Subharmonic Functions

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Problem 5.5.9 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 5.5 "Growth Conditions"

Let the continuous real-valued function $\varphi$ on the complex plane satisfy the Sub--Mean--Value Property: for any point $z_{0}$ ,

\varphi (z_{0}) \leq \frac{1}{2\pi } \int _{0}^{2\pi } \varphi (z_{0}+re^{i\theta }) \, d\theta \qquad (0<r<1).

Prove that $\varphi$ obeys the Maximum Modulus Principle: the maximum of $\varphi$ over any compact set $K$ is attained on the boundary of $K$ .

Problem 4.171 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 1 "Various Propositions"

Find, if possible, $n+1$ analytic functions $f_{1}(z), f_{2}(z), \ldots , f_{n}(z), f(z)$ , which differ from each other not merely by constant factors, are regular in a domain $\mathfrak {D}$ and for which the relation

\left|f_{1}(z)\right| + \left|f_{2}(z)\right| + \cdots + \left|f_{n}(z)\right| = \left|f(z)\right|

holds in $\mathfrak {D}$ .

Problem 2.17.1 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.17 "$H^p$ Spaces"

Prove Theorems 17.4 and 17.5 for upper semicontinuous subharmonic functions.

Problem 2.17.2 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.17 "$H^p$ Spaces"

Assume $f \in H(\Omega )$ and prove that $\log (1+\left|f\right|)$ is subharmonic in $\Omega$ .

Problem 10.16 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 10 "The Maximum Principle"

Let the functions $f_{1}(z), \ldots , f_{m}(z)$ be regular in a domain $G$ , and let $M = \varlimsup_{z \rightarrow \partial G}\left\{ \left|f_{1}(z)\right|+\ldots +\left|f_{m}(z)\right|\right\}$ . Prove that if at least one of the $f_{k}(z)$ is not identically constant, then at every point of $G$ the inequality $\left|f_{1}(z)\right|+\ldots +\left|f_{m}(z)\right| < M$ holds.

Problem 10.18 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 10 "The Maximum Principle"

Prove that a function $u(z)$ , harmonic in a domain $G$ , is subharmonic in this domain.

Problem 10.20 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 10 "The Maximum Principle"

Prove that the function $u(z) = \left|f(z)\right|$ is subharmonic in a domain $G$ , if the function $f(z)$ is regular in the domain $G$ .

Problem 10.21 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 10 "The Maximum Principle"

Let the function $u(z)$ have, in a domain $G \subset \mathbb {C}$ , continuous partial derivatives of the second order (with respect to $x = \mathrm{Re}\left(z\right)$ and $y = \mathrm{Im}\left(z\right)$ ), and let $\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}} \geqslant 0$ in the domain $G$ . Prove that the function $u(z)$ is subharmonic in the domain $G$ .

Problem 10.22 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 10 "The Maximum Principle"

Let the function $f(z)$ be regular in a domain $G$ . Prove that the functions $u(z) = \ln \left|f(z)\right|$ and $u(z) = \left|f(z)\right|^{\alpha }, \alpha > 0$ , are subharmonic in the domain $G$ .

Problem 10.23 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 10 "The Maximum Principle"

(1)(2)(3)(4)

Let the functions $u_{1}(z)$ and $u_{2}(z)$ be subharmonic in a domain $G$ . Prove that:

(1)

if the numbers $a$ and $b$ are positive, then the function $a u_{1}(z)+b u_{2}(z)$ is subharmonic in the domain $G$ ;

(2)

the function $u(z) = \max \left\{ u_{1}(z), u_{2}(z)\right\}$ is subharmonic in the domain $G$ ;

(3)

the function $u(z) = \left|u_{1}(z)\right|^{\alpha }$ is subharmonic in the domain $G$ for $\alpha > 1$ ;

(4)

for $\alpha > 0$ the function $u(z) = e^{\alpha u_{1}(z)}$ is subharmonic in the domain $G$ .

Problem 10.24 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 10 "The Maximum Principle"

Let the functions $f_{1}(z), \ldots , f_{m}(z)$ be regular in a domain $G$ . Prove that for $\alpha > 0$ the function $u(z) = \left|f_{1}(z)\right|^{\alpha }+\ldots +\left|f_{m}(z)\right|^{\alpha }$ is subharmonic in the domain $G$ .

Problem 1.7.14 from Lehmann E.L., Casella G. "Theory of Point Estimation" [LeCa98]More problems in Section 1.7 "Convex Loss Functions"

(a)(b)(c)

Determine whether the following functions are super- or subharmonic:

(a)

$\sum_{i=1}^{k} x_{i}^{p}, \quad p<1, x_{i}>0$ .

(b)

$e^{-\sum_{i=1}^{k} x_{i}^{2}}$ .

(c)

$\log \left(\prod_{i=1}^{k} x_{i}\right)$ .

Problem 1.7.16 from Lehmann E.L., Casella G. "Theory of Point Estimation" [LeCa98]More problems in Section 1.7 "Convex Loss Functions"

(a)(b)(c)

(a)

If $f: \Re^{p} \rightarrow \Re$ is superharmonic, then $\varphi (f(\cdot ))$ is also superharmonic, where $\varphi : \Re \rightarrow \Re$ is a twice-differentiable increasing concave function.

(b)

If $h$ is superharmonic, then $h^{*}(x)=\int g(x-y) h(y) d y$ is also superharmonic, where $g(\cdot )$ is a density.

(c)

If $h_{\gamma }$ is superharmonic, then so is $h^{*}(x)=\int h_{\gamma }(x) d G(\gamma )$ where $G(\gamma )$ is a distribution function.

(Assume that all necessary integrals exist, and that derivatives may be taken inside the integrals.)

References

[deSi04] de Souza P.N., Silva J. Berkeley Problems in Mathematics. 3rd ed., Springer, 2004.

[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.

[LeCa98] Lehmann E.L., Casella G. Theory of Point Estimation. 2nd ed., Springer, 1998.