Problem related with univalent function I am stuck with the following problem: As wikipedia says : "In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is one-to-one." A related example is : > Any mapping $\phi_a$ of the open unit disc to itself : $$\phi_a(z)=\frac{z-a}{1-\bar a z}\,\,\,\text{where}\,\,|a| \leq 1$$ is $\color{red}{\text{univalent}}$. According to the definition,to be univalent $\phi_a(z)$ has to one-one. But if I take $a=1,$then for $z_1 \neq z_2,\,\,\,\text{where}\,\,z_1,z_2 \,\,\text{are any 2 points in the open unit disc,}$ we see $f(z_1)=f(z_2)$. So, How this function can be 1-1. Where I went wrong? Can someone help me? Thanks in advance for your time.

As Daniel Fischer said, the inequality should have been $|a|<1$. I edited the article.