Normal Families by Joel L. Schiff (auth.)

By Joel L. Schiff (auth.)

A ebook just about common households greater than sixty years after the booklet of Montel's treatise Ler;ons sur les familles normales de fonc­ tions analytiques et leurs purposes is unquestionably lengthy past due. yet, in a feeling, it's nearly untimely, as lots modern paintings continues to be being produced. To misquote Dickens, this can be the simplest of occasions, this is often the worst of occasions. The intervening years have visible advancements on a huge entrance, lots of that are taken up during this quantity. A unified therapy of the classical thought is additionally awarded, with a few try out made to maintain its classical flavour. due to the fact that its inception early this century the concept of a regular kin has performed a valuable function within the improvement of advanced functionality concept. actually, it's a thought mendacity on the very center of the topic, weaving a line of concept via Picard's theorems, Schottky's theorem, and the Riemann mapping theorem, to many sleek effects on meromorphic services through the Bloch precept. it truly is this latter that has supplied substantial impetus through the years to the research of ordinary households, and keeps to function a guiding hand to destiny paintings. essentially, it asserts family members of analytic (meromorphic) capabilities outlined by way of a specific estate, P, might be an ordinary kinfolk if a whole (meromorphic in

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Proof. As Un} is normal and Ifni:::; M, say, there is at least one limit function f(z), and If I :::; M in U. If the whole sequence Un} does not converge normally to f, there must be another limit function g, with f(a) ig(a) for some a E U. But f and 9 are bounded in U with f(a n ) -g(an ) = 0, which, by the preceding remarks, implies f == g, a contradiction, and the result is proved. 2, the uniform boundedness of Un} may be replaced by the Nevanlinna condition 0:::; r < 1, for n = 1,2,3, ... (Nevanlinna and Nevanlinna [1922]), or indeed by mer, fn) < ¢(r), where ¢ is an increasing function of r, as r -+ 1.

Then there is a sequence {fn} ~ F for which f~(zo) --+ A. Since F is a uniformly bounded family, and hence normal, there is a subsequence {fnk} which converges locally uniformly in n to an analytic function F that satisfies IF{z)1 ~ 1 in n, F{zo) = O. By Cauchy's formula, f~k (zo) --+ F'(zo), implying 0 < a: ~ F'(Zo) = A < 00. 1, and so F E F, thereby solving our extremal problem. 50 2. Analytic Functions Actually, F is a compact family, and extremal problems for compact normal families, such as considered above, can be treated in far greater generality, but by similar methods (cf.

AN, then N Llog i=1 N I~'I 2)og ~ = ti i=1 t 1271' log ~ dn(t, 0) t 0 [log ~n(t, 0)] r _1271' n(t, 0) dlog ~ = 1 r n(t, 0) dt, tot too where we have applied integration by parts to the Stieltjes integral. 5) becomes 11271' 27l' 0 log IJ(rei¢)1 d4> = log 11(0)1 Setting log+ X t = max(logx,O) t = t. J og 1 I' i=1 ai 1) + N (r, f (log+ 0= 0), so that log x = log+ X -log+ ~ for x> 0, we then obtain - 1 1271' log IJ(rei¢)1 d4> 27l' 0 = N(r, 1). 8. , m(r,1) + N(r, I) = m(r, -1) + N(r, -1 ) + log 11(0)1· The characteristic function of I (z) is then given by the quantity T(r, I) = mer, I) + N(r, I).

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