Intrinsic interpolation, near-circularity and maximal convergence

Let E be compact and connected with cap E > 0 and connected complement = C \ E, let g (z, ) be the Green's function of with pole at infinity and letbe the Green domains with boundaries .Let f be holomorphic on E and let ( f ) denote the maximal parameter of holomorphy of f and let { p n } nN be a sequence of polynomials converging maximally to f on E. If , 1 < < ( f ) < , is fixed and if m n ( ) denotes the number of interpolation points of p n to f in E with normalized counting measure ,n , then there exists a subset N such thatMoreover, there exists a sequence { n } n converging to such that the closed curves n = ( fp n )( n ) do not pass through the point 0 and the winding numbers Ind n (0) satisfy