By Qitai Zhuang
This ebook offers in a transparent and systematic demeanour the final conception of standard households, quasi-normal households and Qm-normal households of meromorphic capabilities, and numerous functions. a lot of this ebook comprises result of the author's examine, between them is the thought of Qm-normality which include the classical notions of normality and quasi-normality brought through Montel as specific circumstances. during this ebook, the proposal of closed households of meromorphic services is usually brought. moreover, purposes in regards to the lifestyles of the answer of varied extremal difficulties for definite sessions of univalent or multivalent capabilities is also stumbled on.
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Extra resources for Normal Families of Meromorphic Functions
JJr, we have _ If'(z)1 , a(z,f)-l+ If(z)12~ If (z)l· 2 CRITERIONS OF NORMALITY OF FAMILIES OF HOLOMORPHIC FUNCTIONS AND APIPLICATIONS 2. 1. 1. Let D be a domain and a, b(a#b) two complex numbers. J7 be the family of the functions fCz) holomorphic in D and such that each of the equations f (z) = a, f (z) = b has no root in D. J7 is normal in D. This theorem is due to Montel . Pr oof. Consider first the case a = D, b = 1. Let Zo be a point of D and r: 1zZo 1
Let $T be the family of the functions f(z) satisfying the following conditions: 10 f(z) is holomorphic in D. 2 0 There are two finite values a(f) and b (O=FO such that Ia (f) I < and that each of the equations M, <5 < Ib(I) I < M (2. 39) = b (f) has no root in D. Then the family $T is normal in D. Proof. Let fn (z)(n = 1,2, ••• ) be a sequence of functions of the family $T. Set a. ), b. (z)-a. (n b. = 1,2, ... ). Since each of the equations 9 • (z) = 0, g. (z)(n= 1,2,···) a subsequence g .. (z)(k=I,2,···) such that as k~+oo, verges locally uniformly to a holomorphic function or the constant 00 g ..
Evidently, when n is sufficiently large, anCr , consequently and If(zj('» < (J(zo,f) ,f(z/'» I + e}(lzj(') - ~ If(zj('» zol + This inequality is incompatible with ,f(zo) I IZ2(') - zol) + ~ If(zo) ,f(Z2('» I (m + e)lzj(') - Z2(') I O. 35), and so we get a contradiction. 31). Theorem 1. 6. Let $T be a family of merom orphic functions in a domain D. 38) is locally uniformly bounded in D. This theorem is due to Marty [23J. Proof. 7 • is locally uniformly bounded in D. Let Zo be a point of D. 7. 39) in 1.
Normal Families of Meromorphic Functions by Qitai Zhuang