# Ngô Quốc Anh

## July 16, 2009

### A beautiful inequality regarding complex variables

The following inequality

$\displaystyle \left|{\frac{{{z_{1}}-{z_{2}}}}{{1-\overline{{z_{1}}}{z_{2}}}}}\right|\geqslant\frac{{\left|{{z_{1}}}\right|-\left|{{z_{2}}}\right|}}{{1-\left|{{z_{1}}}\right|\left|{{z_{2}}}\right|}},\quad\forall{z_{1}},{z_{2}}\in D\left({0,1}\right)$

holds true. To prove, we do as follows: By a direct computation, we get

$\displaystyle {\left|{\frac{{{z_{1}}-{z_{2}}}}{{1-\overline{{z_{1}}}{z_{2}}}}}\right|^{2}}= 1-\frac{{\left({1-{{\left|{{z_{1}}}\right|}^{2}}}\right)\left({1-{{\left|{{z_{2}}}\right|}^{2}}}\right)}}{{{{\left|{1-\overline{{z_{1}}}{z_{2}}}\right|}^{2}}}}$

and

$\displaystyle 1-\frac{{\left({1-{{\left|{{z_{1}}}\right|}^{2}}}\right)\left({1-{{\left|{{z_{2}}}\right|}^{2}}}\right)}}{{{{\left|{1-\overline{{z_{1}}}{z_{2}}}\right|}^{2}}}}\geqslant 1-\frac{{\left({1-{{\left|{{z_{1}}}\right|}^{2}}}\right)\left({1-{{\left|{{z_{2}}}\right|}^{2}}}\right)}}{{{{\left|{1-\left|{{z_{1}}}\right|\left|{{z_{2}}}\right|}\right|}^{2}}}}.$

Similarly,

$\displaystyle 1-\frac{{\left({1-{{\left|{{z_{1}}}\right|}^{2}}}\right)\left({1-{{\left|{{z_{2}}}\right|}^{2}}}\right)}}{{{{\left|{1-\left|{{z_{1}}}\right|\left|{{z_{2}}}\right|}\right|}^{2}}}}=\frac{{{{\left({\left|{{z_{1}}}\right|-\left|{{z_{2}}}\right|}\right)}^{2}}}}{{{{\left|{1-\left|{{z_{1}}}\right|\left|{{z_{2}}}\right|}\right|}^{2}}}}.$

Thus,

$\displaystyle\left|{\frac{{{z_{1}}-{z_{2}}}}{{1-\overline{{z_{1}}}{z_{2}}}}}\right|\geqslant\frac{{\left|{{z_{1}}}\right|-\left|{{z_{2}}}\right|}}{{1-\left|{{z_{1}}}\right|\left|{{z_{2}}}\right|}}.$

As an application we can prove the following Lindelof theorem. It says that if $f$ is assumed to be holomorphic and bounded by $1$ in $D(0, 1)$. Then

$\displaystyle\left|{f\left( z\right)}\right|\leqslant\frac{{\left|{f\left( 0\right)}\right|+\left| z\right|}}{{1+\left|{f\left( 0\right)}\right|\left| z\right|}},\quad\forall z\in D\left({0,1}\right).$