this paper we denote by P the polynomial P(z) = P n k=0 a k z k , where z and the a k 's are complex numbers. Moreover, we assume n 1 and a 0 a n 6 = 0. In 1914 E. Landau 2] presented the following lower bound for the moduli of the zeros of P: Theorem A. If z is a zero of P and t is any positive real number, then (1) jzj ja 0 jt ja 0 j + max 1pn ja p jt p : The same result was also obtained by J. Karamata 1] and D. Markovitch 3]. In 1967 D. M. Simeunovi c 5] proved an interesting reenement of inequality (1) (see also 4, pp. 222{223]):