  The Malmquist-Takenaka system is a perturbation of the classical
trigonometric system, where powers of $z$ are replaced by products of other
M\"obius transforms of the disc. The system is also inherently connected to the
so-called nonlinear phase unwinding decomposition which has been in the center
of some recent activity. We prove $L^p$ bounds for the maximal partial sum
operator of the Malmquist-Takenaka series under additional assumptions on the
zeros of the M\"obius transforms. We locate the problem in the time-frequency
setting and, in particular, we connect it to the polynomial Carleson theorem.
