  We consider the complex case of the so-called S-inequality. It concerns the
behaviour of the Gaussian measures of dilations of convex and rotationally
symmetric sets in C^n (rotational symmetry is invariance under the
multiplication by $e^{it}$, for any real t). We pose and discuss a conjecture
that among all such sets the measure of cylinders (i.e. the sets $\{z \in C^n:
|z_1| \leq p\}$) decrease the fastest under dilations.
  Our main result of the paper is that this conjecture holds under the
additional assumption that the Gaussian measure of considered sets is not
greater than some constant c > 0.64.
