By W. K. Hayman

The category of multivalent services is a crucial one in complicated research. They happen for instance within the evidence of De Branges' Theorem, which in 1985 settled the long-standing Bieberbach conjecture. the second one variation of Professor Hayman's celebrated booklet incorporates a complete and self-contained facts of this outcome, with a brand new bankruptcy dedicated to it. one other new bankruptcy bargains with coefficient transformations. The textual content has been up-to-date in numerous alternative ways, with fresh theorems of Baernstein and Pommerenke on univalent features of constrained progress, and an account of the speculation of suggest p-valent services. additionally, the various unique proofs were simplified. every one bankruptcy comprises examples and workouts of various levels of hassle designed either to check realizing and illustrate the cloth.

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Helv. R. T. Yau; of manifolds lemma in several 55(1980), Compact with non-positive complex vari- 547-558. group actions curvature, and the topology Topology 18(1979), 361- 380. [26] T. Sunada; Holomorphic mappings symmetric bounded domain, [27] T. Sunada; 51(1979), Rigidity 297-307. into a compact quotient Nagoya Math. J. 64(1976), of certain harmonic mappings, of 159-175. Invent. Math. 36 [28] T. Sunada; Tchebotarev~s in a compact locally density symmetric theorem space for closed of negative geodesics curvature, preprint.

Then, ~ - JT(~). d. existence problem. 2. Theorem. moreover ( [12] ) Suppose T < T*. Then, JT(UT) Proof. 1 and such that ; u ~ BV(~) } . sequence {JT(Uj) } is bounded. {uj} of BV(~). 4) we can d e d u c e that U s i n g the r e s u l t s {u~}j w h i c h c o n v e r g e s F r o m lower topology ( see [3], {uj} is b o u n d e d [ii] where For existence inequality for Iy

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