Complex numbers inequality

From now on,i will post here as much inequalities involving complex numbers as possible,they seem to relate with classic geometrical inequalities,the only job to do is to find a meaning for this abstract terminology
For example,prove that $\frac{\overline{a}b+a\overline{b}}{2|a||b|}\in [-1,1]$
Let be A(a),B(b),and $\measuredangle AOB=\measuredangle XOB-\measuredangle XOA$.But $cos\measuredangle XOA=\frac{a+\overline{a}}{2|a|},cos\measuredangle XOB=\frac{b+\overline{b}}{2|b|}$ and $cos(a-b)=cosa\cdot cosb+sina\cdot sinb$ and after a few working you'll discover what means $\frac{\overline{a}b+a\overline{b}}{2|a||b|}$