  We analyze Bergman spaces A p f (D) of generalized analytic functions of
solutions to the Vekua equation $\partial$w = ($\partial$f /f)w in the unit
disc of the complex plane, for Lipschitz-smooth non-vanishing real valued
functions f and 1 < p < $\infty$. We consider a family of bounded extremal
problems (best constrained approximation) in the Bergman space A p (D) and in
its generalized version A p f (D), that consists in approximating a function in
subsets of D by the restriction of a function belonging to A p (D) or A p f (D)
subject to a norm constraint. Preliminary constructive results are provided for
p = 2.
