By G. M. Goluzin
This publication is predicated on lectures on geometric functionality idea given by means of the writer at Leningrad kingdom collage. It reviews univalent conformal mapping of easily and multiply hooked up domain names, conformal mapping of multiply hooked up domain names onto a disk, purposes of conformal mapping to the learn of inside and boundary homes of analytic services, and normal questions of a geometrical nature facing analytic services. the second one Russian version upon which this English translation relies differs from the 1st typically within the growth of 2 chapters and within the addition of an extended survey of more moderen advancements. The e-book is meant for readers who're already accustomed to the fundamentals of the speculation of capabilities of 1 advanced variable.
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Extra resources for Geometric Theory of Functions of a Complex Variable (Translations of Mathematical Monographs)
Sample text
Suppose that n (0) = 0 and ['(0) > 0 for 11, = 1, 2,' . '. ell, 1\2 ~ O. Then, it follows from the inequality ~ Ifn (q I ~f~ (0) ~- ... , that the numbers f~ (0) are bounded and that the functions f n(') are uniformly 1) Caratheodory [1912]; Bieberbach [19131. Marku~evi-:: [19361 showed rhar the condi tions of 'Theorem 1 are also nece ssary and sufficien r for con vergence in mean of z) I to n f ' (z), that is, necessary and sufficient for It'( lim ~ ~ n--+OJ B where r'n (z) I f' n (z) - I' (z) 12 da = is taken equal to 0 outside the domain B n" 0, rm .
Let :4. 1 and A* denote respectively the preimages of the under the mapping z = f«() in is contained in the disk S*. 1(' < 1. Let 8 1 >0 1'1 < 1. Then, :4. C A*, and denote the distance from S· - zo I ~ 8 1 for any point Zo E: S*. On the other hand, there exists an N> 0 such that, for n> N, we have Ifn«() - f«()1 < 8 1 on the boundary of A. since the sequence of the functions f n «() converges uniformly inside the disk \'1 < 1. Consequently, for every n > N, Rouch!? s theorem can be applied to the f«(o)' Let ( denote a positive number such that the circle ,,- (0 \ = ( is con tained in the disk 1'1 < 1.
Consequently, the functions ¢ f are regular in B and they satisfy in B the inequality l¢fz)1 < 1. ('). There fore, let us normalize ¢ z) in such a way that the point ¢ / zo) will, for given Zo € B and an arbitrary function f(z) E: 311, lie in the basic (regular) circular triangle with zero angles inscribed in the circle r< = triangle or in one of the adjacent triangles. We apply to the functions principle of condensation of analytic functions. Now consider an arbitrary sequence Ifn (z)\ of functions fn (z) € ¢ /z) the 311.