Meromorphic functions with bounded boundary rotation

In this article, we generalize the class of meromorphic functions with bounded boundary rotation and related classes. Characterizations and some properties of these classes of functions are given.     

Coefficients of multivalent symmetric functions of bounded boundary rotation

In this paper the sharp coefficient estimate problem for the classesC _p(β, m) andV _p(k,m) of multivalent close-to-convex functions of order β and multivalent functions of bounded boundary rotation of at mostkπ, whose functions are given bym-fold symmetric gap series, have been discussed respectively for β≥1?p/m>0 andk≥2(m/p). Moreover, it is shown that every function inV _p(k,m) arep-valent close-to-convex; hencep-valent; ifk<2 (1+m/p).     

Applications of subordination to functions with bounded boundary rotation

Classes of analytic functions with bounded boundary rotation are considered. For these classes, we use subordination techniques to obtain known as well as new results related to Marx-Strohhäcker-type problems.     

Some critical minimization problems for functions of bounded variations

Using a new elementary method, we prove the existence of minimizers for various critical problems in BV(Ω) and also in W1,p(Ω), 1<p<∞.     

Some extremal problems for Pairs of functions with no common values

By the method of the moduli of families of curves one solves some extremal problems in the family of pairs {f₁, f₂} of functions f₁(z)=αz + ..., f₂(z)=βz^-1 + β⁰ +β¹z+..., with real coefficients, univalent, regular, resp. meromorphic in the circle Δ={∥z∥<1} and mapping onto nonoverlapping domains. As a special case the solution of a problem, posed by V. M. Miklyukov in Sib. Mat. Zh., Vol. 18, 1977, No. 5, pp. 1111–1124, is obtained. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 159–169, 1987.     

Functions of bounded boundary rotation

Classes of functionsU _k, which generalize starlike functions in the same manner that the classV _k of functions with boundary rotation bounded bykπ generalizes convex functions, are defined. The radius of univalence and starlikeness is determined. The behavior off _α(z) = ∫ ₀ ^z [f'(t)]^α dt is determined for various classes of functions. It is shown that the image of |z|<1 underV _kfunctions contains the disc of radius 1/k centered at the origin, andV _k functions are continuous in |z|≦1 with the exception of at most [k/2+1] points on |z|=1.     

Some overdetermined boundary value problems for harmonic functions

Summary In this paper we study various overdetermined problems involving harmonic functions. In particular, we show that if the second eigenfunctionu ₂ of the Stekloff eigenvalue problem in a bounded simply connected plane domain has a constant value of u ₂ on , then is a disk

---

Résumé Cet article est consacré à l'étude de certains problèmes surdéterminés pour des fonctions harmoniques. En particulier, nous montrons que si le gradient de la seconde fonction propre du problème de Stekloff défini dans un domaine borné, simplement connexe du plan, a son module constant sur la frontière , alors est nécessairement un disque.

     

Some extremal problems of theory of quadrature and approximation of functions

This is the author-review of the dissertation presented for the degree of the Doctor of Physicomathematical Sciences. The dissertation was defended on March 13, 1975 in a meeting of the scientists of the V. A. Steklov Mathematics Institute of the Academy of Sciences of the USSR. The official opponents were Doctor of Physicomathematical Sciences Prof. N. S. Bakhvalov, Doctor of Physicomathematical Sciences Prof. P. K.Suetin, and Doctor of Physicomathematical Sciences Prof. S. B. Stechkin.Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 299–311, February, 1976.     

The boundary distortion and extremal problems in certain classes of univalent functions

We consider the class S₁(τ), 0<τ<1, of functions f(z)=rz+a₂z²+... that are regular and univalent in the unit disk U and have |f(z)|<1. We obtain sharp estimates for the 1-measure of the sets {θ: |f(e^iθ)|=1}. As a corollary, for the familiar class S we find Kolmogorov-type estimates for the sets {θ: |f(e^iθ)|>M}, M>1, and prove inequalities for the harmonic measure, which are similar to those by Carleman-Milloux and Baernstein. We also consider problems on distortion of fixed systems of boundary arcs in the classes of functions that are regular (or meromorphic) and univalent in the disk or circular annulus. Bibliography: 23 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 115–142. Translated by A. Yu. solynin.