Let be a complex number. The power series expansion of about is given by

which has radius of convergence 1. If then

If we further assume then

This gives,

I just wanted to see how this inequality will appear graphically, so here are the plots made using SageMath 7.5.1 (by fixing the real part of z to zero and varying the imaginary part till 1/2)

I tried to get a graph where this inequlaity fails (i.e. the plots intersect), but failed to do so.