Advances in Complex Function Theory: Proceedings of Seminars by Carlos A. Berenstein (auth.), William E. Kirwan, Lawrence

By Carlos A. Berenstein (auth.), William E. Kirwan, Lawrence Zalcman (eds.)

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Extra info for Advances in Complex Function Theory: Proceedings of Seminars Held at Maryland University, 1973/74

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U P Up+ 1 = I, We see that 2k 2 so that Up+l(Xl,X2,''',Xp+ I) where has + 1 for We obtain the as nearly equal as pos- 38 sible. 1). 3). 2) Then setting to of the p, where p = i, q = 0 q = n- true p, equal q for and If m > 2, l, k = 2n. and k = 2q+l : 2 n, p = 2, In v i e w o f t h e a r i t h m e t i c - g e o m e t r ~ mean theorem, 21n+m-21 m-I is ,divisible b y x = m- n+m-2 m-I suppose I m---m-~--; ~ 1. inequality k so t h a t : e x > 2. 4). Equality bound if for k we set S : 1 e, q m-i ~ : holds Then 2(x-e)(m-l)8(x+B) (m-l)a - ~ We note = log 2 + (m-1){log )8 8 )a } m - i (i + ~ x + 8 log(l - ~) + m log(1 + ~) that log(l+t) t - [1 t 2 + ~ t 3 - - = - > t-t 2 if Thus log k ~8(~+8) > log 2 + (m-l) = ~8 = ~ ( I - ~ ) < 1 is q = 0.

The lower order If r n sin and lower ' 1 is p r e c i s e l y inequality Heins the ~ k is e v e n he, where u = examples now when extend if pDints u k 8 is sharp. u : r 89 is odd, since is h a r m o n i c (gig) = x + iy, For by 8 + 2~. except is e v i d e n t l y tracts, where k > i, v : I m ( x + iy) n = polynomials 8 + i sin e) k. k 1 ~ ~ ~ k. h. However, Heins' theorem lies m u c h deeper than the corresponding 32 result for harmonic polynomials. Denjoy conjecture proved by Ahlfors totic values of an entire k Thus there exist paths f(z)--+a Since the values ferent 9 a9 in the open plane as z-~ F are disjoint paths going from zV lies in from 0 to z We complete in common.

We dis- since in t h e p > I, p : i. since latter. 6). and negative 1 r is 2e). to our equation corresponds y = t = cos e , which = and that seem to be an elementary case 0 < 8 < ~ u = i, p(S) while Sturm's method the s i (~ _ 1 ~ ~- s i n not Since 1 ~ ~ k, find that i, = does we can apply tinguish ~(i-~) ~ direction. yields we p - ~ S In other in t h i s as 8 increases from 0 In f a c t , 0 d-~ sin2e = ~ < 0 < ~. 1o) L--: e + 1 ~ I < If m = 3, so t h a t (p+u)2 say. , [14, function p. 127, of o r d e r formula k.

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