By Qitai Zhuang

This publication offers in a transparent and systematic demeanour the overall idea of ordinary households, quasi-normal households and Qm-normal households of meromorphic features, and diverse purposes. a lot of this booklet includes result of the author's examine, between them is the proposal of Q_{m}-normality including the classical notions of normality and quasi-normality brought through Montel as specific circumstances. during this booklet, the suggestion of closed households of meromorphic capabilities can be brought. moreover, functions about the lifestyles of the answer of assorted extremal difficulties for definite sessions of univalent or multivalent features is usually discovered.

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7. Let Hz) be a transcendental entire function. Then given any curve z=qJ(t) satisfying the condition (C), we can find a point Zo of the domain 1/2< Izl<4, such that the curve z=zoqJ(t) is a Julia curve of Hz). 1. Theorem 2. 8. Let D be domain and M, 6 two positive numbers. Let $T be the family of the functions Hz) satisfying the following conditions: 1 0 Hz) is holomorphic in D. 2 0 There are two finite values a (f), b (f) such that la(f) 1< M, Ib(f) 1< M, la(f)-b(f) 1< 0 and that each of the equations f (z) = a (f), f (z) = b (f) has no root in D.

And that the circle Izl<1 is a (R(£),11(O;v)-filling circle of f(z). Proof. Assume that such a number A. does not exist. 41), such that there do not exist two positive numbers R, 11 such that R-11>I/n and that the circle Izl<1 is a (R,11;v)-filling circle of fn (z), it fortiori, the circle Iz I <1 is not a (3,1; v)-filling circle of fn(z). 16, the family fn(z)(n=1,2,···) is normal in the circle Izl

Sr. This number A has then the required property. Given a number r(O