A Note on the Removability of Totally Disconnected Sets for Analytic Functions
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BRIEF COMMUNICATIONS A NOTE ON THE REMOVABILITY OF TOTALLY DISCONNECTED SETS FOR ANALYTIC FUNCTIONS A. V. Pokrovskii
UDC 517.537.38
We prove that each totally disconnected closed subset E of a domain G in the complex plane is removable for analytic functions f (z) defined in G \ E and such that, for any point z0 2 E, the real or imaginary part of f (z) vanishes at z0 .
Let G be a domain in the complex plane C, let E be a totally disconnected closed subset of G, and let f (z) = u(z) + iv(z) be an analytic function in G \ E (u(z) = Re f (z)
and
v(z) = Im f (z)).
In [1], Fedorov proved that if f (z) is continuously extended from G \ E to G and u(z) vanishes on E, then this extension is an analytic function in G. Ischanov [2] (see also [3, 4]) generalized this result as follows: If u(z) vanishes on E, then f (z) can be analytically extended from G \ E to G. The aim of the present paper is to prove the following generalization of the outlined results: Theorem 1. Let G be a domain in C, let E a totally disconnected closed subset of G, and let f (z) = u(z) + iv(z) be an analytic function in G \ E such that, for any z0 2 E, either u(z) ! 0 or v(z) ! 0 as z ! z0 , z 2 G \ E. Then the function f (z) can be analytically extended from G \ E to G. Proof. Assume that the conditions of Theorem 1 are satisfied and that z0 2 E. Then we have one of the following cases: (a) the function f (z) is bounded in the intersection of G \ E with some neighborhood of the point z0 ; (b) u(z) ! 0 as z ! z0 , z 2 G \ E, and
z!z0 ,z2G\E
(c) v(z) ! 0 as z ! z0 , z 2 G \ E, and
z!z0 ,z2G\E
lim sup |v(z)| = +1;
lim sup |u(z)| = +1.
Consider the case (a). Then there is r > 0 such that the disk D(z0 , r) := {z 2 C : |z − z0 | < r} Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine; e-mail: [email protected]. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 3, pp. 425–426, March, 2020. Original article submitted April 19, 2017. 0041-5995/20/7203–0485
© 2020
Springer Science+Business Media, LLC
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A. V. P OKROVSKII
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is contained in G and the function f (z) is bounded in D(z0 , r) \ E. We define a function f1 (z):= − if 2 (z),
z 2 D(z0 , r) \ E.
Thus, we get u1 (z) := Re f1 (z) = 2u(z)v(z),
v1 (z) = Im f1 (z) = v 2 (z) − u2 (z).
Since the functions u(z) and v(z) are bounded in D(z0 , r) \ E, for any ⇣ 2 D(z0 , r) \ E we obtain u1 (z) ! 0
as z ! ⇣,
z 2 D(z0 , r) \ E.
The Ischanov theorem implies the existence of an analytic extension F (z) of the function f1 (z) from D(z0 , r) \ E to D(z0 , r). Let ⇣ 2 E \ D(z0 , r). Suppose that F (⇣) 6= 0 and take " 2 (0, r) such that D(⇣, ") ⇢ D(z0 , r)
and
|F (z) − F (⇣)| < |F (⇣)|/2
for all z 2 D(⇣, ").
p Then iF (z) is a univalent analytic function in D(⇣, "), where the branch of the square root in D(iF (⇣), |F (⇣)|/2) is fixed by the condition p iF (z) = f (z)
for all z 2 D(⇣, ") \ E. Thus, we have justified the existence of an analytic continuation f¯(z) of the function f (z) from D(z0 , r) \ E to D(z0 , r) \ (F −1 (0) \ E), where the
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