Normal criterion concerning shared values (Q1760640)

Summary: We prove the following normality criterion: Let \(F\) be a family of meromorphic functions in a domain \(D\) whose zeros have multiplicity at least 2. Assume that there exists nonzero complex numbers \(b_f\), \(c_f\) depending on \(f\) satisfying: (i) \(b_f/c_f\) is a constant; (ii) \(\min \{\sigma(0, b_f), \sigma(0, c_f), \sigma(b_f, c_f)\} \geq m\) for some \(m > 0\); (iii) if \((1/c^{k-1}_f)(f')^k(z) + f(z) \neq b^k_f/c^{k-1}_f\) or \((1/c^{k-1}_f)(f')^k(z) + f(z) = b^k_f/c^{k-1}_f\) then \(f(z) = b_f\). Then \(F\) is normal.