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By W. E. Kirwan, L. Zalcman

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ISBN-13: 9783540075486

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16) This is e q u i v a l e n t > 2(I-S), 0 < S < I. 13). 15) $m(S) 2 < m < ~. 16) are Also that W(S) Theorem is d e f i n e d 6, C o r o l l a r y we h a v e THEOREM 7. We h a v e with s If m = 2 k = 2, sharp > we d e d u c e s as is s h o w n by the a b o v e In o t h e r cases d e r of m a g n i t u d e . 11), , and ~ r189 We r e c a l l m > m0, for R TM 2 ~ k < ~, and which are not as above. Thus 2 ~ m < ~. is a c t u a l l y this has sharp result sharp. If is a l s o 2 tracts and order 1. but the right or- give that 1 JK ~ < ~ ~ m, Thus Lemma S notation for s u(x) from of to Cm(~), we o b t a i n in r as in T h e o r e m leads functions it f o l l o w s u(S) where convex m0 as k --*~.

55 (seventh printing), May 1968. 2. C. Bandle, Konstruktion isoperimetrischen Ungleichungen der mathematischen Physik aus solchen der Geometric, Comment. Math. Helv. 46 (1971), 182-213. 3. A. J. K. Hayman and U. Kuran, A characterization of harmonic polynomials in the plane, to appear. 4. A. Dinghas, Das Denjoy-Carlemansche Problem fur harmonische Funktionen in E ~ Det Kal Norske Videnskabers Selskabs Skrifter 1962, no. 7, 13 p. 5. S. K. Hayman, Eigenvalue inequalities for the Dirichlet problem on the sphere and the growth of subharmonic functions, to appear.

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.

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Advances in complex function theory: Proceedings of seminars held at Maryland University 1973 74 by W. E. Kirwan, L. Zalcman

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