Department of Mathematics University of Utah Ph.D. Preliminary Examination in Real/Complex Analysis January 7, 2004 INSTRUCTIONS

Let \((X,\mathfrak M,\mu )\) be a positive measure space, and let \(f\in L^1(X)\). Let \(X_n=\{x\in X: |f(x)|>n\}\). Show that \[ \lim _{n\to \infty } n\mu (X_n)=0. \]

Let \(\{f_n\}\) be a sequence of complex valued measurable functions defined on a finite interval \((a,b)\subset \mathbb R\) (”measurable” means with respect to the Lebesgue measure). Suppose that there exists a constant \(M<\infty \) such that \(\int _a^b |f_n(x)|^2\,dx<M\) for all \(n\), and \(f_n(x)\to 0\) a. e. Prove that \[ \lim _{n\to \infty }\int _a^b f_n(x)\,dx=0. \] Show also that this assertion fails in general if one replaces the interval \((a,b)\) by \(\mathbb R\) (give a counterexample).

Let \(X=[0,1]\), and \(C(X)\) the space of complex valued continuous functions on \(X\), equipped with the sup norm: \(\|f\|_\infty =\sup _{x\in X}|f(x)|\). Let \(f_n\) be a sequence in \(C(X)\). Prove that the following statements are equivalent:

(a) For each bounded linear functional \(\lambda \) on \(C(X)\), \(\lambda (f_n)\to 0\);

(b) \(f_n(x)\to 0\) for each \(x\in X\), and \(\sup _n\|f_n\|_\infty \) is finite.

Let \((X,\mathfrak M,\mu )\) be a positive measure space, and let \(f\ge 0\) be a measurable function on \(X\). Define \(g(t)=\mu (\{x:f(x)\ge t\})\). Show that \[ \int _X f(x)\,d\mu (x)=\int _0^\infty g(t)\,dt. \]

Let \(\lambda \) be a bounded linear functional on a linear subspace \(V\) of a Hilbert space \(H\). Show that there exists a unique bounded linear functional \(\Lambda \) on \(H\) such that: (a) \(\|\Lambda \|=\|\lambda \|\), and (b) \(\Lambda (x)=\lambda (x)\) for \(x\in V\).

Let \(f:D\to D\) be holomorphic. Show that \[ |f(z)-f(0)|\le |z(1-\overline {f(0)}f(z))| \] for all \(z\in D\).

Let \(f\) be an entire function such that \(|\Re f(z)|+|\Im f(z)|\ge 1\) for each \(z\in \mathbb C\). Prove that \(f\) is a constant function.

Characterize all entire functions \(f\) which satisfy the property: \[ \lim _{z\to \infty }\frac {1}{f(z)}=0. \]

Let \(\Omega \) open in \(\mathbb C\), \(b\in \Omega \), \(f\in H(\Omega )\), and assume that \(f'(b)\ne 0\). Show that \[ \frac {2\pi i}{f'(b)}=\int _C \frac {1}{f(z)-f(b)}\,dz \] for sufficiently small positively oriented simple circles centered at \(b\).

Determine the poles and their orders of the function \[ \frac {1}{e^z-1}-\frac {1}{z}. \]