View source for Kellogg theorem

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]]:

<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>

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" />:

<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>

<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>

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" />.

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

====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>



====Comments====
See also [[Conformal mapping, boundary properties of a|Conformal mapping, boundary properties of a]].

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Brélot,   "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)</TD></TR></table>