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Advances in complex function theory: Proceedings of seminars by W. E. Kirwan, L. Zalcman

By W. E. Kirwan, L. Zalcman

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Milin, Univalent Functions and Orthonormal "Nauka", Moscow, 1971 (in Russian). Functions, Cambridge University Systems, Press, Izdat. K, i. PLANE RESULTS, % R e c e n t l y If v both is h a r m o n i c connected, a positive result Kuran raised in t h e p l a n e is it the HAYMAN and the case that previously [i0]. the following sets v v > 0, is l i n e a r ? A complete question. v < 0 He h a d answer are obtained is c o n t a i n e d in THEOREM k Let i. v be the number nite of harmonic of components if and only the degree be a real if v, v of the set is a polynomial.

The polynomials nomials. 4), ultraspherical is discussed poly- on p. 117 of [14] on p. 5). The last p. 80. We deduce LEMMA 5. u = U(Xl,X2,''',x m) Let nomial of degree xI in m+l x~) 89 R = ([v=l and set v v ~ = be a homogeneous x m. to Let t = Xm+i/R. be a positive integer Then if ~k~(1) ~ ~k (t)u(x), is a homogeneous k harmonic poly- 1 1 ~ + ~ (m-l] = harmonic p o l y n o m i a l of degree s + k in xI to Xm+ 1 9 Suppose first that ~i k degree in k is even. t 2 = x ~+I/R2 Rkp~l)(t ) is clearly Similarly a homogeneous if k Rk_(~ r k )(t) where ~ homogeneous is a polynomial polynomial It remains x = Xm+ I, : = is a polynomial of Thus 89 cv x2vm+IRk-2v ~=0 of degree k in xI to Xm+ I.

U(x) and I = Z = n. h. by u = Ivl, Evidently components make ~ io~ + B(r) log r rlim ~ the of d e g r e e are order u > 0. h. , and the result the left of [8]. THEOREM If 4. 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.

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