View source for 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 [[#References|[1]]]).

Theorem 1) If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c0269701.png" /> is regular and univalent in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c0269702.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c0269703.png" />), then the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c0269704.png" /> is entirely covered by the image of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c0269705.png" /> under the mapping of this function. On the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c0269706.png" /> there are points not belonging to the image only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c0269707.png" /> has the form:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c0269708.png" /></td> </tr></table>

Theorem 2) If a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c0269709.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697010.png" /> univalently, then the entire boundary of the image lies in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697011.png" />.

Theorem 3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697012.png" />, then at least one of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697013.png" /> points nearest to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697014.png" /> on the boundary of the image of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697015.png" /> under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697016.png" /> lying on any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697017.png" /> rays arising from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697018.png" /> at equal angles will have distance from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697019.png" /> not less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697020.png" />.

Theorem 4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697021.png" />, the image of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697022.png" /> under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697023.png" /> contains a set consisting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697024.png" /> open rectilinear segments with the sum of the lengths not less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697025.png" />, which emanate from the origin under equal angles of value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697026.png" />.

For functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697027.png" /> that in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697028.png" /> satisfy the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697030.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697031.png" /> that are regular and univalent in an annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697032.png" />, that map it into regions lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697033.png" />, and that map the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697034.png" /> into the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697035.png" />.

For the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697036.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697037.png" /> regular in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697038.png" />, there is no disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697040.png" />, that is entirely covered by the values of each of the functions in this class. For the functions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697041.png" /></td> </tr></table>

that are regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697042.png" />, each image of this disc entirely covers a certain segment of arbitrary given slope, containing the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697043.png" /> inside it and of length not less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697044.png" />, where the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697045.png" /> cannot be increased without imposing additional restrictions. In this class of functions, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697046.png" /> in the annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697047.png" />, each image of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697048.png" /> entirely covers the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697049.png" />, but does not always cover a greater disc with its centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697050.png" />.

In the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697051.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697052.png" />, regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697053.png" />, such that each value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697054.png" /> is taken at at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697055.png" /> points in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697056.png" /> one has an analogue of theorem 1 with corresponding disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697057.png" />. If moreover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697058.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697059.png" />, the corresponding discs will be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697060.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697061.png" />. Analogous results apply for functions that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697062.png" />-valent in the mean over a circle, over a region, etc. Covering theorem 3 can also be transferred to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697063.png" />.

See also Bloch's theorem in [[Bloch constant|Bloch constant]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Goluzin,   "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR></table>



====Comments====
Theorem 1 is also called Koebe's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026970/c02697065.png" />-theorem. Covering theorems are related to exceptional values (i.e. values not taken by a function, cf. [[Exceptional value|Exceptional value]]). Besides Bloch's theorem one should mention Landau's theorems, and the related constants; cf. [[Landau theorems|Landau theorems]].