Some theorems of Kühnau on the mutual position of the boundary components of images of a multiply connected domain under single-sheeted conformal mappings are generalized to p-sheeted conformal mappings of a multiply connected domain with given singularities at given points of the domain. Concrete estimates for certain functionals are obtained for an annulus.

One of the basic extremal problems of the geometric theory of functions of a complex variable is the best possible estimate above of the modulus of the derivative in a given class of functions, analytic in a finitely connected domain.A result of Ahlfors [I] is well known and concerns the solution of this problem for the class of bounded functions: if D is a finite n-connected domain of the ~-plane, bounded by closed analytic curves, and ~(g) is the class of all functions ~(~) which are regular* in D !

At the end of 1964, the well-known Soviet mathematician Ivan Evgen' evich Bazilevich reached the age of 60. He was born on November 29 1904 in the village of Godunovka in the Glukhovsk district of the province of Sumsk. After finishing secondary school in 1922, I. E. Bazilevich went to study first at the Moscow Electrical Engineering Institute, but soon transferred to the Physics and Mathematics Faculty of the Moscow State University, graduating in applied mathematics in 1930. After postgraduate work at the University of Moscow, Bazilevich submitted in 1935 a thesis for the degree of Master of the Physical and Mathematical Sciences. At the very beginning of his scientific work, he became deeply interested in the geometric theory of functions and in particular in the study of the extremal properties of classes of univalent functions. He obtained a number of deep results in this field which he presented in his Doctor' s dissertation at the University of Moscow in 1949. Bazilevich devoted the whole of his scientific endeavour to the theory of univalent functions and quickly won recognition as one of the foremost specialists in this field. Bazilevich' s work has been mainly concerned with the well-known class 5 of functions f(z) = ζ + c 2 z 2 ..., regular and univalent in the circle μ I < 1, and the class S p of functions from θ of the type f(z) = ζ + c p +iz p+1 + ..., which have p-fold rotational symmetry. One of the best known problems of the theory of univalent functions is that of determining the range of values of the systems of initial coefficients of functions of these classes. Bazilevich' s first paper, published in 1936, gives a complete solution of the problem about the range of values of the system i|c p+1 |, |cap+iIi in the class S p . He has returned to this type of problem in one of his recent papers (1957); with the help of a number of his results dating from 1936-1937, he completely solves the considerably more general problem of determining the range of values of the systems Π c p+i I. I c 2p +i I i and iRe c p+1 , Re c 2p +i\ in the class S p (M) of functions from S p satisfying the condition |/(z)| < Μ (Μ > 1).