Department of Mathematics University of Utah Written Qualifying Exam in Real and Complex Analysis Fall 1997 Instructions

Let \(f(x)=x^{-1/2}\) if \(0<x<1\), \(f(x)=0\) if \(x\le 0\) or \(x\ge 1\), and \(\{r_n\}_{n=1}^{\infty }\) be an enumeration of the rationals. Show that if \[ g(x)=\sum _{n=1}^{\infty }\frac {f(x-r_n)}{2^n} \] then \(g\in L^1(\mathbb R)\) and hence \(g(x)<\infty \) \(m\)-a.e. Show however, that even though \(g^2(x)<\infty \) \(m\)-a.e., that \(g\notin L^2(\mathbb R)\).

Let \(0<p<r<q\). Show that if \((X,\mathfrak M,\mu )\) is a (positive) measure space, then \[ L^r(\mu )\subset L^p(\mu )+L^q(\mu ); \] more precisely, for every \(f\in L^r(\mu )\) there exists \(g\in L^p(\mu )\) and \(h\in L^q(\mu )\) such that \(f=g+h\).

Suppose that \(f\in L^1(\mathbb R^n)\) and \(\hat f\in L^1(\mathbb R^n)\) (\(\hat f\) denoting the Fourier transform of \(f\)). Then show that \[ f\in L^1(\mathbb R^n)\cap L^2(\mathbb R^n). \]

Let \(H\) be a Hilbert space, \(M\) a closed subspace, and \(N\) a finite dimensional subspace of \(H\). Show that \(M+N\) is a closed subspace of \(H\).

Let \(X_1,X_2\) be Banach spaces, \(T\) a linear map of \(X_1\) into \(X_2\), and \(\{\lambda _\alpha \}_{\alpha \in A}\subset X_2^*\) be a subset which separates points of \(X_2\); that is, if \(v_1,v_2\in X_2\), \(v_1\ne v_2\), then there exists \(\alpha \in A\) such that \(\lambda _\alpha (v_1)\ne \lambda _\alpha (v_2)\). Suppose that \(\lambda _\alpha \circ T\in X_1^*\) for all \(\alpha \in A\). Then show that \(T\) is continuous.

Let \(f(z)\) be an entire function and let \(F_n(z)=\sum _{k=1}^{n}\frac {f^k(z)}{k!}\). Show that for every \(R>0\) there exists \(N\) such that all the zeros of \(F_n\), \(n\ge N\), are outside the circle of radius \(R\).

Let \(W=\{z\mid z\in \mathbb C,\ z=x+iy\text { and }y^2-x^2<1\}\). Let \(f\) be an entire function such that \(f(\mathbb C)\subset W\). Show that \(f\) is a constant function.

a. Let \(\Omega \) be an open set containing the closed unit disk, \(f\) holomorphic on \(\Omega \), and suppose that \(f\) has an analytic logarithm on \(\Omega \). Show that if \(z=re^{i\theta }\) with \(0\le r<1\) then \[ \ln |f(z)|=\frac {1}{2\pi }\int _0^{2\pi }P_r(\theta -t)\ln |f(e^{it})|\,dt; \] \(P_r(\theta )\) denoting the Poisson kernel.

b. Let \(g\) be holomorphic on \(\Omega \) and assume that \(g(z)\ne 0\) for all \(|z|=1\). If \(a_1,\ldots ,a_n\) are the zeros of \(g\) in the open unit disk, repeated according to their multiplicities, \(z=re^{i\theta }\), \(0\le r<1\), \(g(z)\ne 0\), then show that \[ \ln |g(z)|=\sum _{j=1}^{n}\ln \left |\frac {z-a_j}{1-\bar a_j z}\right | +\frac {1}{2\pi }\int _0^{2\pi }P_r(\theta -t)\ln |g(e^{it})|\,dt. \]

Let \[ f(z)=\sum _{n=1}^{\infty }\frac {1}{n^z} \] where for \(a>0\), \(a^z\) is defined to be \(e^{z\ln a}\), and \(R=\{z\mid z\in \mathbb C\text { and }\Re z>1\}\). Prove that \(f\) is holomorphic on \(R\).

a. Let \(\Re \alpha <0\), and \(g(z)=\frac {\alpha -z}{\bar \alpha +z}\). Show that \(g\) is a conformal map of \(\{z\mid z\in \mathbb C\text { and }\Re z<0\}\) onto the open unit disk. (Hint: the ordered triple \(\{\alpha ,\frac {\alpha -\bar \alpha }{2},-\bar \alpha \}\) maps to \(\{0,1,\infty \}\) under \(g\).)

b. Let \(f\) be holomorphic on the closed unit disk and suppose that \(\Re f(z)<0\) for all \(z\) in the closed disk. Then show that \[ |f'(0)|\le 2|\Re f(0)|. \]