Talk:Maximum modulus principle

'''By switching to the reciprocal, we can get the minimum modulus principle. It states that if f is holomorphic within a bounded domain D, continuous up to the boundary of D, and non-zero at all points, then the modulus |f (z)| takes its minimum value on the boundary of D.'''

The reciprocal might not be holomorphic in the same domain. This causes confusion (even considering it's only a sketch). The same goes for log|f(z)|.

cvalente 17:38, 12 July 2006 (UTC)


 * if f is holomorphic within a bounded domain D, continuous up to the boundary of D, and non-zero at all points, then how can the reciprocal of f fail to be holomorhic in the domain D? Madmath789 17:49, 12 July 2006 (UTC)


 * For the minimum modulus principle you are right. It is a legitimate argument.
 * For the maximum modulus principle if $$f(z_0)=0$$ for $$z_0$$ in the domain, $$\log|f(z)|$$ won't even be differentiable at $$z_0$$, let alone twice to make it an harmonic function as stated.
 * There are other heuristic ways to try and understand this theorem without this problem.
 * cvalente 22:35, 13 July 2006 (UTC)


 * At the risk of restating: "if f is holomorphic within a bounded domain D, continuous up to the boundary of D, and non-zero at all points" - how can you then have $$f(z_0)=0$$??? Madmath789 22:50, 13 July 2006 (UTC)
 * Because "if f is holomorphic within a bounded domain D, continuous up to the boundary of D, and non-zero at all points" is assumed in the minimum theorem, not the maximum.
 * The maximum states "Mathematically, this can be formulated as follows. Let f be defined on some open subset D of the complex plane C. If z0 is a point in D such that $$|f(z_0)|\ge |f(z)|$$ for all z in a neighborhood of z0, then the function f is constant on D."
 * I already said that for the minimum everything is fine. For the maximum f can have a zero.
 * Even if this weren't stated, the Maximum modulus principle is valid for functions that have zeros in the domain. Functions with zeros in the domain don't result in log|f| harmonic functions (not the definition given on that page at least because they are not C^2).
 * cvalente 08:56, 14 July 2006 (UTC)


 * Sorry - I misunderstood your point earlier. Yes, you are right about the sketch proof, and I think it should be changed. The sketch proof could be corrected by using the fact that log|f| is subharmonic (then use the maximum principle for subharmonic functions) - but I think that would be unnecesarily complex for this article, so I am maybe suggesting that we don't have a sketch proof at all (unless anyone has a nice short one that doesn't use any heavy machinery). Madmath789 09:37, 14 July 2006 (UTC)


 * I have an intuitive one but it's considerable longer. It uses Cauchy's formula, the intuitive fact that the modulus of the integral of a function is the max of the modulus of the function times the length of the interval where it is integrated iff the function is constant and later that 2 holomorphic functions equal along a curve are equal where defined (this not so intuitive but a very standard result). The theorem would follow. I don't agree using that highlevel machinery such as subharmonic functions. Maybe the proof should be removed or at least explain it is only valid for functions with no zeros as it stands. cvalente 10:20, 14 July 2006 (UTC)

Strong Form of the Maximum Modulus Principle
What about including the Strong Form of the Maximum Modulus Principle:

D a bounded connected open set in \C, f holomorphic on D, w on the boundary of D, then if M(w) = limsup_{z\to w, z\in D} |f(z)|, and M=sup_{w\in boundary} M(w), THEN if f not constant, |f(z)|< M for all z in D.

(N.B. This is stronger as it lets us make the inequality strict). -unsigned


 * Sure. I think it is good to have it in its own section, say after the applications section, so that it does not interfere with existing stuff. How's that? Oleg Alexandrov (talk) 22:09, 1 December 2007 (UTC)

Is the boundedness missing at the moment?
It says only connected open subset of C, but it should be bounded, isn`t it? —Preceding unsigned comment added by 141.20.12.157 (talk) 16:44, 4 February 2011 (UTC)

sketch of proof
if the maximum modulus is at $$z_0 =x+iy$$ not at the boundary, then $$\frac{d}{dx}|f(x+iy)|^2 = 2 \overline{f(x+iy)}\frac{d}{dx}f(x+iy) = 2 f'(z_0) \overline{f(z_0)} = 0$$ so $$f(z_0) = 0$$ or $$f'(z_0) = 0$$. if $$f(z_0) = 0$$ we simply consider the function $$g(z) = f(z)+1$$ whose maximum is at the same point, so without loss of generality we can suppose that $$f'(z_0) = 0, f(z_0) \ne 0$$.

now if $$f'(z_0) = 0, f(z_0) \ne 0$$ and $$f(z)$$ is a non-constant holomorphic function (in a neighborhood of $$z_0$$) then there exists $$\alpha \ne 0,n \ge 2$$ such that $$f(z_0+z) = f(z_0)+\alpha z^n+\mathcal{O}(|z|^{n+1})$$ : in particular with $$\epsilon > 0$$ small enough $$|f(z_0 + (\epsilon \overline{\alpha} f(z_0))^{1/n})| = |f(z_0) + \epsilon |\alpha|^2 f(z_0)| + \mathcal{O}(\epsilon^{(n+1)/n})> |f(z_0)| $$ so that $$z_0$$ cannot be a local maximum of $$|f(z)|$$.

78.227.78.135 (talk) 07:07, 19 January 2016 (UTC)