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Proof of a conditional theorem of Littlewood on the value distribution of entire functions. (Russian) Zbl 0627.30025

The authors present full details of their short note [Funkts. Anal. Prilozh. 20, No.1, 71-72 (1986)]. In that note, they proved that \[ (*)\quad \sup_{P}\{\iint \rho_ P(z)dx dy\}=o(n^{1/2}),\quad n\to \infty, \] where the sup is over all polynomials P of degree n, \(\rho\) is the spherical derivative, and integration is over \(\{| z| <1\}\). With “O” in place of “o”, (*) is Schwarz’s inequality, and many years ago Littlewood conjectured that the bound in (*) can be improved to \(An^{-\alpha}\) where A and \(\alpha\) are positive absolute constants.
This work has been the first positive evidence of Littlewood’s conjecture; however, Littelwood’s conjecture has since been proved by J. L. Lewis and J.-M. Wu in 1987 (to appear in J. Anal. Math.). The present paper still retains some interest, as the authors present an elegant application (only sketched by Littlewood): that an entire function assumes most values in the range in a small subset of the z-plane.
Reviewer: D.Drasin

MSC:

30D20 Entire functions of one complex variable (general theory)
30C10 Polynomials and rational functions of one complex variable
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