A hypothesis of Littlewood and the distribution of values of entire functions
Let \(\rho (f)=| f'| /(1+| f|^ 2)\) be the spherical derivative of a polynomial f. Denote \[ \phi (n)=\sup \{\iint_{x^ 2+y^ 2\leq 1}\rho (f) dx dy: \deg f=n\}. \] J. Littlewood proved that \(\phi\) (n)\(\leq \pi \sqrt{n}\) and conjectured that \(\phi (n)\leq An^{\alpha}\) with absolute constants \(A>0\), \(\alpha <\). The authors prove that \(\phi (n)=o(n)\), \(n\to \infty\). This result has an interesting counterpart in the theory of entire functions: let f be an entire function of finite positive order. Then there exists an infinitesimal part of the plane which contains almost all solutions of \(f(z)=a\) for all \(a\in {\mathbb{C}}\).
scientific article; zbMATH DE number 3969015