Limit sets and indicators of an entire function (Q1060314)

Let A(\(\rho)\) be the class of entire functions f(z), \(z\in {\mathbb{C}}\), of finite order \(\rho >0\) and of normal type under the proximate order \(\rho\) (r)\(\to \rho\). Let further Fr[f] be the limit set for \(f\in A(\rho)\), i.e. the set of subharmonic functions v(z) being the limit in \(D'({\mathbb{C}})\) of sequences of the form \(t_ j^{-\rho (t_ j)}\ell n| f(t_ jz)|,\quad t_ j\to \infty.\) If Fr[f] is known the authors define following the previous work of one of the authors, the upper and lower indicator of f: \[ \bar h(\phi,f)=\sup \{v(e^{i\phi}):\quad v\in Fr[f]\}, \] \[ \underline h(\phi,f)=\inf \{v(e^{i\phi}):\quad v\in Fr[f]\}. \] Other definitons of the lower indicator are also known. The authors prove that they are all equivalent and also that \(\bar h(\phi,f)=h_ f(\phi)\), where \(h_ f\) is the usual indicator of f. Further on the authors construct \(f\in A(\rho)\) with a given upper semicontinuous indicator ẖ. They also obtain the conditions for ẖ(\(\phi)\) being bounded at a point. These conditions are formulated in terms of values which are determined by the zeroes of f in the halfplane \(Re ze^{-i\phi}>0.\) When \(\phi\) is not an integer the upper bound for \(h_ f(\phi)\) is obtained in the paper. This bound is determined by zeroes of f and is uninprovable in some sense.