DEPARTMENT OF MATHEMATICS University of Utah Real and Complex Analysis Preliminary Examination August 21, 2013

Let \((X,M,\mu )\) be a measure space, and suppose that \(1\le p<q<r\). Consider the vector space \(L^p\cap L^r\), with norm \(\lVert f\rVert =\lVert f\rVert _p+\lVert f\rVert _r\). Show that \(L^q\subset L^p\cap L^r\), and that the inclusion map \(i:L^q\to L^p\cap L^r\) is continuous.

Let \(V\) and \(W\) be Banach spaces, and \(L(V,W)\) the space of bounded linear maps from \(V\) to \(W\). Suppose that \(T_n\in L(V,W)\), \(n=1,2,\ldots \), and that \(\lim T_nx\) exists for each \(x\in V\). Show that \(T:V\to W\) defined by \(Tx=\lim T_nx\) is in \(L(V,W)\).

Consider \(\mathbb R\) with the Lebesgue measure. Show that \(L^2(\mathbb R)\) is a complete Hilbert space (hint: find a countable, complete orthonormal system).

Let \((X,M,\mu )\) be a measure space, and suppose that \(f:X\times [0,1]\to \mathbb R\) such that \(f(\cdot ,t):X\to \mathbb R\) is integrable for all \(t\). Define \[ F(t)=\int _X f(x,t)\,d\mu (x) \] for \(t\in [0,1]\). Suppose that \(\frac {\partial f}{\partial t}(x,t)\) exists for all \(x\) and \(t\), and that for some integrable function \(g\), \(\left |\frac {\partial f}{\partial t}(x,t)\right |\le g(x)\) for all \(x\) and \(t\). Show that \(F\) is differentiable and that \[ F'(t)=\int _X \frac {\partial f}{\partial t}(x,t)\,d\mu (x) \]

Let \((X,M,\mu )\) be a \(\sigma \)-finite measure space, and consider \(X\times X\) with the product measure. Suppose that \(K\in L^2(X\times X)\). Show that if \(f\in L^2(X)\), then \(Tf\in L^2(X)\) and \(\lVert Tf\rVert _2\le \lVert K\rVert _2\lVert f\rVert _2\) where \[ Tf(x)=\int _X K(x,y)f(y)\,d\mu (y). \]

Let \(f\) and \(g\) be analytic functions defined on an open set \(U\subset \mathbb C\). Assume that \(g\) is not identically equal to zero. Prove that \(F=f/g\) does not have essential singularities in \(U\).

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

Determine the numbers of zeroes of the polynomial \(z^4+2x^2+5z+1\) in the set \(|z|\le 1\).

Let \(D=\{z\in \mathbb C:|z|<1\}\), and let \(f:D\to D\) be an analytic function. Show that \[ |f(z)-f(0)|\le |z(1-\overline {f(0)}f(z))| \] for all \(z\in D\).

Evaluate \[ \int _\Gamma \frac {\sin z}{z^6}\,dz \] where \(\Gamma \) is a positively oriented simple unit circle \(\{z\in \mathbb C:|z|=1\}\).