Illinois Journal of Mathematics

On the exceptional set in a conditional theorem of Littlewood

Lukas Geyer

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 1 April 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30D35: Distribution of values, Nevanlinna theory
Secondary: 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04]


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.

Export citation


  • 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://
  • 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
  • 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.