On Conformal Radii of Non-Overlapping Simply Connected Domains

The paper deals with the following open problem stated by V.N. Dubinin. Let a0 = 0, |a1| = . . . = |an| = 1, ak ∈ Bk ⊂ C, where B0, . . . , Bn are disjoint domains. For all values of the parameter γ ∈ (0, n] find the exact upper bound for r(B0, 0) n ∏ k=1 r(Bk, ak), where r(Bk, ak) is the conformal radius of Bk with respect to ak. For γ = 1 and n > 2 the problem was solved by V.N. Dubinin. In the paper the problem is solved for γ ∈ (0, √ n ] and n > 2 for simply connected domains. Subject Classification Numbers: 30C75. Introduction In geometric function theory of complex variable extremal problems on non-overlapping domains are well-known classic direction (see, for example, [1–26]). A lot of such problems are reduced to determination of the maximum of product of inner radii on the system of non-overlapping domains satisfying a certain conditions. Paper of M.A. Lavrent’ev ”On the theory of conformal mappings” [10] was initial impetus for such direction, in which, was first proposed and solved the problem of maximizing the product conformal radii of two non-overlapping simply connected domains. Namely, he proved the following assertion [10]: let a1 and a2 be some fixed points in the complex planeC,Bk, ak ∈ Bk, k = 1, 2 be any non-overlapping domains in C. Then for functions w = fk(0), k = 1, 2, which regular in the circle |z| < 1 and univalently mapping it to the domain Bk such that fk(0) = ak, we have inequality |f ′ 1(0)| · |f ′ 2(0)| 6 |a1 − a2|. Moreover, for domainsBk, which have classical Green’s function equality in this inequality is attended if and only if domainsB1,B2 are limited by circle z−a1 z−a2 = C, whereC is an arbitrary positive constant. Lavrent’ev used this result to some aerodynamics problems. It follows from the proof of this theorem, as a corollary, the well-known statement of KoebeBieberbach in theory of univalent functions. Based on these elementary estimates are obtained a number of new estimates for functions realizing a conformal mapping of a disc onto domains with certain special properties. Estimates of this type are fundamental to solving some metric problems arising when considering the correspondence of boundaries under a conformal mapping. Also, on the basis of the results concerning various extremal properties of conformal mappings, the problem of the representability of functions of a complex variable by a uniformly convergent series of polynomials is solved. Themes connected with the study of problems on non-overlapping domains was developed in papers [1–26]. Until 1974 a system of points ak, k = 1, n, of the complex planewere fixed. In 1968 P.M. Tamrazov put forward the idea, that we can provide to the points ak, k = 1, n, some freedom. In 1975, in accordance with this idea, G.P. Bakhtina first set and solved a number of extremal problems in classroom mutually non-overlapping domains with so-called free poles in her dissertation. The considering problem in the paper is the problem of this kind. International Journal of Advanced Research in Mathematics Submitted: 2017-09-26 ISSN: 2297-6213, Vol. 11, pp 1-7 Revised: 2017-11-27 doi:10.18052/www.scipress.com/IJARM.11.1 Accepted: 2018-04-05 2018 SciPress Ltd, Switzerland Online: 2018-05-02 SciPress applies the CC-BY 4.0 license to works we publish: https://creativecommons.org/licenses/by/4.0/ Let N and R be the sets of natural and real numbers, respectively, let C be the complex plane, and let C = C ∪ {∞} be its one-point compactification, R = (0,∞). Let r(B, a) = |f ′(0)| be the conformal radius of the simply connected domain B ⊂ C relative to a point a ∈ B, where w = f(z) is a univalent conformal mapping of the unit circle onto the domain B ⊂ C, a = f(0) (see, for example, [1, 10, 11, 12, 13, 14]). Further we consider the following system of points An := {ak ∈ C, k = 1, n}, n ∈ N, n > 2, satisfying the conditions |ak| ∈ R, k = 1, n and 0 = arg a1 < arg a2 < · · · < arg an < 2π. Define the numbers αk, k = 1, n, as follows α1 := 1 π (arg a2 − arg a1), α2 := 1 π (arg a3 − arg a2), . . ., αn := 1 π (2π − arg an). And let α0 = max k αk. Consider one open an extremal problem which was formulated in [1] in the list of unsolved problems and then repeated in monograph [14]. Problem. Consider the product In(γ) = r γ (B0, 0) n ∏


Introduction
In geometric function theory of complex variable extremal problems on non-overlapping domains are well-known classic direction (see, for example, ). A lot of such problems are reduced to determination of the maximum of product of inner radii on the system of non-overlapping domains satisfying a certain conditions. Paper of M.A. Lavrent'ev "On the theory of conformal mappings" [10] was initial impetus for such direction, in which, was first proposed and solved the problem of maximizing the product conformal radii of two non-overlapping simply connected domains. Namely, he proved the following assertion [10]: let a 1 and a 2 be some fixed points in the complex plane C, B k , a k ∈ B k , k = 1, 2 be any non-overlapping domains in C. Then for functions w = f k (0), k = 1, 2, which regular in the circle |z| < 1 and univalently mapping it to the domain B k such that f k (0) = a k , we have inequality |f ′ 1 (0)| · |f ′ 2 (0)| |a 1 − a 2 | 2 . Moreover, for domains B k , which have classical Green's function equality in this inequality is attended if and only if domains B 1 , B 2 are limited by circle z−a 1 z−a 2 = C, where C is an arbitrary positive constant. Lavrent'ev used this result to some aerodynamics problems.
It follows from the proof of this theorem, as a corollary, the well-known statement of Koebe-Bieberbach in theory of univalent functions. Based on these elementary estimates are obtained a number of new estimates for functions realizing a conformal mapping of a disc onto domains with certain special properties. Estimates of this type are fundamental to solving some metric problems arising when considering the correspondence of boundaries under a conformal mapping. Also, on the basis of the results concerning various extremal properties of conformal mappings, the problem of the representability of functions of a complex variable by a uniformly convergent series of polynomials is solved. Themes connected with the study of problems on non-overlapping domains was developed in papers .
Until 1974 a system of points a k , k = 1, n, of the complex plane were fixed. In 1968 P.M. Tamrazov put forward the idea, that we can provide to the points a k , k = 1, n, some freedom. In 1975, in accordance with this idea, G.P. Bakhtina first set and solved a number of extremal problems in classroom mutually non-overlapping domains with so-called free poles in her dissertation. The considering problem in the paper is the problem of this kind.
Let N and R be the sets of natural and real numbers, respectively, let C be the complex plane, and let C = C ∪ {∞} be its one-point compactification, R + = (0, ∞).
Let r(B, a) = |f ′ (0)| be the conformal radius of the simply connected domain B ⊂ C relative to a point a ∈ B, where w = f (z) is a univalent conformal mapping of the unit circle onto the domain B ⊂ C, a = f (0) (see, for example, [1,10,11,12,13,14]).
Define the numbers α k , k = 1, n, as follows Consider one open an extremal problem which was formulated in [1] in the list of unsolved problems and then repeated in monograph [14].
Problem. Consider the product Show that it attains its maximum at a configuration of domains B k and points a k possessing rotational n-symmetry.
Presently, this task is not completely solved, its solutions for certain particular cases are only known. In [1] the problem was solved for any n 2 and γ = 1. In [15] -for any γ > 1 but starting with some unknown number n in advance. The next step in the study of this problem was finding possible solutions for type of restrictions 1 < γ n α where 0 < α < 1. In [16] the problem was solved for n 8 and 1 < γ 4 √ n, in [17] -for n 12 and 1 < γ n 0.45 . In paper [18] this problem was solved for any natural n 5 and 0 < γ n but for condition α 0 2 √ γ . So we will consider only configuration of domains D k and points d k for which α 0 > 2 √ γ . Results of this paper are addendum to the theorem in [18].

Results and Proofs
For simply connected domains we obtained the following result. Theorem 1. Let n ∈ N, n 2 and γ ∈ (1, √ n]. Then for any different points a k , k = 1, n, which lie on the unit circle |w| = 1 and any system of non-overlapping simply connected domains B k , a k ∈ B k ⊂ C, k = 0, n, a 0 = 0, the following inequality holds Equality in (1) is attained, when a k and B k , k = 0, n, are, respectively, the poles and circular domains of the quadratic differential Proof of the theorem 1. Note, that the cases n = 2 and n = 3 were considered in the paper [19], the case n = 4 was considered in [20] for a more general case of multiply connected domains. Therefore, we prove theorem 1 for n 5.

IJARM Volume 11
It is important for us to know numerical value of I n (γ) on the system of circular domains D k and the system of poles d k , k = 0, n of the quadratic differential (2). Let where {d k } n k=0 and {D k } n k=0 are, respectively, the poles and circular domains of the quadratic differential (2). Using theorem 5.2.3 [15] we have The following assertion is known.

Lemma 1. [19]
Let n ∈ N, n 2, γ > 0. And let {B 0 , B 1 , B 2 , . . . , B n } be the system of pairwise non-overlapping simply connected domains such that 0 . Then the following inequality holds Note, that in [21] the problem was solved for n 4 and 0 < γ n but for condition α 0 2 √ γ (see also, [22,25]). Thus taking Lemma 1 into account we consider the case when Taking theorem 5.2.3 [15] into account we obtain Thus, we have the following inequality Note, to prove the theorem 1 for some n and γ, it suffices to show that the inequality G n (γ) < 1 holds. So, we show that for n 5 the inequality G n ( √ n) < 1 holds.
Obviously, for n 5 we have decreases with increasing n, so for n 5 we have ) n−1 also decreases with increasing n, and we obtain that for n 5 it does not exceed 0.01. Summing obtained estimates we have G n ( √ n) < e · 30 · 0.01 < 1.
Thus, for γ = √ n the theorem 1 is proved. Further consider the validity of the theorem for 1 < γ < √ n. The following equality holds It is not difficult to obtain the following lemma.
Proof of the lemma 2. Using the logarithmic derivative, we investigate the monotonicity of the function G n (γ).
It is easily seen that From Lemma 1 we obtain ) .
Proof of the theorem 2. In order to prove the theorem we use the following inequality where the value I 0 n (γ) is determined by the formula (3). Substituting in (8)  Further, using Lemma 2 we have statement of theorem 2. Thus theorem 2 is proved.
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