Meromorphic Functions and Rational Functions

Show that if \(f\) is holomorphic on an open neighborhood of \(\overline {\mathbb {D}}\) and \(|f(z)|=1\) on \(|z|=1\) then \(f\) is a rational function.

Prove the following special case of Runge’s theorem: Suppose that \(f:D(0,4)\setminus (\overline {D(1,.1)}\cup \overline {D(-1,.1)})\to \mathbb {C}\) is holomorphic and \(\epsilon >0\) is given. Then there is a rational function \(q\) such that \(|f(z)-q(z)|\le \epsilon \) for \(z\in K=\) \[ \overline {D(0,2)}\setminus (\overline {D(1,.2)}\cup \overline {D(-1,.2)}). \]

Prove the following special case of Runge’s approximation theorem: If

is holomorphic and \(\epsilon >0\) is given then there exists \(q\), a rational function, so that

Show that any function which is meromorphic on the Riemann sphere, \(\widehat {\mathbb {C}}=\mathbb {C}\cup \{\infty \}\), is either constant or can be written as the sum of the singular parts of its Laurent series at its poles, in particular conclude that the function is rational.

Show that if \(f:\{z:0<|z|<1\}\to \mathbb {C}\) is meromorphic and has poles at \(\frac {1}{2},\frac {1}{3},\frac {1}{4},\ldots \) then the range of \(f\) is dense in \(\mathbb {C}\).

If \(f(z)\) is analytic in \(|z|\le 1\) and satisfies \(|f|=1\) on \(|z|=1\), show that \(f(z)\) is rational.

Prove the following special case of Runge’s approximation theorem: If \(f:B(0,4)\setminus (B(0,.1)\cup B(1,.1))\) is holomorphic and \(\epsilon >0\) is given, then there exists a rational function \(q\) so that \(|f(z)-q(z)|<\epsilon \) for all \[ z\in \overline {B(0,2)\setminus (B(0,.2)\cup B(1,.2))}. \]

If \(f(z)\) is analytic in \(|z|\le 1\) and satisfies \(|f|=1\) on \(|z|=1\), show that \(f(z)\) is rational.

Let \(f\) be a meromorphic function on \(\mathbb C\) such that there exist \(R,C>0\) and a positive integer \(k\) such that \(|f(z)|\le C|z|^k\) for all \(|z|\ge R\). Prove that \(f\) is a rational function.

Let \(f\) be a meromorphic function on \(\mathbb C\) such that \(|f(z)|<B\) if \(|z|>R\). Show that \(f\) is a rational function (a quotient of polynomials).

Let \(a\) be an isolated singularity of the meromorphic function \(f:\Omega \to \mathbb C\). Prove that if \(a\) is an essential singularity, then in any neighborhood of \(a\), the function \(f\) takes values arbitrarily close to any complex number.

Let \(f\) be a function holomorphic on \(D\) and continuous on \(\overline {D}\). Assume that \(|f(z)|=1\) whenever \(|z|=1\). Show that \(f\) can be extended to a meromorphic function on the whole \(\mathbb C\), with at most finitely many poles.

(Hint: Starting with the Schwarz reflection principle for the upper half plane, deduce that an appropriate reflection continuation that can be used in this setting is \(z\mapsto \frac {1}{f(1/\bar z)}\).)

Let \(\Delta \) be the open unit disk and \(\gamma \) the unit circle in \(\mathbb C\) and \(\phi :\gamma \to \mathbb C\) a continuous function. Let \(g\) be a meromorphic function on \(\mathbb C\) that has a single simple pole at \(0\). Define a function \(f:\Delta \to \mathbb C\) by the formula \[ f(z)=\int _\gamma \phi (w)g(w-z)\,dw. \] Show that \(f\) is holomorphic.

Let \(f\) be a non-constant meromorphic function on \(\mathbb C\). Show that either there exists a sequence \(z_n\to \infty \) with \(f(z_n)\to 0\) as \(n\to \infty \) or there is a \(z\in \mathbb C\) such \(f(z)=0\).

Fix positive integers \(k\) and \(l\). Describe the space of meromorphic functions on the Riemann sphere that have a have a pole of order at most \(k\) at \(\infty \) and a zero of order at least \(l\) at the origin.

Let \(f\) be a meromorphic function on the Riemann sphere. Show that \(f\) is a rational function.

Suppose that \(f\) is meromorphic on the plane, is never zero on the plane, and that \[ \lim _{z\to \infty }|f(z)|=0. \] Show that \(f\) is the inverse of a polynomial.

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 \(g\notin L^2(\mathbb R)\).

(a) Let \(f\) be an entire function. Assume that there exists a polynomial \(g\) such that for each \(z\in \mathbb C\), \(|f(z)|\le |g(z)|\). Show that \(f\) is also a polynomial.

(b) Can we draw the same conclusion if the function \(g\) is rational? In other words, let \(f\) be an entire function and \(g\) a rational function such that for each \(z\in \mathbb C\), which is not a pole of \(g\), \(|f(z)|\le |g(z)|\). Does it follow that \(f\) is a polynomial?

Show that if \(f\) is a meromorphic function in \(\Delta (0;1+h)=\{z:|z|<1+h\}\), where \(h>0\), and \(|f(z)|=1\) for \(|z|=1\), then \(f\) is a rational function.