Normal families of holomorphic functions (Q1425781)

A classical result of \textit{P. Montel} [Ann. de l'Éc. Norm. (3) 29, 487-535. (1912; JFM 43.0509.05)] asserts that a family \(\mathcal F\) of holomorphic functions in a domain \(D\) in \(\mathbb C\) which omits two complex values is a normal family. This result was improved by \textit{C. Miranda} in [Bull. Soc. Math. Fr. 63, 185--196 (1935; Zbl 0013.27202)] who proved that if \(\mathcal F\) is a family of holomorphic functions in a domain \(D\); \(a\) and \(b\) are complex numbers with \(b\neq 0\) and \(k\) is a positive integer, and for every \(f\in \mathcal F\), \(f\neq a\) and \(f^{(k)}\neq b\) in \(D\), then \(\mathcal F\) is normal. In this very interesting paper, the authors extend Miranda's theorem. Indeed, Theorem~1 of the paper asserts that if \(k\) is a positive integer, \(k\neq 2\), \(\mathcal F\) is a family of holomorphic functions in a domain \(D\subset \mathbb C\), \(h\) is a positive number, and \(a\) is a non-vanishing holomorphic function in \(D\) then \(\mathcal F\) is normal if all the functions \(f\in \mathcal F\) satisfy the following conditions: (i) All the zeros of \(f\) have multiplicity at least \(k\); (ii) \(f(z)=0 \Rightarrow f^{(k)} (z)=a(z)\); and (iii) \(f^{(k)} (z)=a(z) \Rightarrow | f^{(k+1)}(z)| \leq h\). The authors also show that this result is not valid for \(k=2\) even if the function \(a\) is a non-zero constant, but prove that it is true if the function \(a\) is a non-zero constant and condition (iii) is replaced by (iii') \(f'' (z)=a \Rightarrow 0<| f'''(z)| \leq h\). In Theorem~3, the authors also obtain a result valid for \(k=2\) and a holomorphic function \(a\).