Inequality involving complex numbers.

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Gabriel T. Prajitura

Is there a typo for the imaginary parts?

Dinu Teodorescu

Gabriel T. Prajitura Yes, it was a typo! Thanks! Now it's correct!

Spiros Konstantogiannis

Setting z=a+ib, we have 1/(1-z)=(1-a)/[(1-a)2+b2]+ib/[(1-a)2+b2].

The inequality we want to prove is thus equivalent to

a(1-a)/[(1-a)2+b2]+b2/[(1-a)2+b2]

Cauchy-Schwarz again!

[ Re(z)Re(1/(1-z))+Im(z)Im(1/(1-z)) ]2