The zeros of entire functions of small positive order and their derivatives (Q1129749)

If \(f\) is an entire function with order \(\rho< {1\over 2}\), W. Fuchs has conjectured that the Nevanlinna deficiency \(\delta(0,{f'\over f})=0\). \textit{A. Eremenko}, \textit{J. Langley} and \textit{J. Rossi} [J. Anal. Math. 62, 271-286 (1994; Zbl 0818.30020)] have shown \(\delta(0,{f'\over f})\leq 1-\cos \pi\rho\). In the article under review the authors obtain a better bound for the ratio lim sup \(n(r,o,f')/n(r,o,f) =R(r\to\infty)\), by showing for small \(\rho>0\) that \(R\geq h(\rho)> \cos\pi \rho\) where \(h(\rho)\) is a function of \(\rho\). The proof emerges from using the Fourier series representation for \(\log| f'(re^{i\theta})|\), where \(f'\) is expressed as a canonical product of genus zero, together with an equality relating the logarithmic derivatives of \(f\) and \(f'\) shown in work of \textit{S. Hellerstein}, \textit{J. Miles}, and \textit{J. Rossi} [Ann. Acad. Sci. Fenn., Ser. AI Math. 17., No. 2, 343-365 (1992; Zbl 0759.34005)].