Picard values and normality criterion (Q2732165)

Let \(F\) be a family of meromorphic functions in a domain \(D\), and let \(a\) be a nonvanishing analytic function in \(D\). If for every \(f\) in \(F\), \(f\) and \(f'\) have the same zeros \([f(z)= 0\) if and only if \(f'(z)= 0]\), and \(f(z)= a(z)\) whenever \(f'(z)= a(z)\), then \(F\) is normal in \(D\). An application of this result is the following interesting theorem: If \(f\) is a transcendental meromorphic function with infinite order for which \(f\) and \(f'\) have the same zeros, then \(f'(z)- b(z)\) has infinitely many zeros for any \(b(z)\) in \(\{az^m\mid a\neq 0, m= 0,1,2,\dots\}\). The latter complements a result of \textit{W. Bergweiler} and \textit{A. Eremenko} [Rev. Mat. Iberoam. 11, No. 2, 355-373 (1995; Zbl 0830.33016)] which the author also extends using a method of \textit{W. Bergweiler} [Arch. Math. 64, No. 3, 199-202 (1995; Zbl 0818.30021)].