Maximum Modulus Principle

Consider the upper half-plane

equipped with the hyperbolic metric. Show that the hyperbolic distance from \(n+in\) to the positive imaginary axis does not depend on \(n\in \{1,2,3,\ldots \}\).

Prove or disprove: There is no nonconstant entire function satisfying

for all \(z\). In your solution don’t quote either Picard theorem.

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 \(L\) be a partial differential operator of the form: \[ L=\sum _{j=0}^{m-1} c_j\partial ^j, \] where the \(c_j\) are arbitrary constants. Show that \(x^m(L\delta )=0\) as distributions in \(\mathcal {D}'(\mathbb {R})\). Here \(\delta \in \mathcal {D}'(\mathbb {R})\) is the usual “delta function”.

Prove or disprove: Let \[ \Omega =\{z:\ |\operatorname {Re}(z)|<1\text { and }|\operatorname {Im}(z)|<1\}. \] If \(f:\mathbb D\to \Omega \) is a biholomorphism so that \(f(z)\) is real for all real \(z\in \mathbb D\) and \(f(0)=0\), then \[ f(iz)=i f(z). \]

Let \[ f(x)=\operatorname {signum}(x)= \begin {cases} 1, & x>0,\\ 0, & x=0,\\ -1, & x<0. \end {cases} \]

1. Find its first and second derivatives using the theory of distributions.
2. Compute \[ \begin {aligned} I_0&=\int _{-1}^{\infty }\log (x+10)f(x)\,dx,\\ I_1&=\int _{-1}^{\infty }\log (x+10)f'(x)\,dx,\\ I_2&=\int _{-1}^{\infty }\log (x+10)f''(x)\,dx. \end {aligned} \]

Prove Schwarz’s lemma: If \(f:\Delta \to \Delta \) is a holomorphic function from the open unit disk \(\Delta \) to itself with \(f(0)=0\) show that \(|f(z)|\le |z|\), \(|f'(0)|\le 1\) and if \(|f(z)|=|z|\) for some \(z\ne 0\) or \(|f'(0)|=1\) then \(f(z)=\lambda z\) with \(|\lambda |=1\).

(Hint: Apply the maximum principle to the function \(f(z)/z\).)

Formulate and prove the Maximum Modulus Theorem.

Assume that \(\mathbb R\) has the Lebesgue measure. Let \(f\in L^\infty (\mathbb R)\). For \(g\in L^2(\mathbb R)\) define \[ M(g)=fg \] Prove that \(M:L^2(\mathbb R)\to L^2(\mathbb R)\) is a bounded linear operator, and that \(\lVert M\rVert =\lVert f\rVert _\infty \).

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 \([0,1)\).

Suppose \(P(z)=a_nz^n+\cdots +a_0\). If \(M(\rho )=\sup _\theta |P(\rho e^{i\theta })|\) then show that \[ \frac {M(r)}{r^n}\ge \frac {M(R)}{R^n} \] if \(0<r<R\).