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

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

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But 1 and We give where Then 2 do some more is a p o l y n o m i a l is t r a n s c e n d e n t a l . i). harmonic the set n IVll vI of the i n e q u a l i t y We n o t e that functions v ~ 0 v in Rm m u s t have at least two c o m p o n e n t s . Example a2 + b2 = 1 4. Consider va positive numbers a, b such that and set va Then by the c o n j u g a t e s of T h e o r e m s to h i g h e r d i m e n s i o n s . Vl(X,y,z) e x a c t l y two tracts. degree in f2 [3]. Example have z f2 and in such e l e m e n t s .

6) becomes d2u + {~ + ~ - ! 14) d2u m -dt- 2 : h It turns out [14, p. 106, - 0. is the one which tends to zero as is the largest S = zero of this solution. 2)] Further ~ e_ 89 2 dt. h that u = e- i/%t2 H (t//~), 50 where H~(t) gives is Hermite's for large t Now Sturm's comparison theorem [14, p. 132] h v-f m where function. 85575. We can also obtain a good convex minorant. i e- %h2 2~ h = i I ~ e-%t2(l (2/[~7 We note that + t~ ) dt h Thus e,h We set sume 2~+i = x, and assume p > i, so that x > 3.

H. in on the Thus the set in each domain has at least D9 k and so 33 tracts. 4, u(z) has and since the lower order of u(z) coincides definition of if Thus, by T h e o r e m of the lower order k ~ 2. in k we deduce Ahlfors' 1 P ~ 7 also yields and which have at least Rm The r e s t r i c t i o n func t i o n in any o r d e r if theorem k = i. m both greater than one, let Rm 1 k > 2 k to What is is e s s e n t i a l has at least subject tracts. since one tract and 0 < I < ~. h. m a y have 4 we have 1 = ~ k.

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