| Let D = {z ∈ C | |z| < 1}, Gamma = {z ∈ D | |z| = 1} and H( D ) = {f : D → C ∣f is analytic on D }, with topology of uniform convergence on compact subsets of D . Let F be a compact subset of H( D ).;A function f ∈ F is an extreme point of F if f can not be written as a proper convex combination of two distinct elements of F, i.e. if f = tg + (1 -- t)h, 0 < t < 1, and g, h ∈ F, then f = g = h. We will investigate extreme points and non-extreme points of compact convex family of H( D ) which are generated by integrating a function 1-xzp 1-yzq against a probability measure on torus = Gamma x Gamma for p > 0 and q > 0; Fp,q = {fmu(z) = 1-xz p1-yz q dmu(x, y) : mu is a probability measure on Gamma x Gamma = torus}.;A function f ∈ F is a generalized exposed point if f uniquely maximizes ReL over F for some generalized linear functional L. We use the fact that if f0 is a generalized exposed point of F, then f0 is an extreme point of F to find extreme points.;On the other hand, to find non-extreme points, we will show the existence of a probability measure mu s.t 1-x0z p1-y0z q=1- xzp1-y zqdm x,y where mu is not an unit point mass at (x0 , y0). We will show that 1-x0z n1-y0z q is not extreme when x0 = y 0 by using Fejer's Kernel. Furthermore, we will show the existence of non-extreme point near x0 = y 0. |
