On a conjecture of Fuchs (Q2771449)

W. Fuchs conjectured that if \(f\) is a meromorphic function in the plane with order \(\rho\) less than one, then the Nevanlinna deficiency of zero for \(f'/f\) is zero. \textit{A. A. Gol'dberg} and \textit{N. E. Korenkov} [Teor. Funkts. Funkts. Anal. Prilozh. 34, 41-46 (1980; Zbl 0441.30039)] showed the conjecture to be false for \(\rho> 1/2\). \textit{A. Eremenko}, \textit{J. Langley} and \textit{J. Rossi} [J. Anal. Math. 62, 271-286 (1994; Zbl 0818.30020)] showed if \(\rho\leq 1/2\), then the Nevanlinna deficiency of zero for \(f'/f\) is less than or equal to \(1-\cos \pi\rho\). The paper under review strengthens this estimate for functions with order \(\rho\) where \(0<\rho <2^{-11}\). The proof is based on an intricate analysis of the Fourier series of \(\log|g(re^{i \theta})|\) for \(g\) an entire function of order \(\rho\) in \((0,1)\).