The Set of Values of Functionals in the Classes of Functions Having Integral Representation

This paper presents a certain method to determine the set of values of the functional J(f)= Σ^ n_k=0 a_k(z) f^k(z) defined on the family of analytic functions in the unit disc possessing an integral representation with kernel q(z,t) in the interval [α ,β ]. It has been shown that the set of values of the linear functional J [f( z )] at the point z_0 is the convex hull of the curve Γ, given by equation Γ: w(t)= Σ^n_k=0 a_k(z_0)q^k ( z_0 ,t ), α ≤ t ≤ β . The Class of Functions with Limited Rotation was cosidered as a special case and some new results were obtained .