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 .