Two normality criteria and the converse of the Bloch principle (Q1014657)

The authors prove two normality criteria of Lahiri's type and using one of them provide another counterexample to the general converse of the Bloch principle. Theorem 1. Let \(\mathcal F\) be a family of meromorphic functions in a complex domain \(D\). Let \(a, b\in\mathbb C\) such that \(a\neq 0\). Let \(m_1\), \(m_2\), \(n_1\), \(n_2\) be nonnegative integers such that \(m_1n_2 -m_2n_1 > 0\), \(m_1 +m_2\geqslant1\), \(n_1+ n_2\geqslant 2\), and put \[ E_f=\left\{z\in D:(f(z))^{n_1}(f'(z))^{m_1}+ \frac{a}{(f(z))^{n_2}(f'(z))^{m_2}}=b\right\}. \] If there exists a positive constant \(M\) such that \(|f(z)|\geqslant M\) for all \(f\in\mathcal F\) whenever \(z\in\mathcal F\), then \(\mathcal F\) is a normal family. To consider the case \(m_1n_2 =m_2n_1\), a value distribution result is needed for the corresponding differential expression. Theorem 2. Let \(a, b\in\mathbb C\) such that \(a\neq 0\) and let \(f\) be a nonconstant meromorphic function. If \(n_1\), \(n_2\), \(m_1\), \(m_2\) are positive integers such that \(m_1n_2=m_2n_1\), then a function \(\psi(z)\) defined by \[ \psi(z):=(f(z))^{n_1}(f'(z))^{m_1}+ \frac{a}{(f(z))^{n_2}(f'(z))^{m_2}}-b \] has a finite zero. Theorem 3. Let \(\mathcal F\) be a family of meromorphic functions in a complex domain \(D\). Let \(a\), \(b\in\mathbb C\) such that \(a\neq 0\). Let \(m_1\), \(m_2\), \(n_1\), \(n_2\) be positive integers such that \(m_1 n_2 = m_2n_1\). If there exists a positive constant \(M\) such that \(|f(z)|\geqslant M\) for all \(f\in E_f\) whenever \(z\in E_f\), then \(\mathcal F\) is a normal family.