We establish that the differential subordinations of the forms p(z)+γzp′(z)≺ h(A1,B1;z) or p(z)+γzp′(z)/p(z) ≺ h(A2,B2;z) implies p(z) ≺ h(A,B;z), where γ ≥ 0 and h(A,B;z)= (1+Az)/(1+Bz) with −1≤ B <A.

A positive linear fractional transformation (PLFT) is a function of the form f(z) = az+b cz+d where a, b, c and d are nonnegative integers with determinant ad− bc 6= 0. Nathanson generalized the notion of the Calkin-Wilf tree to PLFTs and used it to partition the set of PLFTs into an infinite forest of rooted trees. The roots of these PLFT Calkin-Wilf trees are called orphans. In this paper, we...

We present two algorithms for computing the root, or equivalently the xed point, of a function in exact real arithmetic. The rst algorithm uses the iteration of the expression tree representing the function in real arithmetic based on linear fractional transformations and exact oating point. The second and more general algorithm is based on a trisection of intervals and can be compared with the...