Holomorphic Roots and Logarithms

Show that there is no proper holomorphic map \(f:\mathbb {D}\to \mathbb {C}\). Recall that “proper” means that \(|z_n|\to 1\) implies \(|f(z_n)|\to \infty \) for any sequence \(z_n\in \mathbb {D}\).

Let \(f:B(0,1)\to \mathbb {C}\) be holomorphic. Describe all such \(f\) that satisfy \[ f\left (\frac {1}{n}\right )=\frac {n^2-2n-1}{n^2} \] for every \(n\in \{2,3,\ldots \}\).

Let \(f:\mathbb {C}\setminus \{0,1,2\}\to \mathbb {C}\) be a holomorphic function. Show that if \(f\) omits at least 4 values then it is constant.

Show that if \(f:B(0,1)\to \mathbb {C}\) is holomorphic and \(\operatorname {Re}(f'(z))>0\) for all \(z\in B(0,1)\) then \(f\) is injective.

Let \(f:\mathbb {D}\to \mathbb {D}\) be holomorphic and \(f(0)=a\). Show that \(f\) has no zeros in \(\{z\in \mathbb {C}\mid |z|<|a|\}\).

Prove the following version of Morera’s theorem. If \(f:\mathbb {D}\to \mathbb {C}\) is a continuous function such that for every rectangle \(R\subset \mathbb {D}\)

then \(f\) is holomorphic. Note: It does not follow from the assumption in any obvious way that integrals over triangles are 0.

Let \(f\) be holomorphic on \(\mathbb {C}\setminus \{0,2\}\) with simple poles at 0 and 2. Suppose that \(\int _{|z|=1} f(z)\,dz=2\pi i\) and \(\int _{|z|=3} f(z)\,dz=0\). Also suppose that \(f\) is bounded on \(|z| > 3\). List all such functions (with a proof that the list is complete).

Suppose a sequence of holomorphic functions \(f_n:\mathbb {D}\to \mathbb {C}\) converges pointwise to a function \(f:\mathbb {D}\to \mathbb {C}\), and that all \(f_n\) are uniformly bounded. Show that convergence is uniform on every disk

for \(r<1\).

Let \(\Omega \) be a simply connected domain in \(\mathbb {C}\). Suppose that \(f:\Omega \to \mathbb {C}\) is holomorphic and has finitely many zeros in \(\Omega \). What is the supremum of \(m\in \mathbb {N}\) so that \(f(z)=g(z)^m\) for some \(g\) holomorphic in \(\Omega \)? Identify this value explicitly in terms of the orders of the zeros of \(f\) and prove that it is correct.

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.

Let \(\Omega \) be a bounded domain in \(\mathbb {C}\) and \(f_n:\Omega \to \mathbb {C}\) a sequence of holomorphic functions such that \[ \int _\Omega |f_n-f_m|^2\,dA(z)\to 0\text { as }n,m\to \infty . \] Here \(dA(z)\) is the Lebesgue area integral. Show that there is \(f\) holomorphic on \(\Omega \) such that \(f_n\to f\) locally uniformly on \(\Omega \).

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 \(z\in K=\) \[ \overline {D(0,2)}\setminus (\overline {D(1,.2)}\cup \overline {D(-1,.2)}). \]

Let \(\Omega \) be a simply connected domain in \(\mathbb {C}\). Show that if a holomorphic function \(f:\Omega \to \mathbb {C}\) has finitely many zeros, all of even order, then \(f\) has a holomorphic square root in \(\Omega \), i.e. there is a holomorphic function \(g:\Omega \to \mathbb {C}\) so that \(f(z)=g(z)^2\).

Show that if \(f\) is holomorphic on \(\mathbb {D}\setminus \{0\}\) and

then \(f\) can be extended to a holomorphic function on \(\mathbb {D}\).

Show that there is no function \(f\) holomorphic on \(\mathbb {D}\) so that \[ f\left (\frac {1}{n}\right )=\frac {1}{n+2} \] for all large \(n\in \mathbb {N}\).

Suppose \(f:\mathbb {D}\setminus \{0\}\to \mathbb {C}\) is holomorphic and has an essential singularity at 0. Show that for all \(\epsilon > 0\) there is \(z\) near 0 so that \[ \left |f(z)-\frac {1}{z^4}\right |\le \epsilon . \]

Suppose that \(f:\mathbb {D}\to \mathbb {D}\) is holomorphic and zero at points \(z_1,\ldots ,z_n\in \mathbb {D}\) counted with multiplicity. Show that

where

Show that if \(D\subsetneq B(0,1)\) is simply connected and \(0\in D\) then there exists an injective, holomorphic function \(f:D\to B(0,1)\) so that \(f'(0)>1\). -Note that this is part of the proof of the Riemann mapping theorem, so please do not use the Riemann mapping theorem in the proof.

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

Let \(D\subset \mathbb {C}\) be a connected open set and \(f_1,\ldots :D\to \mathbb {C}\) be a sequence of holomorphic functions that converges uniformly on compact sets to a function \(g\). Show that \(g\) is holomorphic and \(g'(z)=\lim _{j\to \infty } f_j'(z)\) for all \(z\in D\).

Show that if \(f,g:B(0,1)\setminus \{0\}\to \mathbb {C}\) are holomorphic, \(f\) has an essential singularity at 0 and \(g\) has a pole at 0 then \(\frac {g}{f}\) has an essential singularity at 0.

Give an example of a domain \(D\) and a holomorphic function \(f:D\to \mathbb {C}\), so that \(f\) is not given by a single Laurent series on \(D\). (Please include a proof.)

Show that if \(f(z)\) is entire and nowhere zero then there is an entire function \(g(z)\) such that \(f(z)=g(z)^2\).

Suppose that \(f_n\) are holomorphic on a domain \(U\subset \mathbb {C}\) and \(f_n(U)\subset \mathbb {C}\setminus \{w\}\) for all \(n\) and some fixed \(w\in \mathbb {C}\). Show that if \(f_n\to f\) locally uniformly in \(U\) then either \(f(z)\equiv w\) or \(f(U)\subset \mathbb {C}\setminus \{w\}\).

Suppose that \(f\) is holomorphic on the upper half plane \(\mathbb {H}=\{\operatorname {Im}(z)>0\}\), \(f\) extends continuously to \(\overline {\mathbb {H}}=\mathbb {H}\cup \mathbb {R}\), and \(f(\mathbb {R})\subset \mathbb {R}\). Show that \(f\) has a unique holomorphic extension to \(\mathbb {C}\) and explicitly identify that extension (in terms of \(f\)).

Let \(f_i:\mathbb {D}\to \mathbb {C}\) be a sequence of injective holomorphic functions so that \(f_i\to f_\infty \) uniformly on compact sets. If \(f_\infty \) is not injective, what can you say about it? Prove this.

Prove that if \(f:\{z:1<|z|<2\}\to \mathbb {C}\) is holomorphic then \[ f(z)=\sum _{j=-\infty }^{\infty } a_j z^j. \] What can you say about the power series \(\sum _{j=0}^{\infty }a_jz^j\) and \(\sum _{j=-\infty }^{0}a_jz^{-j}\)? Prove this.

Let \(f:\mathbb {H}\to \mathbb {D}\) be a holomorphic function so that \(f(i)=0\). What can you say about \(|f(2i)|\)? (Of course with proof.)

State the Cauchy Riemann equations and prove that a function is holomorphic on an open set iff it satisfies the Cauchy Riemann equations.

Give an example of a sequence of numbers \(a_1,\ldots \) so that there is no holomorphic function \(f:\mathbb D\to \mathbb C\) so that \(a_1=f'(0)\), \(a_2=f''(0)\), \(a_3=f'''(0)\), \(\ldots \). Justify your answer.

Prove or disprove: \[ \{f:\mathbb H\to \mathbb H\mid f\text { is holomorphic}\} \] is a normal family.

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 \[ z\in \overline {B(0,2)\setminus (B(0,.2)\cup B(1,.2))}. \]

Show that if \(f:\mathbb D\to \mathbb D\) is holomorphic then there exist \(n_1,n_2,\ldots \) so that \(f^{n_i}(z)\) converges pointwise for all \(z\in \mathbb D\). Note that \(f^{n_i}\) denotes \[ \underbrace {f\circ \cdots \circ f}_{n_i\ \text {times}}. \]

Show that if \(f:\mathbb D\to \mathbb C\) is holomorphic and \(f'(0)\ne 0\) then there is a neighborhood of \(0\) where \(f\) is injective. Do not use the inverse function theorem.

Let \(f:\mathbb H\to \mathbb D\) be a bijective holomorphic map. Are we guaranteed that \[ -f(\bar z)=-\overline {f(z)}? \] In words, is negative \(f\) at the conjugate input equal to the negative complex conjugate of the original output?

Let \(f:D\to D\) be holomorphic. Show that \[ |f(z)-f(0)|\le |z(1-\overline {f(0)}f(z))| \] for all \(z\in D\). In words, the coefficient multiplying \(f(z)\) in the parentheses is the complex conjugate of \(f\) at zero.

Let \(f\) be a function holomorphic for \(z\) in the annulus \(r<z<R\). Show that if for some \(\rho \) with \(r<\rho <R\), \[ \int _{|z|=\rho } z^n f(z)\,dz=0 \] for all nonnegative \(n\), then \(f\) extends to a holomorphic function on the disk \(|z|<R\).

Let \(f:\mathbb C\to \mathbb C\) be a function such that:

(a) \(f\) is continuous;

(b) \(f\) is holomorphic at all points \(z\) not on the real axis;

(c) if \(z\) is on the real axis then \(f(z)\) is real.

Show that \(f(\bar z)=\overline {f(z)}\) and that \(f\) is holomorphic on all of \(\mathbb C\). In words, applying \(f\) to the conjugate input gives the complex conjugate of the original output.

Let \(f\) be a holomorphic function on \(D\) such that \(|f(1/n)|\le 1/2^n\) for all positive integers \(n\). Show that \(f\) is the zero function.

Let \(f\) be a holomorphic function on \(D\) such that \(f^{(n)}(0)\ne 0\) for an infinite number of positive integers \(n\). Show that there exists a point \(z_0\in D\) such that \(f^{(n)}(z_0)\ne 0\) for all \(n>0\).

What are the possible ranges of holomorphic functions from \(\mathbb C\setminus \{\pi \}\) to \(\mathbb C\)?

Let \(\mathbb D\) be the unit disk \(\{z\in \mathbb C\mid |z|<1\}\). Let \(f\) be a holomorphic function on \(\mathbb D\setminus \{0\}\) such that \[ |f(z)|\le \log \left (\frac {1}{|z|}\right ) \quad \text {for all }z\in \mathbb D\setminus \{0\}. \] Show that \(f=0\).

Let \(f\) be a holomorphic function on the unit disk \(\mathbb D\). Then there exists \(n\in \mathbb N\) so that \[ f\left (\frac {1}{n}\right )\ne \frac {1}{n+2}. \]

Let \(f:U\to \mathbb C\) be a holomorphic function, where \(U\) is an open set. Let \(a\in U\). A standard result in complex analysis is that if \(f'(a)\ne 0\), then there is an open neighborhood \(V\subset U\) of \(a\) on which \(f\) is injective and the image \(f(V)\) is a neighborhood of \(f(a)\). Prove this result!

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 \(\gamma \).

Classify entire biholomorphisms. That is, what are all \(f:\mathbb C\to \mathbb C\) holomorphic with a holomorphic inverse \(g:\mathbb C\to \mathbb C\)?

Does there exist \(f:\mathbb D\to \mathbb C\) holomorphic so that \[ f\left (\frac {1}{n}\right )f\left (\frac {1}{n+1}\right )=\frac {1}{n} \] for all \(n\in \{2,3,\ldots \}\)?

Let \(\mathcal F\) be a family of holomorphic functions on \(\mathbb D\) so that there exists \(C\) with \(|f(z)|<C\) for all \(z\in \mathbb D\).

(a) If \(D\subset \mathbb D\) satisfies that the closure \(\overline D\) is contained in \(\mathbb D\), then \(\mathcal F\) is uniformly Lipschitz on \(D\). That is, there exists \(u\) so that for any \(f\in \mathcal F\) and \(w,z\in D\) we have \[ |f(z)-f(w)|<u|z-w|. \]

(b) Show that \(\mathcal F\) is not necessarily a uniformly Lipschitz family on \(\mathbb D\).

Let \(f\) be a holomorphic function defined on a domain \(\Omega \) in \(\mathbb C\). Show that \(f\) is either a polynomial, or there exists \(z\in \Omega \) such that \[ f^{(n)}(z)\ne 0 \] for all integers \(n\ge 0\).

Let \(f\) be an a holomorphic function defined on an open set \(U\subset \mathbb C\) and \(a\in U\). Assume that \(f'(a)\) is nonzero. Show that for sufficiently small circles \(C\) centered at \(a\) \[ \frac {2\pi i}{f'(a)}=\int _C \frac {dz}{f(z)-f(a)} \]

Let \(f\) be a holomorphic function and \(a\) an isolated singularity of \(f\). Prove that the following statements are equivalent:

(a) there exists a disk \(D(a,\epsilon )=\{z\in \mathbb C\mid |z-a|<\epsilon \}\) and a constant \(C>0\) such that \[ |f(z)|\ge \frac {C}{|z-a|^n} \] for all \(z\in D(a,\epsilon )-\{a\}\);

(b) \(f\) has a pole of order \(m\ge n\) at \(a\).

Let \(f\) be a holomorphic function in a neighborhood of the closed unit disk centered at \(0\). Assume that \(|f(z)|<1\) for any \(z\) on the unit circle centered at \(0\). How many fixed points does \(f\) have inside this unit circle?

Let \(f\) be a holomorphic function defined on a domain \(U\subset \mathbb C\). Let \[ \bar U=\{z\in \mathbb C\mid \bar z\in U\}. \] Consider the function \(g\) on \(\bar U\) defined by \[ g(z)=\overline {f(\bar z)}. \] Equivalently, \(g(z)\) is the complex conjugate of \(f(\bar z)\). Is \(g\) holomorphic or not? Explain your answer!

Let \(f\) be a holomorphic function on \(U\subset \mathbb C\). Let \(a\in U\) and let \(k\) be a positive integer. Prove that the following statements are equivalent:

(i) \(a\) is a zero of order \(\ge k\) of \(f\);

(ii) there exist \(C,\epsilon >0\) such that \[ |f(z)|\le C|z-a|^k \] for all \(z\) such that \(|z-a|<\epsilon \).

Let \(f:\mathbb C\to \mathbb C\) be a function such that:

(a) \(f\) is continuous;

(b) \(f\) is a holomorphic at all points \(z\) not on the real axis;

(c) if \(z\) is on the real axis then \(f(z)\) is real.

Show that \(f(\bar z)=\overline {f(z)}\) and that \(f\) is holomorphic on all of \(\mathbb C\).

Let \(f\) be a function holomorphic for \(z\) in the annulus \(r<z<R\). Show that if for some \(\rho \) with \(r<\rho <R\), \[ \int _{|z|=\rho } z^n f(z)\,dz=0 \] for all integers \(n\) with \(n\ge 0\) then \(f\) extends to a holomorphic function on the disk \(|z|<R\).

Let \(f\) be a holomorphic function on \(\mathbb C\setminus \{0\}\) such that \(f(z)\sin z\to 1\) as \(z\to 0\). Calculate \[ \int _\gamma f(z)\,dz \] where \(\gamma \) is the unit circle traversed counter-clockwise.

Let \(Q\) be a rectangle in \(\mathbb C\) and \(f:\mathbb C\to \mathbb C\) a holomorphic map. Assume that there exists a \(z\in Q\) such that \(f(z)=z\) and \(f'(z)=1\). Show that \(f\) is the identity map.

Let \(\Omega \subsetneq \mathbb C\) be a simply connected region, and fix \(z_0\in \Omega \). If \(\phi :\Omega \to D\) is a conformal map such that \(\phi (z_0)=0\), show that \[ |\phi '(z_0)| =\sup \{|f'(z_0)|:\ f:\Omega \to D\text { holomorphic}, f(z_0)=0\}. \]

Let \(Q\) be a square in \(\mathbb C\) and \(f:Q\to \mathbb C\) a continuous map that is holomorphic on the interior of \(Q\). Also assume that if \(z\in \partial Q\) then \(|f(z)|=1\). Show that \(f\) extends to a holomorphic map on a neighborhood of \(Q\).

Show that the function \[ f(z)=\frac {\cos z}{z^2} \] is the complex derivative of a holomorphic function \(F\) on \(\mathbb C\setminus \{0\}\). Write down a Laurent series for \(F\).

Let \(\Omega \) be an open subset of \(\mathbb C\) and \(f:\Omega \to \mathbb C\) a holomorphic function. Let \(z\) be a point in \(\Omega \) and assume that \(\Omega \) contains a disk \(D\) centered at \(z\) of radius \(R\). Also assume that \(f(D)\) is contained in a disk \(D'\) centered at \(f(z)\) of radius \(r\). Show that \(|f'(z)|\le r/R\).

Let \(\Omega \) be an open connected subset of \(\mathbb C\) and \(\mathcal F\) a family of holomorphic functions with domain \(\Omega \) such that for every compact subset \(K\) of \(\Omega \) there is a positive real constant \(M_K\) such that for all \(f\in \mathcal F\) and \(z\in K\) we have \(|f(z)|<M_K\). Show that for every \(z\in \Omega \) there exists a positive constant \(B_z\) such that \(|f'(z)|<B_z\) for all \(f\in \mathcal F\).

Let \(f\) be a holomorphic function on an open set containing \(\overline {D_1}(0)\). Prove that if \(z\) is in \(D_1(0)\), then \[ f(z)=\frac {1}{2\pi i}\int _\gamma \frac {f(w)}{\sin (w-z)}\,dw \] where \(z\in D_1(0)\) and \(\gamma \) traverses the unit circle once in the positive direction.

Suppose \(f\) is holomorphic in an open set \(\Omega \) and \(\overline {D_R}(z_0)\subset \Omega \). If \(f\) has a zero of order \(k\) at \(z_0\) and no other zeroes in \(\overline {D_R}(z_0)\), then prove that there is an \(\epsilon >0\) such that, if \(|\lambda |<\epsilon \), then \(f\) takes on \(\lambda \) as a value \(k\) times (counting multiplicity) in the disc \(D_R(z_0)\).

Let \(f\) be a holomorphic function with isolated singularity at point \(a\). Assume that \(g=\frac {1}{f}\) also has an isolated singularity at \(a\), and that \(a\) is not a removable singularity for either \(f\) or \(g\). Determine what type of isolated singularity \(a\) is for \(f\) and \(g\).

Do you know an example of such a function \(f\)?

The series \[ f(z)=\sum _{n=1}^{\infty }\frac {1}{n^2}z^n \] defines a holomorphic function near the origin. Describe a maximal open subset of the complex plane on which \(f\) can be analytically continued.

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 a holomorphic automorphism of \(\mathbb C^*\). Show that either \(f(z)=\alpha z\) or \(f(z)=\frac {\alpha }{z}\) for some complex number \(\alpha \ne 0\).

Let \(f\) be holomorphic on the unit disc \(U\). Let \(h(z,w)\) be the function on \(U\times U\) defined by \[ h(z,w)= \begin {cases} \dfrac {f(z)-f(w)}{z-w} & \text {if } z\ne w,\\[0.6ex] f'(z) & \text {if } z=w \end {cases} \] Prove that \(f\) is continuous as a function of two variables on \(U\times U\). You need only check the continuity at points of the diagonal. At other points it is obvious.

Let \(\Omega \) be an unbounded domain in the plane with the property that there exists a sequence \(\{h_n\}\) of functions holomorphic on a neighborhood of \(\overline {\Omega }\) such that each \(h_n\) vanishes at infinity on \(\Omega \), \(|h_n|\le 1\) on \(\overline {\Omega }\) for each \(n\), and \(\{h_n\}\) converges uniformly to \(1\) on each compact subset of \(\overline {\Omega }\). Prove that if \(f\) is a function which is bounded and holomorphic in a neighborhood of \(\overline {\Omega }\) and if \(|f|\le 1\) on \(\partial \Omega \), then \(|f|\le 1\) on all of \(\overline {\Omega }\).

Suppose \(f\) is continuous on \(U=\{z\in \mathbb C: |z|<1\}\) and holomorphic on the complement in \(U\) of the interval \((-1,1)\). Prove that \(f\) is actually holomorphic on \(U\).

Let \(f\) be a complex valued function defined on the domain \(D\subset \mathbb C\). Suppose, that as a mapping of \(\mathbb R^2\) to \(\mathbb R^2\), \(f=u+iv\) is twice differentiable, and that the jacobian \[ \begin {pmatrix} \dfrac {\partial u}{\partial x} & \dfrac {\partial u}{\partial y}\\[0.6ex] \dfrac {\partial v}{\partial x} & \dfrac {\partial v}{\partial y} \end {pmatrix} \] is everywhere, as a linear transformation, a composition of a dilation and a rotation. Show that \(f\) is a holomorphic function.

Let \(D\) be a domain in the plane, and \(\{f_n\}\) a sequence of functions holomorphic on \(D\). Suppose the sequence is uniformly bounded; that is, there is a positive number \(M\) such that \[ |f_n(z)|<M \quad \text {for all } n \text { and } z\in D. \] Suppose, in addition, that there is an infinite compact set \(K\) in \(D\) such that the sequence \(f_n|K\) converges uniformly on \(K\). Show that the sequence \(f_n\) converges uniformly on all compact subsets of \(D\).

Let \(f\) be a holomorphic function defined on a neighbourhood of the closed disc \(|z|\le 1\). Assume that \(f(0)=1\), and \(|f(z)|>1\) if \(|z|=1\). Must \(f\) have a zero in the open disc \(|z|<1\)? Justify your answer.

Suppose \(f\) is holomorphic in \(r<|z|<R\), and suppose that for some \(\rho \), \(r<\rho <R\), \[ \int _{|z|=\rho } f(z)z^n\,dz=0 \] for all negative integers \(n\). Show that \(f\) has a holomorphic extension to the set \(|z|>r\).

Let \(f\) be a holomorphic function on the unit disc. For every \(0\le r<1\) define \[ M(r)=\max _{|z|=r}|f(z)|. \] Show that \(M(r)\) is an increasing, continuous function on \([0,1)\).

Let \(f\) be a non-vanishing holomorphic function defined in a neighborhood of the closed unit disc \(\overline U\) such that \(|f(z)|=1\) if \(|z|=1\). Show that \(f\) is constant.

Let \(f\) be holomorphic in \(\Delta -\{0\}\). Assume that \(f'\) is bounded in \(\Delta -\{0\}\). Show that \(f\) extends to a holomorphic function on \(\Delta \).

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 \[ |f'(0)|\le 2|\Re f(0)|. \]

Let \(D\) be an open set in \(\mathbb C\). Let \(\mathcal F\) be a collection of holomorphic functions defined on \(D\).

(a) Assume \(\mathcal F\) is uniformly bounded on the boundary \(\partial \Delta \) of each triangle \(\Delta \) contained in \(D\). Show that \(\mathcal F\) forms a normal family.

(b) Is \(\mathcal F\) a normal family if we assume it is uniformly bounded when restricted to any interval contained in \(D\)?

If \(f\) is holomorphic in \(\Delta (0;1)=\{z\in \mathbb C:|z|<1\}\) and \(r<1\) prove that there exists a constant \(M\) so that \[ |f(z)-f(w)|\le M|z-w| \] for all \(z,w\in \Delta (0;r)=\{z\in \mathbb C:|z|<r\}\).

Let \(P(z)=a_nz^n+\cdots +a_0\) and suppose that \(P\) has zeroes at \(z_1,\ldots ,z_r\) each with multiplicity \(m_1,\ldots ,m_r\) respectively. Let \(\Omega \) be some domain not containing \(z_1,\ldots ,z_r\), and \(\gamma \) a cycle in \(\Omega \). Suppose further that there exists \(g\) holomorphic on \(\Omega \) such that \(g^p=P\). Show that there exists \(k\in \mathbb Z\) such that \[ m_1\operatorname {Ind}(\gamma ,z_1)+\cdots +m_r\operatorname {Ind}(\gamma ,z_r)=kp. \]

(a) Let \(D\) be an open connected set in \(\mathbb C\), and \(f\) a holomorphic function on \(D\). Show that if either the real or imaginary part of \(f\) is constant, then \(f\) is constant.

(b) Let \(f\) be an entire function. Show that if either the real or imaginary part of \(f\) is a bounded function then \(f\) is a constant.

Let \(D\) be an open connected set in \(\mathbb C\). Let \(\mathcal F\) be a collection of holomorphic functions defined on \(D\), which is uniformly bounded on each circle contained in \(D\). Show that \(\mathcal F\) forms a normal family.

Show that every continuous function on \(\overline {\Delta }=\{z:|z|\le 1\}\) which is holomorphic on \(\Delta =\{z:|z|<1\}\) can be uniformly approximated on \(\overline {\Delta }\) by holomorphic polynomials.

Suppose \(f\) is holomorphic in \(\{z:|z|<1\}\) and satisfies \(|f(\frac {1}{n})|\le (\frac {1}{n})^n\) for \(n=2,3,\ldots \). Can you identify \(f\)? Justify your answer.

Let \(H(D)\) be the space of holomorphic functions on an open connected set \(D\subset \mathbb C\). Suppose \(f_n\) is a sequence in \(H(D)\) converging to a function \(f\) uniformly on compact subsets. Suppose also that \(f\) is a nonconstant function. Let \(a\in D\), \(\alpha =f(a)\).

Show there is a sequence \(a_n\in D\) which converges to \(a\), such that \(f_n(a_n)=\alpha \) for each \(n\).