DEPARTMENT OF MATHEMATICS UNIVERSITY OF UTAH REAL AND COMPLEX PRELIMINARY EXAM, 1999

Let \(f\) be a non-negative measurable function and \(\mu \) a finite positive measure on a measure space \((X,\Sigma )\). Prove that \[ \lim _{n\to \infty }\int \frac {f^n(x)}{1+f^n(x)}\,d\mu (x) =\mu (E)+\frac {1}{2}\mu (F), \] where \(E=\{x\in X; f(x)>1\}\) and \(F=\{x\in X; f(x)=1\}\).

Let \(\mu \) be a measure on a measure space \(X\) and let \(f\ge 0\) be a measurable function on \(X\). Prove that \[ \int _X f(x)\,d\mu (x)=\int _0^\infty \phi (y)\,dy \] where \(\phi (y)=\mu (\{x:f(x)\ge y\})\).

Let \(D\) be the closed unit disc in \(\mathbb C\) and \(T\) the unit circle. Prove that for every \(\mu \) positive measure of total mass one on \(D\) there is a positive measure \(\nu \) of total mass one on \(T\) such that \[ \int f\,d\mu =\int f\,d\nu \] for every \(f\) which is continuous on \(D\) and holomorphic on the interior of \(D\).

A bounded linear transformation \(A\) from a Banach space \(X\) to a Banach space \(Y\) is said to be ”compact” if the image under \(A\) of the unit ball in \(X\) has compact closure in \(Y\). Prove that if a compact linear transformation is also surjective, then \(Y\) is finite dimensional (you may assume that a locally compact Banach space must be finite dimensional).

Let \(H\) be a Hilbert space with inner product \((\cdot ,\cdot )\) and \(\{x_n\}\) a sequence of elements of \(H\) with the property that the sequence \(\{(x_n,y)\}\) converges for each \(y\in H\). Prove that there is an element \(x\in H\) so that \(\{(x_n,y)\}\) converges to \((x,y)\) for each \(y\in H\).

Let \(U\) be the open unit disc in \(\mathbb C\). Let \(U^+\) be the intersection of \(U\) with the upper half plane.

(a) Let \(f(z)=\frac {1}{2}\left (z+\frac {1}{z}\right )\). Describe \(f(U^+)\).

(b) Give a conformal equivalence of \(U^+\) and \(U\).

Let \(\lambda >1\). Use Rouche’s Theorem to show that the equation \(\lambda -z-e^{-z}=0\) has one solution for \(\operatorname {Re}(z)>0\).

Let \(f\) be a non-vanishing holomorphic function defined in a neighborhood of the closed unit disc \(\overline U\) such that \(|f(z)|=1\) if \(|z|=1\). Show that \(f\) is constant.

Let \(f\) be an injective entire function. Put \(g(z)=f(1/z)\). Show that \(g\) does not have an essential singularity at \(0\). In particular, it follows that \(f(z)=az+b\). Why?

Compute \[ \int _0^\infty \frac {x^2}{x^4+1}\,dx. \]