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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Computer investigation of Landau's theorem

Author: P. S. Chiang
Journal: Math. Comp. 23 (1969), 185-188
MSC: Primary 30.20
MathSciNet review: 0241611
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Abstract: Let $ f(z) = {a_0} + {a_1}z + \cdots $ be regular for $ \left\vert z \right\vert < 1$ and never take the values 0 and $ 1$; then $ \left\vert {{a_1}} \right\vert$ has a bound depending only on $ {a_0}$. J. A. Jenkins gave an explicit bound (Canad. J. Math. 8 (1956), 423-425) $ \left\vert {{a_1}} \right\vert \leqq 2\left\vert {{a_0}} \right\vert\left\{ {\left\vert {\log } \right\vert\left. {{a_0}} \right\Vert + 5.94} \right\}$. The author investigates the shapes for the curves $ \left\vert {{a_1}} \right\vert \leqq L{\text{(}}{a_0}{\text{)}}$ for given $ {a_0}$ by the aid of a computer and shows that although Jenkins' result is about right when $ {a_0}$ is negative, 4.38 will be the best possible constant in his form and that a much smaller estimate should be available when $ {a_0}$ is positive or complex.

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Article copyright: © Copyright 1969 American Mathematical Society

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