ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. 62, 1 (1993)
pp. 13-16

REFINEMENT OF AN INEQUALITY OF E. LANDAU
H. ALZER

Abstract. We prove: Let $P(z)=\sum^n_k=0 a_kz^k$ be a complex polynomial with $n\geq 1$ and $a_0a_n\ne0$. If $z$ is a zero of $P$, then we have for all real numbers $t>0$: |z|>\frac|a_0|t|a_0|+K_n(t)\tag* with \align K_n(t)&=\frac11-\alpha _n(t)^n\min_1\le m \le n\Bigl[(\alpha _n(t)^m-\alpha _n(t)^n)\max_m\le p\le nA_p(t)\\\vspace5truept &\qquad\qquad+(1-\alpha _n(t)^m)\max_1\le p\le n A_p(t)\Bigr],\ \alpha _n(t)&=\frac|a_0||a_0|+\max\limits_1\le p\le n A_p(t)\,,\ A_p(t)&=\frac1p\sum^p_k=1|a_k|t^k\,. \endalign Inequality (*) sharpens a result of E. Landau.

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