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 = CDF.0; p1 = plot(lambda t: abs(log(1+t*i)), 0, 0.5, rgbcolor=(0.8,0,0),legend_label=’ $|log(1+z)|$’, thickness=2); p2 = plot(lambda t: abs((t*i)/2), 0, 0.5, rgbcolor=(0,0.8,0), legend_label=’$|z|/2$’, thickness=2); p3 = plot(lambda t: abs(3*(t*i)/2), 0, 0.5, rgbcolor=(0,0,0.8), legend_label=’ $3|z|/2$’, thickness=2); p1+p2+p3
I tried to get a graph where this inequlaity fails (i.e. the plots intersect), but failed to do so.
This is what Richard Feynman used to call “piddling” or “playing” with smallish problems…Your graphs and observation(s) do kindle some thoughts…:-)
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Feynman called his work on Quantum Electrodynamics to be a result of “fooling around” 🙂
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