Harnack's inequality and principle

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Problem 5.5.20 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 5.5 "Series and Normal Families"

Let $D$ be a domain (connected open set) in $\mathbb {C}$ and let $\left\{ u_n\right\}$ be a sequence of harmonic functions $u_n : D \to (0,+\infty )$ . Show that if $u_n(z_0) \to 0$ for some $z_0 \in D$ , then $u_n \to 0$ uniformly on compact subsets of $D$ .

Problem 3.257 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 5 "Propagation of Convergence"

The harmonic functions

u_{0}(x,y), u_{1}(x,y), u_{2}(x,y), \ldots , u_{n}(x,y), \ldots

are assumed to be regular and positive in a certain open region $\mathfrak {R}$ of the $x,y$ -plane. If the infinite series

u_{0}(x,y) + u_{1}(x,y) + u_{2}(x,y) + \cdots + u_{n}(x,y) + \cdots

converges at a single point of $\mathfrak {R}$ it converges everywhere in $\mathfrak {R}$ ; in fact, it converges uniformly in any closed subdomain of $\mathfrak {R}$ .

Problem 3.258 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 5 "Propagation of Convergence"

We suppose that the functions of the sequence $f_{0}(z),f_{1}(z),f_{2}(z),\ldots ,f_{n}(z),\ldots$ are analytic in the open region $\mathfrak {R}$ and that the sequence of their real parts converges uniformly in every closed subdomain of $\mathfrak {R}$ . Then the sequence of their imaginary parts either diverges at all points or it converges uniformly in any closed subdomain of $\mathfrak {R}$ .

Problem 2.11.12 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.11 "Harmonic Functions"

(Harnack's Inequalities) Suppose $\Omega$ is a region, $K$ is a compact subset of $\Omega , z_{0} \in \Omega$ . Prove that there exist positive numbers $\alpha$ and $\beta$ (depending on $z_{0}, K$ , and $\Omega$ ) such that

\alpha u\left(z_{0}\right) \leq u(z) \leq \beta u\left(z_{0}\right)

for every positive harmonic function $u$ in $\Omega$ and for all $z \in K$ .

If $\left\{ u_{n}\right\}$ is a sequence of positive harmonic functions in $\Omega$ and if $u_{n}\left(z_{0}\right) \rightarrow 0$ , describe the behavior of $\left\{ u_{n}\right\}$ in the rest of $\Omega$ . Do the same if $u_{n}\left(z_{0}\right) \rightarrow \infty$ . Show that the assumed positivity of $\left\{ u_{n}\right\}$ is essential for these results.

Problem 2.11.13 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.11 "Harmonic Functions"

Suppose $u$ is a positive harmonic function in $U$ and $u(0) = 1$ . How large can $u\left(\frac{1}{2}\right)$ be? How small? Get the best possible bounds.

Problem 2.14.26 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.14 "Conformal Mapping"

(a)(b)(c)(d)

Iteration of Step 2 in the proof of Theorem 14.8 leads to a proof (due to Koebe) of the Riemann mapping theorem which is constructive in the sense that it makes no appeal to the theory of normal families and so does not depend on the existence of some unspecified subsequence. For the final step of the proof it is convenient to assume that $\Omega$ has property $(h)$ of Theorem 13.11. Then any region conformally equivalent to $\Omega$ will satisfy $(h)$ . Recall also that $(h)$ implies $(j)$ , trivially.

By Step 1 in Theorem 14.8 we may assume, without loss of generality, that $0 \in \Omega , \Omega \subset U$ , and $\Omega \neq U$ . Put $\Omega = \Omega_{0}$ . The proof consists in the construction of regions $\Omega_{1}, \Omega_{2}, \Omega_{3}, \ldots$ and of functions $f_{1}, f_{2}, f_{3}, \ldots$ , so that $f_{n}\left(\Omega_{n-1}\right) = \Omega_{n}$ and so that the functions $f_{n} \circ f_{n-1} \circ \cdots \circ f_{2} \circ f_{1}$ converge to a conformal mapping of $\Omega$ onto $U$ .

Complete the details in the following outline.

(a)

Suppose $\Omega_{n-1}$ is constructed, let $r_{n}$ be the largest number such that $D\left(0 ; r_{n}\right) \subset \Omega_{n-1}$ , let $\alpha_{n}$ be a boundary point of $\Omega_{n-1}$ with $\left|\alpha_{n}\right| = r_{n}$ , choose $\beta_{n}$ so that $\beta_{n}^{2} = -\alpha_{n}$ , and put

F_{n} = \varphi _{-\alpha _{n}} \circ S \circ \varphi _{-\beta _{n}}.

(The notation is as in the proof of Theorem 14.8). Show that $F_{n}$ has a holomorphic inverse $G_{n}$ in $\Omega_{n-1}$ , and put $f_{n} = \lambda_{n} G_{n}$ , where $\lambda_{n} = \left|c\right| / c$ and $c = G_{n}^{\prime }(0)$ . (This $f_{n}$ is the Koebe mapping associated with $\Omega_{n-1}$ . Note that $f_{n}$ is an elementary function. It involves only two linear fractional transformations and a square root).

(b)

Compute that $f_{n}^{\prime }(0) = \left(1+r_{n}\right) / 2 \sqrt{r_{n}} > 1$ .

(c)

Put $\psi_{0}(z) = z$ and $\psi_{n}(z) = f_{n}\left(\psi_{n-1}(z)\right)$ . Show that $\psi_{n}$ is a one-to-one mapping of $\Omega$ onto a region $\Omega_{n} \subset U$ , that $\left\{ \psi_{n}^{\prime }(0)\right\}$ is bounded, that

\psi _{n}^{\prime }(0) = \prod _{k = 1}^{n} \frac{1+r_{k}}{2 \sqrt{r_{k}}}

and that therefore $r_{n} \rightarrow 1$ as $n \rightarrow \infty$ .

(d)

Write $\psi_{n}(z) = z h_{n}(z)$ , for $z \in \Omega$ , show that $\left|h_{n}\right| \leq \left|h_{n+1}\right|$ , apply Harnack's theorem and Exercise 8 of Chap. 11 to $\left\{ \log \left|h_{n}\right|\right\}$ to prove that $\left\{ \psi_{n}\right\}$ converges uniformly on compact subsets of $\Omega$ , and show that $\lim \psi_{n}$ is a one-to-one mapping of $\Omega$ onto $U$ .

Problem 8.16 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 8 "Sequences and Series of Regular Functions. Integrals Depending on a Parameter."

Let the functions $u_{n}(x, y), n = 1,2, \ldots$ , be harmonic in a domain $D$ and satisfy the conditions

u_{n+1}(x, y) \geqslant u_{n}(x, y) \quad ((x, y) \in D, n = 1,2, \ldots )

Prove that the sequence $\left\{ u_{n}(x, y)\right\}$ converges uniformly on every compact subset of the domain $D$ (possibly to $+\infty$ ).

References

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.

[Rud87] Rudin W. Real and Complex Analysis. 3rd ed., McGraw-Hill, 1987.

[ШПК15] Шабунин М.И., Половинкин Е.С., Карлов М.И. Сборник задач по теории функций комплексного переменного. 4-е изд., БИНОМ. Лаборатория знаний, 2015.