I have a group of complex functions of two complex variables, $f(z_g, z_h)$, with finite series of:
$f(z_g,z_h) = \frac{c^1_{10}}{1-a_gz_g} + \frac{c^1_{01}}{1-a_hz_h} + \frac{c^2_{20}}{(1-a_gz_g)^2} + \frac{c^2_{11}}{(1-a_gz_g)(1-a_hz_h)} + \frac{c^2_{02}}{(1-a_hz_h)^2} + \sum_{m,p} \frac{c^m_{p,m-p}}{(1-a_gz_g)^p(1-a_hz_h)^{m-p}} +... $
And we have: $f(0,0)=1 $
And $a_g$, $a_h$, $c^m_{p,m-p}$ are real numbers, and $0<a_g,a_h<=1$, ( not sure if these are neccessary )
?:
Can we say that on the imaginary axis, $z_g=x_gi, z_h=x_hi$,
the maximum magnitude of $f(x_gi, x_hi)$ is 1
when and only when:
$x_g=0, x_h=0$