Examples of RP-measures (Q1088033)

The author continues his search for the extreme rays of the cone of functions holomorphic in the bidisc with positive real part there. In this note he gives examples of (i) inner functions whose Cayley transforms are extreme with boundary measures of full support, i.e., the boundary measure of the Cayley transform vanishes in no open set, and (ii) functions that are not extreme, but that he believes, I think, suggest there are extreme functions whose Cayley transforms are not inner, alternatively, extreme functions whose boundary measures are not singular. In fact, he believes the functions in (ii) suggest there are extreme functions whose boundary measures are absolutely continuous. (If \(| g| <1\), its Cayley transform is \((1+g)/(1-g)\), while if \(Re f>0,\) its Cayley transform is \((f-1)/(f+1).)\)