Extremal problems in classes of univalent functions,omitting prescribed values

Section 1 of the paper is devoted to extremal problems in the classes of conformal homeomorphisms of the circle and the annulus, connected directly with the problem on the maximum of the conformal modulus in the family of doubly connected domains. In Secs. 2 and 3 one considers the class R of functions f()=c₁+c₂²+... regular and univalent in the circleU={||<1} and such that f(₁)f(₂)=1 for ₁₂U (the class of Bieberbach-Eilenberg functions). Here one solves the problem of the maximum of |f(₀)| in the class of functions f()R with a fixed value f(₀, where ₀ is an arbitrary point U, and of the maximum of |f(₀)| in the entire class R. For the proof one makes use of the method of the moduli of families of curves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 94–114, 1985.     

Some properties for certain classes of univalent functions defined by differential inequalities

Let A be the space of functions analytic in the unit disk D = {z:|z| 1}.Let U denote the set of all functions f ∈ A satisfying the conditions f(0) = f'(0)-1 = 0 and|f'(z)(z/f(z))~2-1|1(|z|1).Also,let Ω denote the set of all functions f ∈ A satisfying the conditions f(0) = f'(0)-1 = 0and|zf'(z)-f(z)|1/2(|z|1).In this article,we discuss the properties of U and Ω.     

Some extremal problems for functions of bounded boundary rotation

A variational method is developed within the class of functions of boundary rotation not exceedingkπ which is based on the fact that the set of representing measuresμ is convex. It shows that an extremal problem related to a functional with Gateaux derivative and some constraints leads to extremal measuresμ ₀ with finite support. The positive and negative part of aμ ₀ is located at points where a functionJ (depending onμ ₀) reaches its maximum and minimum respectively. The method is tested successfully on various problems.     

Some extremal bounds for subclasses of univalent functions

In this paper we obtain the sharp lower bound for , for functions f that are k-uniformly convex in the unit disk U. Next we consider the problem of finding the minimum of for functions f that are k-uniformly convex in the disk of radius r. Corresponding results for the class of starlike functions related to the class of k-uniformly convex functions are presented.     

Quasi-Hadamard product of certain univalent functions

The author improves some recent results due to Shigeyoshi Owa (Tamkang J. Math., 14 1983, 15–21) concerning the quasi-Hadamard product of certain starlike and convex univalent functions.     

Perfect splines with boundary conditions and their application to certain extremal problems

In this paper we consider two problems. The first is connected with the optimal recovery of functions satis fyiog boundary conditions. The second is the characterization of the unique function whose r-th derivative has minimum L_∞-norm, taking given values of alternating signs and satisfying boundary conditions. Partially supported by Ministry of Science under Project MM-414.     

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.     

Variational problems of nonlinear elasticity in certain classes of mappings with finite distortion

We study the problem of minimizing the functional \(I(\phi ) = \int\limits_\Omega {W(x,D\phi )dx}\) on a new class of mappings. We relax summability conditions for admissible deformations to φ ∈ W _n ¹ (Ω) and growth conditions on the integrand W(x, F). To compensate for that, we require the condition \(\frac{{\left| {D\phi (x)} \right|^n }} {{J(x,\phi )}} \leqslant M(x) \in L_s (\Omega )\), s > n ? 1, on the characteristic of distortion. On assuming that the integrand W(x, F) is polyconvex and coercive, we obtain an existence theorem for the problem of minimizing the functional I(φ) on a new family of admissible deformations A.     

Extremal solutions of Nevanlinna-Pick problems and certain classes of inner functions

Consider a scaled Nevanlinna-Pick interpolation problem and let ∏ be the Blaschke product whose zeros are the nodes of the problem. It is proved that if ∏ belongs to a certain class of inner functions, then the extremal solutions of the problem or most of them are in the same class. Three different classical classes are considered: inner functions whose derivative is in a certain Hardy space, exponential Blaschke products and the well-known class of α-Blaschke products, for 0 < α < 1.