Extreme points of integral families of analytic functions (Q2861576)

Let \(\mathbb{D}\), \(\Gamma\) denote the open unit disk and the unit circle, respectively. Let \(H(\mathbb{D})\) denote the space of analytic functions in \(\mathbb{D}\), and let \(\mathbb{T}=\Gamma\times \Gamma\). For \(p,q >0\), let NEWLINE\[NEWLINEF_{p,q} = \left\{f_\mu (z) = \int_\mathbb{T}\frac{(1-xz)^p}{(1-yz)^q} d\mu(x,y)\right\},NEWLINE\]NEWLINE where \(\mu\) is a probability measure on \(\mathbb{T}\). For a specific choice of parameters, the family \(F_{p,q}\) is the closed convex hull of several classes of analytic functions. For example, the set \(F_{1,3}\) is the closed convex hull of the derivatives of normalized close-to-convex functions on \(\mathbb{D}\).NEWLINENEWLINEIn this paper the general case \(F_{p,q}\) is investigated. The authors give an affirmative answer to the question whether any points \((1-x)^p\), with \(|\arg (x)| < \pi(p-1)/(p+1)\), yield kernel functions that are the extreme points of \(F_{p,q}\), and answer whether the parameter \(q\) plays a role in a determination of extreme points.