On the growth of the logarithmic derivative (Q1851465)

Let \(f\) be a meromorphic function on \(\mathbb{C}\) with \(f(0)=1\). \textit{A. Gol'dberg} and \textit{V. Grinshtein} [Math. Notes 19, 320-323 (1976; Zbl 0338.30020)] sharpened the lemma of logarithmic derivative in Nevanlinna theory as follows: \[ m\left(r,\frac{f'}{f}\right)\leq \log^+\frac{\rho T(\rho,f)}{r(\rho-r)}+ c, \] where \(\rho>r\), \(c\leq 5.8501\). In this paper, the author proved \(c\leq 5.3078\).