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)