Problem of the maximum of the product of the conformal radii of nonoverlapping domains in a circle

Let a_k, k=1,2,3, be distinct points of the circle U={z:¦z¦<1}, a_3+k=1/¯a_k, k=1,2,3. Let D₁,...,D₆ be a system of nonoverlapping simply connected domains D₁,...,D₆ on,a_k D_k, k=1,...,6. Let R(D_k,a_k) be the conformal radius of the domain D_k with respect to the point a_k. One formulates the following theorem. For any points a_k U, k=1,2,3, and any system of the indicated domains one has the sharp inequality One points out all the cases when equality prevails in (1). One indicates the main steps of the proof of this theorem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 99–113, 1983.     

The maximum of the product of conformal, radii of nonoverlapping domains in the disk

Two problems on the extremal decomposition of the unit disk are considered. It is shown that the associated quadratic differentials in these problems have at least two distinct zeros on the unit circle. Bibliography: 7 titles.Dedicated to the 90th anniversary of G. M. Goluzin's birthTranslated fromZapiski Nauchnykh Seminarov POMI, Vol. 237, 1997, pp. 105–118.     

The problem of product of conformal radii of nonoverlapping domains

Let a₁, a₂, a₃, b be distinct points in and let D be the family of all triples of nonoverlapping domains D₁, D₂, D₃ in \ {b} such that a_k∈ D_k, k=1,2,3. For this family we consider the problem on the maximum of the functional I=R₁R₂R₃, where R_k=R(D_k, a_k) is the conformal radius of D_k with respect to a_k. Geometrical properties of the extremal triple of domains are described. We prove that the maximum of I monotonically depends on the position of the point b and find the maximum in some special cases Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 114–128 Translated by N. Yu. Netsvetaev     

Maximum of the product of the conformal radii of four nonoverlapping domains

Let a_k, k=1,...,4, be given distinct points of . Let D_k, k=1,...,4, be a system of simply connected domains in the closed plane such that a_kD_k, k=1,...,4, D_kD=for k,=1, ...,4, k. Let R(D_k,a_k) be the conformal radius of the domain D_k relative to the point a_k. In this paper we obtain an explicit expression for the maximum of the product in the family of all indicated system of domains in terms of elliptic functions. In the proof one makes directly use of the property of the extremal family of domains of the problem under consideration in terms of the associated quadratic differential.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 100, pp. 146–165, 1980.     

On the product of inner radii of symmetric nonoverlapping domains

Some results concerning extremal problems for nonoverlapping domains with free poles on the unit circle, known for the simply connected case, are generalized to the case of multiply connected domains.     

Estimation of the product of inner radii of partially nonoverlapping domains

We present new results on the maximization of products of positive powers of inner radii for several special systems of domains in the extended complex plane {ie1176-01} with respect to points of finite sets such that any two different points z ₁, z ₂ ∈ ℂ \ {0} of such a set lie on different rays that emerge from the origin of coordinates. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 7, pp. 1004–1008, July, 2008.     

Inequalities for the inner radii of nonoverlapping domains and open sets

We generalize some classical results in the theory of extreme problems for nonoverlapping domains.     

Maximum of a conformal,invariant in the problem of nonoverlapping domains

The paper is devoted to the well-known circle of problems on the maximum of the products of powers of conformal radii of nonoverlapping domains. Let a₁, ..., a_n be distinct points of and let D₁, ..., D_n be a system of simply connected domains in, pairwise disjoint and such that a_kD_k, k=1, ..., n. By R(D_ka_k) we denote the conformal radius of the domain D_k relative to the point a_k. One considers the problem on the maximum of the product in the family of all indicated systems of domains, under the condition that a₁, ..., a_n runs over all systems of distinct points in (n4) and one finds the geometric characteristic of the extremal configurations of this problem in terms of the associated quadratic differential.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 172–183, 1981.     

Sharp estimates for inner radii of systems of nonoverlapping domains and open sets

We study extremal problems of the geometric theory of functions of a complex variable. Sharp upper estimates are obtained for the product of inner radii of disjoint domains and open sets with respect to equiradial systems of points. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1601–1618, December, 2007.     

Supplement to G. V. Kuz'mina's paper: “On the problem of the maximum of the product of the conformal radii of nonoverlapping domains” (Zap. Nauchn. Sem. Lomi,100, 131–145, 1980)

One gives a generalization of a result of G. V. Kuz'mina.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 22–23, 1983.     

On a problem of nonoverlapping domains

We solve the problem of finding the range E of some functional on the class of pairs of functions univalent in the system of the disk and the interior of the disk for the arbitrary parameters characterizing the functional. We prove that E is connected and bounded. Using the method of internal variations and the parametric method, we find the equation of the boundary of E. The obtained results extend Lebedev’s study [1].     

The maximum of the conformal radius in the families of domains satifying additional conditions

We solve the problems on the maximum of the conformal radius R(D,1) in the family D(R₀) of all simply connected domains D ⊃ ℂ containing the points 0 and 1 and having a fixed value of the conformal radius R(D,0)=R₀, and in the family D(R₀, ρ) of domains from D(R₀) with given hyperbolic distance ρ=ρ_D(0,1) between 0 and 1. Analogs of the mentioned problems for doubly-connected domains with given conformal module are considered. Solution of the above problems is based on results of general character in the theory of problems of extremal decomposition and related module problems. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 93–108.     

Some problems in the theory of nonoverlapping domains

We generalize some results concerning extremal problems of nonoverlapping domains with free poles on the unit circle. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 6, pp. 723–731, June, 1999.     

The problem of conformal transformations of a circle into nonoverlapping regions

Let a, a0, a, be a fixed point in the z-plane, (a, 0, ), the class of all systemsf _k()_l ³ of functions z=f ^k(), k=1, 2, 3, of which the first two map conformally and in a s ingle-sheeted manner the circle ¦¦<1, and the third maps in a similar manner the region ¦¦>1, into pair-wise nonintersecting regions B_k, k=1, 2, 3, containing the points a, 0, and , respectively, so thatf ₁(0)=a,f ₂(0)=0 andf ₃()=. The region of values (a, 0, ) of the system M(¦f ₁'(0)¦, ¦f ₂'(0)¦, 1/¦f ₃'()¦) in the class (a, 0, ) is determined.Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 417–424, October, 1969.     

Lengths of radii under conformal maps of the unit disc

If is the set of endpoints of radii which have length greater than or equal to under a conformal map of the unit disc, then as for the logarithmic capacity of . The exponent is sharp.

     

Decompositions into nonoverlapping domains and extremal properties of univalent functions

We apply the method of extremal metrics and certain symmetrization approaches to study problems on conformal mappings of a disk and circular annulus. For instance, we solve the problem on the maximal conformal module in the family of all doubly-connected domains of the form \(E₁∪E₂) with E₁∩E₂=0, r₁E₁, 0≤r₁·r₁≤∞, and diam E₂∩{z:|z|<1}. This generalizes the classical result by A. Mori. We also give a new solution of a problem by P. M. Tamrazov, which was initially solved by V. A. Shlyk. Some new theorems on the covering of a regular system of n rays are obtained for certain classes of convex mappings. Bibliography: 22 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 139–163. Translated by A. Yu. Solynin.