View source for Landau theorems

Theorems for functions regular in a disc, which establish some connections between geometrical properties of the conformal mapping that is induced by these functions and the initial coefficients of the power series that represent them.

In 1904 E. Landau showed [[#References|[1]]] that if a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l0574301.png" /> is regular in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l0574302.png" /> and does not take the values 0 and 1 in it, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l0574303.png" /> is bounded from above by a positive constant that depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l0574304.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l0574305.png" />. In 1905 C. Carathéodory established that the role of extremal function in this theorem is played by a [[Modular function|modular function]]. These results of Landau and Carathéodory are known in the form of the following theorem.

The Landau–Carathéodory theorem. If the function

<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/l/l057/l057430/l0574306.png" /></td> </tr></table>

is regular and does not take the values 0 and 1 in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l0574307.png" />, then

<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/l/l057/l057430/l0574308.png" /></td> </tr></table>

here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l0574309.png" /> is a branch of the function inverse to the classical modular function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743010.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743011.png" /> of fractional-linear transformations

<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/l/l057/l057430/l05743012.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743014.png" /> are odd numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743016.png" /> are even numbers. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743017.png" /> maps the [[Fundamental domain|fundamental domain]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743019.png" />:

<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/l/l057/l057430/l05743020.png" /></td> </tr></table>

( <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743021.png" /> is obtained by adjoining to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743022.png" /> that part of the boundary for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743023.png" />) onto the whole extended <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743024.png" />-plane in such a way that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743027.png" />. For each value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743028.png" /> the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743029.png" /> has one and only one solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743030.png" /> belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743031.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743032.png" /> in the Landau–Carathéodory theorem can be understood as the branch of the inverse function that maps the extended <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743033.png" />-plane onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743034.png" />.

The example of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743035.png" />, regular in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743036.png" /> and not equal to 0 or 1 for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743037.png" />, shows that the Landau–Carathéodory theorem cannot be improved. The Landau–Carathéodory theorem implies the [[Picard theorem|Picard theorem]] on values that cannot be taken by entire functions.

Landau found the exact value of the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743038.png" /> that occurs in the following formulation of the Cauchy theorem on inverse functions. Suppose that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743039.png" /> is regular in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743040.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743043.png" /> in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743044.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743045.png" />; then there is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743046.png" /> such that the inverse function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743047.png" />, which vanishes at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743048.png" />, is regular in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743050.png" /> in this disc. Landau established that

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The extremal function attaining this bound is

<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/l/l057/l057430/l05743052.png" /></td> </tr></table>

This function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743053.png" /> is extremal in the following theorem of Landau. If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743054.png" /> satisfies the conditions mentioned above, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743055.png" /> is single-valued in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743056.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743057.png" />.

Landau has also established a number of [[Covering theorems|covering theorems]] in the theory of conformal mapping that establish the existence of and bounds for the corresponding constants. One of them is given below. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743058.png" /> be the class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743059.png" /> regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743060.png" /> and normalized by the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743062.png" />. Bloch's theorem (see [[Bloch constant|Bloch constant]]) implies the following theorem of Landau: There is an absolute constant

<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/l/l057/l057430/l05743063.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743064.png" /> is the radius of the largest disc in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743065.png" />-plane that is entirely covered by the image of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743066.png" /> under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743067.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743068.png" /> is Bloch's constant. The constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743069.png" /> is called Landau's constant. The following bounds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743070.png" /> are known (see [[#References|[5]]], [[#References|[8]]]): <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743071.png" />. The Picard theorem again follows from this theorem.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Landau,   "Ueber eine Verallgemeinerung des Picardschen Satzes"  ''Sitzungsber. Preuss. Akad. Wiss.'' , '''38'''  (1904)  pp. 1118–1133</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Landau,   D. Gaier,   "Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie" , Springer, reprint  (1986)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Landau,   "Zum Koebeschen Verzerrungssatz"  ''Rend. Circ. Mat. Palermo'' , '''46'''  (1922)  pp. 347–348</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E. Landau,   "Der Picard–Schottkysche Satz und die Blochsche Konstante"  ''Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl.'' , '''32'''  (1926)  pp. 467–474</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E. Landau,   "Ueber die Blochsche Konstante und zwei verwandte Weltkonstanten"  ''Math. Z.'' , '''30'''  (1929)  pp. 608–634</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E. Landau,   "Ansgewählte Kapitel der Funktionentheorie"  ''Trudy Tbilis. Mat. Inst. Akad. Nauk. SSSR'' , '''8'''  (1940)  pp. 23–68</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S. Stoilov,   "The theory of functions of a complex variable" , '''1–2''' , Moscow  (1962)  (In Russian; translated from Rumanian)</TD></TR><TR><TD valign="top">[8]</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><TR><TD valign="top">[9]</TD> <TD valign="top">  G. Valiron,   "Les fonctions analytiques" , '''Paris'''  (1954)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A. Bermant,   "Dilatation of a modular function and reconstruction problems"  ''Mat. Sb.'' , '''15''' :  2  (1944)  pp. 285–318  (In Russian)  (French abstract)</TD></TR></table>



====Comments====
It is now (1989) known that the Landau constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057430/l05743072.png" /> satisfies

<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/l/l057/l057430/l05743073.png" /></td> </tr></table>

The upper bound is over half a century old and is due to R. Robinson and, independently, H. Rademacher [[#References|[a1]]]. See [[#References|[a2]]] for more detailed information on these and related questions.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Rademacher,   "On the Bloch–Landau constant"  ''Amer. J. Math.'' , '''65'''  (1943)  pp. 387–390</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.D. Minda,   "Bloch constants"  ''J. d'Anal. Math.'' , '''41'''  (1982)  pp. 54–84</TD></TR></table>