### A Criterion for a Meromorphic Function to be Normal

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For a ${C}^{1}$-function $f$ on the unit ball $\mathbb{B}\subset {\u2102}^{n}$ we define the Bloch norm by ${\parallel f\parallel}_{\U0001d505}=sup\parallel \tilde{d}f\parallel ,$ where $\tilde{d}f$ is the invariant derivative of $f,$ and then show that $${\parallel f\parallel}_{\U0001d505}=\underset{\genfrac{}{}{0pt}{}{z,w\in \mathbb{B}}{z\ne w}}{sup}{(1-|z|}^{2}{)}^{1/2}{(1-|w|}^{2}{)}^{1/2}\frac{\left|f\right(z)-f(w\left)\right|}{|w-{P}_{w}z-{s}_{w}{Q}_{w}z|}.$$

We prove a normality criterion for a family of meromorphic functions having multiple zeros which involves sharing of a non-zero value by the product of functions and their linear differential polynomials.

For sequences of rational functions, analytic in some domain, a theorem of Montel’s type is proved. As an application, sequences of rational functions of the best ${L}_{p}$-approximation with an unbounded number of finite poles are considered.

In this paper, the definition of the derivative of meromorphic functions is extended to holomorphic maps from a plane domain into the complex projective space. We then use it to study the normality criteria for families of holomorphic maps. The results obtained generalize and improve Schwick's theorem for normal families.

The little Bloch space, B0, is the space of all holomorphic functions f on the unit disk such that limlzl→1 lf'(z)l (1- lzl2) = 0. Finite Blaschke products are clearly in B0, but examples of infinite products in B0 are more difficult to obtain (there are now several constructions due to Sarason, Stephenson and the author, among others). Stephenson has asked whether B0 contains an infinite, indestructible Blaschke product, i.e., a Blaschke product B so that (B(z) - a)/(1 - âB(z)), is also a Blaschke...