CC BY
55Issues of Analysis
Vol. 2(20), No. 1, 2013
UDK 517.54
E. G. Ganenkova, V. V. Starkov
ASYMPTOTIC VALUES OF FUNCTIONS, ANALYTIC IN PLANAR DOMAINS1
Abstract. In [1] W. Gross constructed the example of an entire function of infinite order whose set of asymptotic values is equal to the extended complex plain. We obtain an analog of Gross’ result for functions, analytic in planar domains of arbitrary connectivity with isolated boundary fragment.
Key words: analytic function, asymptotic value, isolated boundary fragment.
2010 Mathematical Subject Classification: 30D40.
In 1918 W. Gross constructed the example of an entire function $ of infinite order whose set of asymptotic values is equal to C (see [1]). This means that for every a E C there exists an open arc Ta with endpoint at infinity such that
lim $(2) = a.
ra Bz^^
In this note, using Gross’ example, we construct a function, analytic in preassigned planar domain, having C as the set of asymptotic values at a given boundary point.
Let D be an arbitrary domain, D C C, z0 E dD is an accessible point (i. e. there exists an open arc r C D with endpoint z0), f is analytic in D function.
Definition 1. [2], [3, p. 8] We say that a E C is an asymptotic value of f at the point z0 if there exists an open arc Ta C D with endpoint z0 such that
lim f(z) = a.
raBz^Zo
1 This work was supported by the Programm of strategic development of the PetrSU and Russian Foundation for Basic Research (project No. 11-01-00952-a).
© Ganenkova E. G., Starkov V. V., 2013
We shall say that Ta is an asymptotic curve corresponding to the asymptotic value a. The set of all asymptotic values of a function f at a point z0 we shall denote by As(f, z0).
This definition restricts the choice of point z0. It is possible to define asymptotic values of function not at all boundary points. In the sequel, we assume that z0 belongs to an isolated boundary fragment.
Definition 2. [4] A domain D C C has an isolated boundary fragment if one of the following conditions holds:
(I) There exists a continuum K C dD, different from a point, and an open set U such that K C U and (dD \ K) n U = №.
(II) There exists a Jordan arc r C dD with distinct ends n and an open disk B such that £,n E dB, r \{£,n] C B and (dD \ r) n B = №.
(III) There exist a point a E dD and an open disc B(a) centered at a such that (B(a) \ {a}) n dD = №, i. e. a is an isolated point of the set dD.
Using Gross’ example we construct a functions f = $ o p, analytic in a domain D with isolated boundary fragment, with condition
As(f, zo) = C, zo E dD.
Here $ is an entire function, p is either injective analytic function from D into C or surjective at most 3-valent analytic function from D onto C.
Theorem 1. Let D be an arbitrary domain with isolated boundary fragment and a point z0 belong to this fragment. If it is the fragment of type (I) we assume in addition that z0 is an accessible point and it corresponds to the prime end, which is equal to z0. Then there exist an injective analytic function p : D — C and an entire function $ such that
. . , T.— lnln |$(p(z))|
As($ o p, z0) = C and lim —-—.—r—-= m.
z^zo ln |p(z)|
Proof. Let $ be Gross’ function. Asymptotic curves ro, corresponding to asymptotic values a E C of function $, are rays, lying in the set
{z E C : | arg zl < n/4 or In — arg zl < n/4}
(see [1]).
Case 1. If z0 is an isolated boundary fragment of type (III) we take
P(z) = —-—. z — zo
By Ya denote a connected subset of p-1(ra n p(D)) with endpoint at z0. Then
lim $(p(z)) = lim $(w) = a Va E C (1)
Ya Bz^Zo ra Bw^^
and
lim ^|$(p(z))| = nm lnln|$(w)i = (2)
Ya Bz^zo ln |p(z)| w^m ln |w|
Case 2. Let z0 belong to an isolated boundary fragment r of type (II). r can be considered as an open arc. By the Riemann mapping theorem there exists a biholomorphic mapping 0 from D into the upper half-plane P such that
{z E C : z < p, Imz > 0} C D1 = 0(D) for some p > 0,
the point 0 E dD1 corresponds to the z0 and some interval (a, 3) 9 0
corresponds to r (see [4] for details). It follows from the Caratheodory theorem that 0 can be extended to homeomorphism from (D U r) onto (0(D) U (a, 3)).
The univalent in D1 function
±i
w± 2
z 2
(the sign will be chosen later) maps D1 onto a domain D2, containing the rays
ra = ra n |w E C : |w| > -121 Va E C.
Since $ is the function of infinite order then there exists a sequence
wn — m as n — m such that
ln ln |$(wn)|
lim — ----:--:— = m.
ln wn |
We can choose a subsequence wn (let us save the notation) that doesn’t belong to one of the following sets {w E C : Re w = 0, Im z > 0} or {w E C : Re w = 0, Im z < 0}. Therefore, we can choose the sign for the function w± such that wn E D2.
Let p be the constructed above univalent function from D onto D2, Ya = p-1(ra), zn = p-1(wn). Then As($ o p, zo) = C and
lnln |$(p(zn))| lim ----II ( ^-- = m.
n^m ln p(zn)|
Case 3. Let z0 belong to isolated boundary fragment K of type (I). Consider the simply connected domain D0 D D, dD0 = K. By the Rie-mann mapping theorem there exists a univalent conformal mapping 0 of the domain D0 onto the upper half-plane P. In addition, the origin corresponds to the accessible boundary point z0 E dD0. It follows from the definition of isolated boundary fragment of type (I) that if D1 = 0(D), then the set dD1 \ R is a subset of some compact set C C P. Hence, we can find p > 0 such that
D1 D {z E C : z < p, Imz > 0}.
Since the point z0 is a prime end of the domain D (this means that z0 is a prime end of D0), then 0-1 can be continuously extended from D1 to D1 U {0}, 0-1(0) = z0 (see [5, ch. II, §3, p. 41]). Consequently, 0 is a homeomorphism from D U {z0} to D1 U {0}. Now, as in Case 2, we put p = w± o 0, choosing an appropriate sign. Then for all a E C conditions
(1) and (2) are fulfilled for p and Ya = p-1(ra). D
It was shown in [4] that the domain D from Theorem 1 can be mapped onto C locally biholomorphically and at most 3-valently. The mapping function F : D — C is the composition of the univalent function p from Theorem 1 and an 3-valent function g. Here 3-valence of a function F means that for every w E C the equation F(z) = w has at most three solutions and it has exactly three solutions for some w E C. For the polynomial Q(z) = z3 — 3z the Riemann surface Q(C) contains all rays ra = ra n {w E C : |w| > p} for sufficiently large p > 2, here ra are rays from Gross’ example. Thus, it follows from the construction of mapping g that the rays ra belong to the Riemann surface F(D). Like in the proof of Case 3, Theorem 1 we can show that F is continuous at the prime end z0 and
lim F(z) = .
DBz——zo
Therefore, for every a E C there exists an open arc Ya C D, F(Ya) = ra such that
lim $(F (z)) = lim $(w) = a
YaBz — zo TaBw — m
and
-— lnln |$(F(z))|
lim —-—, ,,---= m.
z—zo ln |F(z)|
Consequently, the univalent function from Theorem 1 can be replaced by
the at most 3-valent function F (it was proved in [6] that there is no
a 2-valent locally biholomorphic mapping of D onto C).
In such way we have obtained
Theorem 2. Let D and z0 be as in Theorem 1, D = C \ {z0}. Then there exist a locally biholomorphic at most 3-valent surjective function F:D — C and an entire function $ of infinite order such that
. N ,— lnln |$(F(z))|
As($ o F, z0) = C and lim —-— — = m.
z—zo ln F (z)|
References
[1] Gross W. Eine ganze Funktion fur die jede Komplexe Zahl Konvergen-zwert ist // Math. Ann. 1918. V. 79. P. 201-208.
[2] Encyclopedia of Mathematics. [S. l.]: Kluwer Academic Publishers, 1987. Vol. 1 (A—B).
[3] Collingwood E. F., Lohwater A. J. The theory of cluster sets. Cambridge: Collingwood Lohwater Cambridge University Press, 1966.
[4] Liczberski P., Starkov V. V. On locally biholomorhic mappings from multi-connected onto simply connected domains // Ann. Polon. Math. 2005. V. 85. No. 2. P. 135-143.
[5] Goluzin G. M. Geometric Theory of Functions of a Complex Variable. Providence, R. I.: American Mathematical Scociety, 1969.
[6] Starkov V. V. Locally biholomorphic mappings of multiconnected domains // Sib. Math. J. 2007. V. 48. No. 4. P. 733-739.
The work is received on May 25, 2013.
Petrozavodsk State University, Department of Mathematics 185910, Petrozavodsk, Lenin Avenue, 33. E-mail: g_ek@inbox.ru,
vstar@psu.karelia.ru
