# Luzin theorem

Luzin's theorem in the theory of functions of a complex variable (the local principle of finite area) is a result of N.N. Luzin that reveals a connection between the boundary properties of an analytic function in the unit disc and the metric of the Riemann surface onto which it maps the disc (see , ).

Let $V$ be any domain inside the unit disc $D= \{ {z } : {| z | < 1 } \}$ of the complex $z$-plane adjoining an arc $\sigma$ of the unit circle $\Gamma = \{ {z } : {| z | = 1 } \}$, and let

$$w = f ( z) = \sum _ { k=0 } ^ \infty c _ {k} z ^ {k}$$

be a regular analytic function in $D$. If the area of the Riemann surface that is the image of $V$ under the mapping $w = f ( z)$ is finite, then the series

$$\sum _ { k=0 } ^ \infty c _ {k} z ^ {k}$$

converges almost-everywhere on $\sigma$.

In connection with this theorem Luzin made a conjecture, also known as Luzin's problem. A point $e ^ {i \theta _ {0} } \in \Gamma$ is called a Luzin point of the function $w = f ( z)$ if $w = f ( z)$ maps every disc touching $\Gamma$ from the inside at $e ^ {i \theta _ {0} }$ onto a domain of infinite area on the Riemann surface of $w = f ( z)$. The Luzin conjecture is that there are bounded analytic functions in $D$ such that every point of $\Gamma$ is a Luzin point for them. The Luzin conjecture was first confirmed completely in 1955 (see ).

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How to Cite This Entry:
Luzin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Luzin_theorem&oldid=51147
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article