Talk:Subharmonic function

Latest comment: 3 years ago by 131.152.55.74 in topic Is this true?

A comment

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The following comment was left in the article proper by Szilard.revesz (talk · contribs):

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. RayTalk 19:23, 10 November 2010 (UTC)Reply

Is this true?

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One stated that for a holomorphic function   the function   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  .

Let   and  , a closed disk centered at 0 with radius a half, then

 , if  

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

I think you confused   and   in your counterexample. If you interpret  , (as one is supposed to do), the inequality really holds. Vigfus (talk) 20:23, 3 August 2012 (UTC)Reply
But in the section Examples it is also stated, that for an analytic function   the function   is a subharmonic function. This is definitely not true. Consider for example the identity function   on  . Then  , which is negative in general. — Preceding unsigned comment added by 131.152.55.74 (talk) 17:37, 11 December 2020 (UTC)Reply

Maximum Principle(with modulus) for harmonic functions.

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Let's suppose that   is a harmonic function non-constant,and   is an open simply connected set.I want to prove that there is not   with  .We observe this equality is equivalent to  .Since   is subharmonic,we find a contradiction with the maximum principle for subharmonic functions.

¿Is it correct?

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