  We study some approximation problems on a strict subset of the circle by
analytic functions of the Hardy space H2 of the unit disk (in C), whose modulus
satisfy a pointwise constraint on the complentary part of the circle. Existence
and uniqueness results, as well as pointwise saturation of the constraint, are
established. We also derive a critical point equation which gives rise to a
dual formulation of the problem. We further compute directional derivatives for
this functional as a computational means to approach the issue. We then
consider a finite-dimensional polynomial version of the bounded extremal
problem.
