Lower estimate in Littlewood's conjecture on the mean spherical derivative of a polynomial and iteration theory

Author:
A. È. Erëmenko

Journal:
Proc. Amer. Math. Soc. **112** (1991), 713-715

MSC:
Primary 30C10; Secondary 30B20, 30D15

DOI:
https://doi.org/10.1090/S0002-9939-1991-1065943-1

MathSciNet review:
1065943

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Abstract: Lewis and Wu have proved J. E. Littlewood's conjecture of the growth of the spherical derivative of polynomials. We apply the theory of complex iteration to show that this conjecture is sharp in a qualitative sense.

**[B]**P. Blanchard,*Complex analytic dynamics on the Riemann sphere*, Bull. Amer. Math. Soc.**11**(1984), 85-14] R. MR**741725 (85h:58001)****[Bo]**R. Bowen,*Equilbrium states and the ergodic theory of Anosov diffeomorphisms*, Lecture Notes in Math., vol. 470, Springer-Verlag, Berlin and New York,1975. MR**0442989 (56:1364)****[EL]**A. Eremenko and M. Lyubich,*Dynamics of analytical transformations*, Algebra and Analysis**1**(1989), 1-70 (Russian); English transi.: Leningrad Math. J.**1**(1990),563-634. MR**1015124 (91b:58109)****[ES1]**A. Eremenko and M. Sodin,*On a conjecture of Littlewood and value distribution of entire functions*, Funct. Anal. Appl.**20**(1986) 71-72. MR**831054 (87f:30070)****[ES2]**-,*Proof of a conditional theorem of Littlewood on the distribution of values of entire functions*, Math. USSR Izvestiya**30**(198), 395-402.**[H]**W. Hayman,*On a conjecture of Littlewood*, J. Anal. Math.**36**(1979), 75-95. MR**581802 (81j:30042)****[L]**J. E. Littlewood,*On some conjectural inequalities with applications to the theory ofintergral functions*, J. London Math. So.**27**(1952), 387-393. MR**0049315 (14:154f)****[LeW]**J. L. Lewis and Jang-Mei Wu,*On conjectures of Arakelyan and Littlewood*, J. Anal.Math.**50**(1988), 259-283. MR**942832 (89i:30038)****[R]**D. Ruelle,*Repellers for real analytic maps*, Ergodic Theoryynamical Systems**2**(1982), 99-107. MR**684247 (84f:58095)**

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DOI:
https://doi.org/10.1090/S0002-9939-1991-1065943-1

Article copyright:
© Copyright 1991
American Mathematical Society