Kellogg theorem

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)