Sets of Values of Systems of Functionals in Classes of Univalent Functions

Asymptotic estimates for the coefficients of bounded univalent functions

“…As before, the functions f in a dense subset of the class S(M ) or S are determined by integrals w(z, t) of the differential equation (19) by (9) and (10) or (11). It was demonstrated in [14] that in this case the representation of f is unique, if we restrict our consideration to a constant coefficient λ and continuous controls u 1 (t) and u 2 (t). The Löwner differential equation (8) induces (12) and (14) and the Hamiltonian (13).…”

“…Theorem B [14]. The Löwner differential equation (8) generates the following differential equation in the vector a(τ ):…”

“…In the extremal problem of maximizing Re a n in the class S(M ) with infinitesimal 1/M , we should look for appropriate variations of the Koebe function K which turn out to be functions of the class S(M ) and provide the boundary points of the set V n (M ). All these variations are produced by variation of the initial data in the conjugate system (14) and integration of (12) and (14) over the interval [0, 1 − 1/M ] with the controls u satisfying the maximum principle.…”

The sum of coefficients of bounded univalent functions

“…Then the control u = zr satisfies the Pontryagin maximum principle for t > 0 in the neighborhood of the initial value t = 0, and the relative solution w(z, t) of the Lhwner differential equation has real coefficients. Thus, the control u = zr is optimal on the entire ray [0, oe) (see, for example, [2]). Integrating the L6wner equation with u = zr over [0, log M] yields the function pM.…”

“…The Pick functions pM, defined by the equation The author has proved [1] w By V M (here n > 2 and 1 < M < oc) denote the set of values of the system of coefficients {a2,..., an} on the class S M . control system [2] da n-1…”