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By W. E. Kirwan, L. Zalcman

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F: S(G)--+S(H) is said to induce and a lift f*: A--+A r if there is an exten- which induces # . Other choices of lifts will induce isomorphisms of the form A ~ T2~(TIAT[I)T~ I __ for and T 1 E G, T 2 E H. O b v i o u s l y such h o m e o m o r p h i s m s *Supported in part by the National Science Foundation. f f* 57 preserve boundary to present a proof is a boundary G and A i8 too. ently less stringent #(A) is a b o u n d a r y famous influence which induces on the theory is by means equivalent transformation, statement A [2] which, although of fuehsian of their intersecting condition implies ~ important groups, in its own right not just Here way.

2. Proof of Theorem The proof 2. of Theorem 2 is l o n g a n d w e m u s t refer for it to o u r 29 forthcoming paper [3]. However I should like to m e n t i o n one key step. LEMMA and 3. Vl, If fl = Ul + ivl' v2 have a common f2 = u2 + iv2 tract D, are entire functions then Vl, v2 have the same sign in the whole plane. The p r o o f of L e m m a analytically related series development 3 is b a s e d on the for fl in t e r m s of obtained from points of b o t h fl f2 2. Rs examples Most satisfactorily 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. 6. K. Hayman, 7. K. Hayman and E. Ortiz, An upper bound for the largest zero of Eermite's functions with applications to subharmonic functions, to appear.

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