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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k0551701.png" /> be a function realizing a univalent [[Conformal mapping|conformal mapping]] of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k0551702.png" /> onto a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k0551703.png" /> bounded by a smooth closed Jordan curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k0551704.png" /> for which the angle of inclination <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k0551705.png" /> of the tangent to the real axis, as a function of the arc length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k0551706.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k0551707.png" />, satisfies a [[Hölder condition|Hölder condition]]:
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<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/k/k055/k055170/k0551708.png" /></td> </tr></table>
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Then the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k0551709.png" /> is continuous in the closed disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k05517010.png" />, and on the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k05517011.png" /> the following Hölder conditions hold, with the same exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k05517012.png" />:
+
Let  $  w = f ( z) $
 +
be a function realizing a univalent [[Conformal mapping|conformal mapping]] of the disc $  \{ {z \in \mathbf C } : {| z | < 1 } \} $
 +
onto a domain  $  D $
 +
bounded by a smooth closed Jordan curve  $  S $
 +
for which the angle of inclination  $  \theta ( l) $
 +
of the tangent to the real axis, as a function of the arc length  $  l $
 +
of  $  S $,
 +
satisfies a [[Hölder condition|Hölder condition]]:
  
<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/k/k055/k055170/k05517013.png" /></td> </tr></table>
+
$$
 +
| \theta ( l _ {1} ) - \theta ( l _ {2} ) |  \leq  \
 +
K  | l _ {1} - l _ {2} |  ^  \alpha  ,\ \
 +
0 < \alpha < 1 .
 +
$$
  
<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/k/k055/k055170/k05517014.png" /></td> </tr></table>
+
Then the derivative  $  f ^ { \prime } ( z) $
 +
is continuous in the closed disc  $  | z | \leq  1 $,
 +
and on the circle  $  | z | = 1 $
 +
the following Hölder conditions hold, with the same exponent  $  \alpha $:
  
Kellogg's theorem is a direct corollary of more general results by O.D. Kellogg (see [[#References|[1]]], [[#References|[2]]]) on the boundary behaviour of the partial derivatives of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k05517015.png" /> of a [[Harmonic function|harmonic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k05517016.png" /> that is a solution of the [[Dirichlet problem|Dirichlet problem]] for a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k05517017.png" /> in Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k05517018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k05517019.png" />, bounded by a sufficiently-smooth Lyapunov surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k05517020.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k05517021.png" />) or a Lyapunov curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k05517022.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k05517023.png" />; see [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]), where the given function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k05517024.png" /> is also assumed to be sufficiently smooth on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055170/k05517025.png" />.
+
$$
 +
| f ^ { \prime } ( e ^ {i \theta _ {1} } )
 +
- f ^ { \prime } ( e ^ {i \theta _ {2} } ) |
 +
\leq  K _ {1} | \theta _ {1} - \theta _ {2} |  ^  \alpha  ,
 +
$$
 +
 
 +
$$
 +
|  \mathop{\rm ln}  f ^ { \prime } ( e ^ {i \theta _ {1} } ) -
 +
\mathop{\rm ln}  f ^ { \prime } ( e ^ {i \theta _ {2} } ) |
 +
\leq  K _ {2} | \theta _ {1} - \theta _ {2} |  ^  \alpha  .
 +
$$
 +
 
 +
Kellogg's theorem is a direct corollary of more general results by O.D. Kellogg (see [[#References|[1]]], [[#References|[2]]]) on the boundary behaviour of the partial derivatives of orders $  r \leq  1 $
 +
of a [[Harmonic function|harmonic function]] $  u $
 +
that is a solution of the [[Dirichlet problem|Dirichlet problem]] for a domain $  D $
 +
in Euclidean space $  \mathbf R  ^ {n} $,  
 +
$  n \geq  2 $,  
 +
bounded by a sufficiently-smooth Lyapunov surface $  S $(
 +
for $  n \geq  3 $)  
 +
or a Lyapunov curve $  S $(
 +
for $  n = 2 $;  
 +
see [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]), where the given function $  f $
 +
is also assumed to be sufficiently smooth on the boundary $  S $.
  
 
Other results on the boundary behaviour of the derivative of the mapping function can be found in [[#References|[3]]], [[#References|[4]]].
 
Other results on the boundary behaviour of the derivative of the mapping function can be found in [[#References|[3]]], [[#References|[4]]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O.D. Kellogg,  "Harmonic functions and Green's integral"  ''Trans. Amer. Math. Soc.'' , '''13''' :  1  (1912)  pp. 109–132</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O.D. Kellogg,  "On the derivatives of harmonic functions on the boundary"  ''Trans. Amer. Math. Soc.'' , '''33''' :  2  (1931)  pp. 486–510</TD></TR><TR><TD valign="top">[3]</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">[4]</TD> <TD valign="top">  S.E. Warschawski,  "On differentiability at the boundary in conformal mapping"  ''Proc. Amer. Math. Soc.'' , '''12'''  (1961)  pp. 614–620</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O.D. Kellogg,  "Harmonic functions and Green's integral"  ''Trans. Amer. Math. Soc.'' , '''13''' :  1  (1912)  pp. 109–132</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O.D. Kellogg,  "On the derivatives of harmonic functions on the boundary"  ''Trans. Amer. Math. Soc.'' , '''33''' :  2  (1931)  pp. 486–510</TD></TR><TR><TD valign="top">[3]</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">[4]</TD> <TD valign="top">  S.E. Warschawski,  "On differentiability at the boundary in conformal mapping"  ''Proc. Amer. Math. Soc.'' , '''12'''  (1961)  pp. 614–620</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 22:14, 5 June 2020


Let $ w = f ( z) $ be a function realizing a univalent conformal mapping of the disc $ \{ {z \in \mathbf C } : {| z | < 1 } \} $ onto a domain $ D $ bounded by a smooth closed Jordan curve $ S $ for which the angle of inclination $ \theta ( l) $ of the tangent to the real axis, as a function of the arc length $ l $ of $ S $, satisfies a Hölder condition:

$$ | \theta ( l _ {1} ) - \theta ( l _ {2} ) | \leq \ K | l _ {1} - l _ {2} | ^ \alpha ,\ \ 0 < \alpha < 1 . $$

Then the derivative $ f ^ { \prime } ( z) $ is continuous in the closed disc $ | z | \leq 1 $, and on the circle $ | z | = 1 $ the following Hölder conditions hold, with the same exponent $ \alpha $:

$$ | f ^ { \prime } ( e ^ {i \theta _ {1} } ) - f ^ { \prime } ( e ^ {i \theta _ {2} } ) | \leq K _ {1} | \theta _ {1} - \theta _ {2} | ^ \alpha , $$

$$ | \mathop{\rm ln} f ^ { \prime } ( e ^ {i \theta _ {1} } ) - \mathop{\rm ln} f ^ { \prime } ( e ^ {i \theta _ {2} } ) | \leq K _ {2} | \theta _ {1} - \theta _ {2} | ^ \alpha . $$

Kellogg's theorem is a direct corollary of more general results by O.D. Kellogg (see [1], [2]) on the boundary behaviour of the partial derivatives of orders $ r \leq 1 $ of a harmonic function $ u $ that is a solution of the Dirichlet problem for a domain $ D $ in Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, bounded by a sufficiently-smooth Lyapunov surface $ S $( for $ n \geq 3 $) or a Lyapunov curve $ S $( for $ n = 2 $; see Lyapunov surfaces and curves), where the given function $ f $ is also assumed to be sufficiently smooth on the boundary $ S $.

Other results on the boundary behaviour of the derivative of the mapping function can be found in [3], [4].

References

[1] O.D. Kellogg, "Harmonic functions and Green's integral" Trans. Amer. Math. Soc. , 13 : 1 (1912) pp. 109–132
[2] O.D. Kellogg, "On the derivatives of harmonic functions on the boundary" Trans. Amer. Math. Soc. , 33 : 2 (1931) pp. 486–510
[3] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[4] S.E. Warschawski, "On differentiability at the boundary in conformal mapping" Proc. Amer. Math. Soc. , 12 (1961) pp. 614–620

Comments

See also Conformal mapping, boundary properties of a.

See [a1], p.15, for a similar problem.

References

[a1] M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)
How to Cite This Entry:
Kellogg theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kellogg_theorem&oldid=15965
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article