Analytic Continuation and Monodromy

Prove that if \((f,B(0,1))\) is a function element, \(\{(g_t,U_t)\}_{t\in [0,1]}\) and \(\{(h_t,V_t)\}_{t\in [0,1]}\) are analytic continuations of \((f,B(0,1))\) along the curve \(\phi : [0,1]\to \mathbb {C}\) defined by \(\phi (t)=4t+it\) then \(g_1=h_1\) in a neighborhood of \(4+i\).

Let \(f\) be an analytic function. Show that the successive derivatives of \(f\) at a point can never satisfy \[ |f^{(n)}(z)|>n!n^n \] for all \(n\).

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

Let \(f(z)\) be an analytic function. Show that the successive derivatives of \(f(z)\) at a point can never satisfy \[ |f^{(n)}(z)|>n!n^n \] for all \(n\).

Let \(f(z)\) be analytic in a deleted neighborhood of \(z=a\) and \[ \lim _{z\to a}(z-a)f(z)=0. \] Does it mean that \(f(z)\) has a removable singularity at \(a\)? If so, prove it. If not, give a counterexample.

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

Prove that in any region \(\Omega \) the family of analytic functions whose values lie on an open half-plane is normal.

Prove or give a counterexample: If \(\Omega \) is a region, \(f:\Omega \to \mathbb C\) admits unrestricted analytic continuation on a region \(D\supset \Omega \), then there exists \(F:D\to \mathbb C\), holomorphic and with \(F|_\Omega =f\).

Recall: If \(f:\Omega \to \mathbb C\) is an analytic function, we say \(f\) admits unrestricted analytic continuation on \(D\) if for any \(\gamma :[0,1]\to D\), a piecewise smooth path, there exists an analytic continuation of \((f,\Omega )\) along the curve \(\gamma \).

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 \(U\).

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 \(z\in D\).

Let \(U\) be an open connected domain in the complex plane, and let \(f\) be analytic on \(U\).

(a) Prove or disprove: if \(f\) is injective on \(U\), then the derivative \(df/dz\) is never zero on \(U\).

(b) Prove or disprove: if the derivative \(df/dz\) is never zero on \(U\), then \(f\) is injective on \(U\).

Let \(f(z)\) be an analytic function in the unit disc \(\{z\mid |z|<1\}\) such that \(f(z)\) is bounded in modulus by \(M\): \(|f(z)|\le M\). Let \(0<r<1\). Find a constant \(C\) depending on \(M\) and \(r\) so that for all \(z,w\) of modulus less than \(r\), \[ |f(z)-f(w)|\le C|z-w|. \]

Suppose that \(f\) and \(g\) are one to one analytic functions of the open unit disk \(U=\{z\in \mathbb C:|z|<1\}\) onto some domain \(\Omega \). Suppose that \(f(0)=g(0)\), and \(f'(0)=g'(0)\ne 0\). Show that \(f(z)=g(z)\).