A problem regarding the exact majorant

19o5.

A PROBLEM REGARDING THE EXACT MAJORANT*

Let G be an arbitrary domain in the complex plane (e.g., G = ~ ) and let h be a nonnegative function defined in G. We consider the class Bh of all single-valued functions f~ analytic in G, for which If(z) l ~ h(z) (z e G) and let the function H be defined in the fol ~ lowing manner:

The problem consists in the following: Give (necessary and sufficient) conditions on h so that h = H. If the indicated identity holds for h, then we shall call h an emac~ m ~ j o r a ~ (e.m.). Clearly, an e.m. must be a logarithmic subharmonic function. However, simple examples show that only this condition is not sufficient. A trivial sufficient condition is h = IFi, where F is a single-valued function, analytic in G; however, this condition is not necessary. At the investigation of the problem one can make additional assumptions about h, for example, one can assume h to be continuous in G or even (at the beginning) in elosG. In principle, there exists the possibility of an approach to the formulated problem on the basis of the duality relations in extremal problems (see, e.g., [I]). However, we have not succeeded in obtaining in this manner interesting information about the characteristics of an e.m. It is possible that an additional useful consideration for this approach will be the fact that h, being an e.m. in G, must be such in any subdomain of the domain G. We denote by Q the class of all possible e.m. for the domains G (consisting of functions, continuous in G or even in closG). One of the conjectures relative to the structure of the class Q is: h ~ Q ~-> (h belongs to the closure of the set of functions of the form: Ifll + ... + [fn [ , where fj are analytic in G, n ~ ); the closure is taken either in the topology of the space C(closG) when one considers Q as consisting of the elements of this space (then all fj are continuous in closG) or in the topology of the projective limit of the spaces C(closGn) , where the domains {Gn} exhaust G. An approach to this conjecture on the basis of convex analysis considerations lead to the following dual formulation: Let be some real Borel measure on G; from the fact that analytic

in G does there follow the inequality ~

0

IGI~!~0 _ for

~

for all functions which are ?

One can attempt to ob-

tain some progress in the last question by investigating measures in the Riesz representation for Ifl (f is analytic) with the aid of the logarithmic potential (namely Ifl and not in Ifl!) In our opinion, answers to the formulated questions would constitute an interesting contribution to the theory of extrema in the space of analytic functions. LITERATURE CITED I.

S. Ya. Khavinson, "The theory of extremal problems for bounded analytic functions, satisfying additional conditions inside the domain," Usp. Mat. Nauk, 18, No. 2, 25-98

(1963).

*S. Ya. KHAVINSON. USSR.

V. V. Kuibyshev Moscow Structural Engineering

Institute, Moscow,

121352,

2207