Covering theorems

Theorems for various classes of regular functions that establish certain properties of sets that are entirely contained in the range of values of each function of the corresponding class. Below some basic covering theorems are presented (see also [1]).

Theorem 1) If a function $ w = f( z) = z + a _ {2} z ^ {2} + \dots $ is regular and univalent in the disc $ | z | < 1 $( i.e. $ f \in S $), then the disc $ | w | < 1/4 $ is entirely covered by the image of the disc $ | z | < 1 $ under the mapping of this function. On the circle $ | w | = 1/4 $ there are points not belonging to the image only if $ f $ has the form:

$$ f( z) = \frac{z}{( 1+ e ^ {i \alpha } z) ^ {2} } ,\ \ 0 \leq \alpha < 2 \pi . $$

Theorem 2) If a meromorphic function $ w = F( \zeta ) = \zeta + \alpha _ {0} + \alpha _ {1} / \zeta + \dots $ maps $ | \zeta | > 1 $ univalently, then the entire boundary of the image lies in the disc $ | w - \alpha _ {0} | \leq 2 $.

Theorem 3) If $ f \in S $, then at least one of the $ n $ points nearest to $ w = 0 $ on the boundary of the image of the disc $ | z | < 1 $ under the mapping $ w = f( z) $ lying on any $ n $ rays arising from $ w = 0 $ at equal angles will have distance from $ w = 0 $ not less than $ ( 1/4) ^ {1/n} $.

Theorem 4) If $ f \in S $, the image of the disc $ | z | < 1 $ under the mapping $ w = f( z) $ contains a set consisting of $ n $ open rectilinear segments with the sum of the lengths not less than $ n $, which emanate from the origin under equal angles of value $ 2 \pi / n $.

For functions $ f \in S $ that in the disc $ | z | < 1 $ satisfy the inequality $ | f( z) | < M $, $ M \geq 1 $, there are covering theorems analogous to theorems 1 and 3 (with corresponding constants). The covering theorems 1 and 3 can also be transferred to the class of functions $ w = f( z) $ that are regular and univalent in an annulus $ 1 < | z | < r $, that map it into regions lying in $ | w | > 1 $, and that map the circle $ | z | = 1 $ into the circle $ | w | = 1 $.

For the class $ R $ of functions $ w = f( z) = z + a _ {2} z ^ {2} + \dots $ regular in the disc $ | z | < 1 $, there is no disc $ | w | < \rho $, $ \rho > 0 $, that is entirely covered by the values of each of the functions in this class. For the functions

$$ w = F( z) = z ^ {q} + a _ {2} z ^ {q+} 1 + \dots ,\ \ q \geq 1, $$

that are regular in $ | z | < 1 $, each image of this disc entirely covers a certain segment of arbitrary given slope, containing the point $ w = 0 $ inside it and of length not less than $ A = 8 \pi ^ {2} / \Gamma ^ { 4 } ( 1/4) = 0.45 \dots $, where the number $ A $ cannot be increased without imposing additional restrictions. In this class of functions, if $ F( z) \neq 0 $ in the annulus $ 0 < | z | < 1 $, each image of the disc $ | z | < 1 $ entirely covers the disc $ | w | < 1/16 $, but does not always cover a greater disc with its centre at $ w = 0 $.

In the class $ S _ {p} $ of functions $ f( z) = z ^ {p} ( 1 + a _ {1} z + a _ {2} z ^ {2} + \dots ) $, regular in $ | z | < 1 $, such that each value $ w $ is taken at at most $ p $ points in the disc $ | z | < 1 $ one has an analogue of theorem 1 with corresponding disc $ | w | < 1/2 ^ {p+} 1 $. If moreover $ a _ {1} = \dots = a _ {p-} 1 = 0 $ or $ a _ {1} = \dots = a _ {p} = 0 $, the corresponding discs will be $ | w | < 1/4 $ or $ | w | < 1/2 $. Analogous results apply for functions that are $ p $- valent in the mean over a circle, over a region, etc. Covering theorem 3 can also be transferred to the class $ S _ {p} $.

See also Bloch's theorem in Bloch constant.

References

[1]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)

Comments

Theorem 1 is also called Koebe's $ {1 / 4 } $- theorem. Covering theorems are related to exceptional values (i.e. values not taken by a function, cf. Exceptional value). Besides Bloch's theorem one should mention Landau's theorems, and the related constants; cf. Landau theorems.