A mapping f [member of] S([OMEGA]) is called starlike (respectively convex) if its image is a starlike domain with respect to origin (respectively

convex domain).

Assume that [OMEGA] is a

convex domain. Let [phi] and [psi] be the solutions of the dual (forward and backward) problems (3.11) and (3.12), respectively.

Instead of enriching with discontinuous function, a mapping technique has been used to partition the given

convex domain and coined almost everywhere partition of unity [22] because of failing partition of unity at some points at the boundary.

A mapping f : D [right arrow] C is called convex harmonic if f (D) is a

convex domain.

In particularly, for each member of [HC.sup.0.sub.p], F maps U onto a

convex domain.

(iii) ([phi] * zf')/([phi] * z[psi]') takes all its values in a

convex domain D if f'/[psi]' takes all its values in D.

where t is the grid step; [GAMMA] is the circulation around the base profile [L.sub.0]; and [D.sup.-.sub.0*] denotes the bounded simple

convex domain:

where F is a Frechet-differential operator defined on an open

convex domain D of a Banach space X with values in a Banach space Y.

By a

convex domain we mean an open set which contains the line segment joining any two points of it.

Namely, for any n-dimensional

convex domain there is the sharp inequality

where F: D [subset or equal to] X 6 X is an operator defined on an open

convex domain D of a Banach space X with values in X.

First we give error estimates in case of a

convex domain, i.e., the results from Section 3 for the semidiscretization will be used.