Computer investigation of Landau's theorem

Author:
P. S. Chiang

Journal:
Math. Comp. **23** (1969), 185-188

MSC:
Primary 30.20

DOI:
https://doi.org/10.1090/S0025-5718-1969-0241611-7

MathSciNet review:
0241611

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Abstract: Let be regular for and never take the values 0 and ; then has a bound depending only on . J. A. Jenkins gave an explicit bound (*Canad. J. Math*. **8** (1956), 423-425) . The author investigates the shapes for the curves for given by the aid of a computer and shows that although Jenkins' result is about right when is negative, 4.38 will be the best possible constant in his form and that a much smaller estimate should be available when is positive or complex.

**[1]**W. K. Hayman, ``Some remarks on Schottky's theorem,''*Proc. Cambridge Philos. Soc.*, v. 43, 1947, pp. 442-454. MR**9**, 84. MR**0021590 (9:84e)****[2]**J. A. Jenkins, ``On explicit bounds in Landau's theorem,''*Canad. J. Math.*, v. 8, 1956, pp. 423-425. MR**18**, 28. MR**0079098 (18:28d)****[3]**J. E. Littlewood,*Lectures on the Theory of Functions*, Oxford Univ. Press, 1944, Theorem 228, p. 196. MR**6**, 261. MR**0012121 (6:261f)****[4]**E. Jahnke & F. Emde,*Tables of Functions with Formulae and Curves*, 4th ed., Dover, New York, 1945. MR**7**, 485. MR**0015900 (7:485b)**

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DOI:
https://doi.org/10.1090/S0025-5718-1969-0241611-7

Article copyright:
© Copyright 1969
American Mathematical Society