91Ó°ÊÓ

Prove that if \(f\) is a continuous complex-valued function in the open subset \(G\) of \(\mathbf{C}\), and if \(\int_{R} f(z) d z=0\) for every rectangle \(R\), with edges parallel to the coordinate axes, contained with its interior in \(G\), then \(f\) is holomorphic.

Let \(f\) be a holomorphic map of the open unit disk into itself. Prove that, for any two points \(z\) and \(w\) in the disk, $$ \left|\frac{f(z)-f(w)}{1-f(z) \overline{f(w)}}\right| \leq\left|\frac{z-w}{1-z \bar{w}}\right| $$ and that the inequality is strict for \(z \neq w\) except when \(f\) is a linear- fractional transformation mapping the disk onto itself.

Prove that an entire function with a positive real part is constant.

Prove that, except for the identity function, a holomorphic map of the open unit disk into itself has at most one fixed point in the disk.