By W. E. Kirwan, L. Zalcman

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**Additional info for Advances in complex function theory: Proceedings of seminars held at Maryland University 1973 74**

**Example text**

Set e, 89 (m-2). } (dmlm-I 2 JK ~-I + F - 1 > < 12 ~I of m. 5) ( lm) 89 i r(~-) m = ( 2 ~ ) ;~. = I 2/U~7 ( Thus ' i S 2 I I h e - 89 ~ dt 0 [ ~ e_ 89 2 dt. h Again U Finally, = since (sin 8)9y cos t--]"~(m- 2 ) y --~ : i = 7 (m-2), e -~

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 --*~. is an a b s o l u t e constant, Theorem 7 yields 1 s > On c o m p a r i n g gives the and when The this correct max{ result bound k < 4 m-l. 3, 2 - ~}. this inequality both when for to be v e r y k > 4m - I m ~ m 0.

L EMM A ber. 2. 2) z onto : w0 v D be any c o m p l e x nump roots in D, as imaginary part. by a c o n f o r m a l mapping t(~), and set F(~) Then U is p o s i t i v e = f{t(~)} and h a r m o n i c in = U + iV. 2). 13, that p. 179) f r o m the imaginary F(~) = ic ~=i where the Thus c are r e a l F(~) evidently tion f(z) Lemma 2. i). , 0, the I~I z [6], ~ = 1 to theorem Theorem < 1. the to equation v equation in the p l a n e f(z) the This that = w0 equa- proves and f(z) for a n y cannot p. F(~) Thus = t(~).