Verification methods for inclusion disks

Let Z = {z : Iz r < r} be a disk in the complex plane with the center ~" = mid Z E C and the radius r = rad Z > O, denoted shorter by parametric notation Z = {r r}. If f is a closed complex function then the complex-valued set f (Z) = { f ( z ) : z E Z} is closed. In general, the range f (Z) is not a disk, which is quite impractical in calculations. In order to remain in the realm of disks, it is convenient to introduce a circular including approximation, denoted by I ( f (Z) ) , which completely includes the range f (Z) , that is, I ( f (Z ) ) D f(Z). The disk I ( f (Z ) ) is called a circular including approximation, or shorter, I-approximation. The practical point is to find an as good as possible /-approximation for given f and Z. Many authors have studied /-approximations, especially for elementary functions, polynomials, rational functions and analytic functions. For a given disk Z = {~'; r} and a function f let us assume that we have found (using some useful technique or an assumption based on geometrical construction) a disk {c; R}. Then the following important question arises: Does disk {c; R} completely contain the complex-valued range f (Z) , that is, is the enclosing condition [ f ( z ) c I < _ n ( z e Z ) (1)