A holomorphic function f on a closed bounded domain D̄ satisfies |f(z)| ≤ 5 for all z on the boundary ∂D. What can you conclude about |f(z)| for z strictly inside D?
ANothing can be concluded — interior values of a holomorphic function are independent of boundary values
B|f(z)| < 5 strictly for all interior points — equality can never be achieved inside the domain
C|f(z)| ≤ 5 for all z in D, with equality at an interior point possible only if f is constant throughout D
D|f(z)| ≤ 5 for all z in D, since holomorphic functions cannot exceed their average value
The Maximum Modulus Principle says |f| achieves its maximum on the boundary (not the interior) unless f is constant. So |f(z)| ≤ 5 everywhere inside — this follows immediately from the boundary bound. But can |f| = 5 at an interior point? Only if f is constant on all of D (in which case |f| = 5 everywhere, including the boundary). Option B is wrong because it rules out the constant function case; option A is the wrong answer about complex analysis behaving like real analysis.
Question 2 Multiple Choice
What is the key mechanism that forces a holomorphic function to be constant if |f| achieves a local maximum at an interior point?
AThe Cauchy-Riemann equations force the gradient of |f| to vanish at any critical point, making all critical points saddle points
BThe mean-value property means f(z₀) equals the average of f on any surrounding circle; for the modulus of this average to equal the maximum, all values on the circle must point the same direction with the same modulus — forcing constancy
CLiouville's theorem directly applies: any bounded analytic function on a domain must be constant
DThe maximum of |f| is always achieved at the boundary because that is where the complex derivative is largest
The mean-value property (from Cauchy's integral formula) is the key: f(z₀) equals the average of f on any circle around z₀. If |f(z₀)| is a maximum, the average of f on the circle has modulus equal to the maximum modulus of all values on the circle. For a complex average to achieve the maximum modulus of its summands, all the complex values being averaged must point in the same direction with the same magnitude — otherwise they partially cancel. This forces f to be constant on that circle, and by the identity theorem, constant everywhere. Liouville's theorem applies to entire functions on the full plane, not to bounded domains.
Question 3 True / False
A real smooth function on an open interval can certainly achieve its maximum in the interior — for example, f(x) = −x² peaks at x = 0. This is possible for real functions but not for the modulus of a holomorphic function on a domain.
TTrue
FFalse
Answer: True
This contrast is the key to understanding why the Maximum Modulus Principle is a deep result about complex analysis rather than a routine fact. Real smooth functions routinely achieve interior maxima — f(x) = −x² is a simple example. Holomorphic functions are far more rigid: their values on any open set determine them everywhere on the connected domain (identity theorem), and the mean-value property prevents |f| from having an interior peak. This global rigidity — where local information forces global conclusions — is the characteristic feature of complex analysis that has no real-variable analogue.
Question 4 True / False
The Maximum Modulus Principle is mainly useful for showing that |f| is large somewhere inside a domain, given information about interior values of f.
TTrue
FFalse
Answer: False
The principle works in the opposite direction: it reduces questions about interior values to questions about boundary values. To bound |f| inside a region, you only need to examine the boundary — if |f| ≤ M on ∂D, then |f| ≤ M everywhere inside D. This is a powerful reduction: analyzing a function's behavior on the boundary (a lower-dimensional object) gives complete control over the interior. The principle is not about showing interior values are large; it is about using boundary information to control the entire interior.
Question 5 Short Answer
Why does the Maximum Modulus Principle reflect a deeper rigidity in holomorphic functions than what is possible for smooth real-valued functions, and what is the key mechanism behind this rigidity?
Think about your answer, then reveal below.
Model answer: Real smooth functions can have interior maxima — they are free to peak and valley anywhere. Holomorphic functions cannot: any interior maximum of |f| forces f to be constant throughout the domain. The mechanism is the mean-value property, inherited from Cauchy's integral formula: a holomorphic function equals the average of its values on any surrounding circle. If |f(z₀)| is maximum, the modulus of the average equals the maximum modulus of all the averaged values — which requires all those values to be identical in direction and magnitude, forcing constancy. This rigidity is not isolated: it reflects that a holomorphic function's values on any open set determine it everywhere, so local information (where |f| is largest) forces global conclusions.
The same principle extends to harmonic functions (the real and imaginary parts of holomorphic functions) and underlies the maximum principle in partial differential equations. The rigidity of complex analysis comes from the Cauchy-Riemann equations, which impose much stronger constraints than mere smoothness — they force a function's behavior at any point to be completely constrained by its behavior on any surrounding curve.