Proof of a conjecture of P. Erdös on the derivative of a polynomial
Proof of a conjecture of P. Erdös on the derivative of a polynomial
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DOI：
10.1090/s0002-9904-1944-08177-9
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发表时间：
1944-08
影响因子：
1.3
通讯作者：
P. Lax
中科院分区：
文献类型：
--
作者：
P. Lax
介绍。我们从S. Bernstein关于三角多项式的著名定理的下列结论出发。设pn(z)为n次多项式，其中| ^(^)| ^ 1为\z\ gal；则当且仅当pn(z)=az, \a\ = 1时，I/>n (s)| ^n = \z\ g l, |^n（«）| -n。前段时间P. Erdös推测，如果\pn(z)\ ^ 1为|z\ ^ 1，且pn(z)在单位圆内没有根，则\pl (z)\ ^n/2为\z\ g1。在本文中，我们给出这个猜想的一个证明。
Introduction. We start out from the following consequence of S. Bernstein's well known theorem on trigonometric polynomials. Let pn(z) be a polynomial of degree n for which | ^(^) | ^ 1 holds as \z\ gal; then I/>n (s)| ^n as \z\ g l with |^n(«)| —n if and only if pn(z)=az , \a\ = 1. Some time ago P. Erdös conjectured that if \pn(z)\ ^ 1 holds as |z\ ^ 1 and pn(z) has no roots inside the unit circle, then \pl (z)\ ^n/2 as \z\ g l . In the present note we give a proof of this conjecture.