## Illinois Journal of Mathematics

### On the exceptional set in a conditional theorem of Littlewood

Lukas Geyer

#### Abstract

In 1952, Littlewood stated a conjecture about the average growth of spherical derivatives of polynomials, and showed that it would imply that for entire function of finite order, “most” preimages of almost all points are concentrated in a small subset of the plane. In 1988, Lewis and Wu proved Littlewood’s conjecture. Using techniques from complex dynamics, we construct entire functions of finite order with a bounded set of singular values for which the set of exceptional preimages is infinite, with logarithmically growing cardinality.

#### Article information

Source
Illinois J. Math., Volume 58, Number 1 (2014), 279-284.

Dates
First available in Project Euclid: 1 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1427897178

Digital Object Identifier
doi:10.1215/ijm/1427897178

Mathematical Reviews number (MathSciNet)
MR3331851

Zentralblatt MATH identifier
1341.30028

#### Citation

Geyer, Lukas. On the exceptional set in a conditional theorem of Littlewood. Illinois J. Math. 58 (2014), no. 1, 279--284. doi:10.1215/ijm/1427897178. https://projecteuclid.org/euclid.ijm/1427897178

#### References

• W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana 11 (1995), no. 2, 355–373.
• D. Beliaev, Integral means spectrum of random conformal snowflakes, Nonlinearity 21 (2008), no. 7, 1435–1442.
• C. Bishop, The order conjecture fails in $\mathcal{S}$, preprint, 2013; available at \surl http:// www.math.sunysb.edu/~bishop/papers/order.pdf.
• D. Beliaev and S. Smirnov, On Littlewoods's constants, Bull. Lond. Math. Soc. 37 (2005), no. 5, 719–726.
• D. Beliaev and S. Smirnov, Random conformal snowflakes, Ann. of Math. (2) 172 (2010), no. 1, 597–615.
• X. Buff, On the Bieberbach conjecture and holomorphic dynamics, Proc. Amer. Math. Soc. 131 (2003), no. 3, 755–759 (electronic).
• L. Carleson and T. W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer, New York, 1993.
• A. È. Eremenko and M. Y. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 4, 989–1020.
• A. Epstein and L. Rempe-Gillen, On the invariance of order for finite-type entire functions, preprint, 2013; available at arXiv:\arxivurl1304.6576.
• A. È. Eremenko, Lower estimate in Littlewood's conjecture on the mean spherical derivative of a polynomial and iteration theory, Proc. Amer. Math. Soc. 112 (1991), no. 3, 713–715.
• A. È. Eremenko, Some constants coming from the work of Littlewood, 2002; available at http://www.math.purdue.edu/~eremenko/dvi/lit.pdf.
• A. È. Eremenko and M. L. Sodin, A conjecture of Littlewood and the distribution of values of entire functions, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 71–72.
• A. È. Eremenko and M. L. Sodin, A proof of the conditional Littlewood theorem on the distribution of the values of entire functions, Izv. Ross. Akad. Nauk Ser. Mat. 51 (1987), no. 2, 421–428, 448.
• H. Hedenmalm and S. Shimorin, Weighted Bergman spaces and the integral means spectrum of conformal mappings, Duke Math. J. 127 (2005), no. 2, 341–393.
• J. K. Langley, On the multiple points of certain meromorphic functions, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1787–1795.
• J. E. Littlewood, On some conjectural inequalities, with applications to the theory of integral functions, J. London Math. Soc. 27 (1952), 387–393.
• J. L. Lewis and J.-M. Wu, On conjectures of Arakelyan and Littlewood, J. Anal. Math. 50 (1988), 259–283.
• H. Mihaljević-Brandt and J. Peter, Poincaré functions with spiders' webs, Proc. Amer. Math. Soc. 140 (2012), no. 9, 3193–3205.
• C. L. Siegel, Iteration of analytic functions, Ann. of Math. (2) 43 (1942), 607–612.
• G. Valiron, Sur les fonctions entières d'ordre nul et d'ordre fini et en particulier les fonctions à correspondance régulière, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (3) 5 (1913), 117–257.