Talk:Subharmonic function

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The following comment was left in the article proper by {{user|Szilard.revesz}}:

{{quote|

For definition of subharmonic functions on Riemannian manifolds, the defining inequalities between the subharmonic function $f$ and the harmonic function $f_1$ should be the reverse, in accordance with the definition given above 8and in accordance with the heuristical content of the name SUBharmonic).}}

I'm not sure what he means precisely, possibly due to notational confusion. Ray^Talk 19:23, 10 November 2010 (UTC)

Is this true?

One stated that for a holomorphic function $f(z)$ the function $\varphi(z)=\log|f(z)|$ is a subharmonic function. I think I have a counterexample, since this is only valid for functions which do not have a local maximum in $G$ .

Let $f(z)=z^2$ and $G=D(0,\tfrac{1}{2})$ , a closed disk centered at 0 with radius a half, then

$0 \leq \frac{1}{2\pi} \int_{0}^{2\pi} \log|r^2e^{2i\theta}|d\theta=2\log r < 0$ , if $0$

Am I wrong? —Preceding unsigned comment added by 94.212.22.34 (talk) 15:41, 20 January 2011 (UTC)

:I think you confused $f(0)$ and $\log|f(0)|$ in your counterexample. If you interpret $\log(0)=-\infty$ , (as one is supposed to do), the inequality really holds. Vigfus (talk) 20:23, 3 August 2012 (UTC)

::But in the section Examples it is also stated, that for an analytic function $f$ the function $\log|f|$ is a subharmonic function. This is definitely not true. Consider for example the identity function $f(x)=x$ on $\mathbb{R}^1$ . Then $\Delta \log|f(x)|=-x^{-2}$ , which is negative in general. — Preceding unsigned comment added by 131.152.55.74 (talk) 17:37, 11 December 2020 (UTC)

Maximum Principle(with modulus) for harmonic functions.

Let's suppose that $u:\Omega\subseteq\mathbb{R}^{2}\longrightarrow \mathbb{R}$ is a harmonic function non-constant,and $\Omega\,$ is an open simply connected set.I want to prove that there is not $z_{0}\in\Omega$ with $|u(z)|\leq |u(z_{0})|\,\forall z\in\Omega$ .We observe this equality is equivalent to $u(z)^{2}\leq u(z_{0})^{2}\,\forall z\in\Omega$ .Since $u^{2}\,$ is subharmonic,we find a contradiction with the maximum principle for subharmonic functions.

¿Is it correct?

Nawiks (talk) 00:34, 11 January 2012 (UTC)