Proof of the Schwarz lemma by the maximum modulus principle

Builds on Holomorphic function Holomorphic functions are analytic Disc of convergence Power series are holomorphic in the disc of convergence Heine–Borel theorem Continuous real function on a compact set attains its bounds Maximum modulus principle

Proof

Let $\mathbb{D}$ and $f$ be as in the statement. By holomorphic functions are analytic with $c = 0$ and radius $1$ , $f(z) = \sum_{n \ge 0} a_n z^n$ for each $z \in \mathbb{D}$ .
We have $a_0 = f(0) = 0$ . By power series are holomorphic, differentiating term by term at $0$ gives $f'(0) = a_1$ .
The series $\sum_{n \ge 1} a_n z^{n-1}$ converges at each $z \in \mathbb{D}$ : for $z \ne 0$ its partial sums are $1/z$ times those of the series for $f(z)$ .
A power series that converges at $w$ has bounded terms there. So, by the definition of the radius of convergence, the series of step 3 has radius at least $1$ .
Let $g(z) = \sum_{n \ge 1} a_n z^{n-1}$ on $\mathbb{D}$ . By power series are holomorphic, $g$ is holomorphic on $\mathbb{D}$ , hence continuous.
By steps 2 and 3, $g(z) = f(z)/z$ for $z \ne 0$ , and $g(0) = a_1 = f'(0)$ .
Claim: for $0 < r < 1$ and $\abs{z} \le r$ , $\abs{g(z)} \le 1/r$ . The disc $K_r = \set{z : \abs{z} \le r}$ is compact by the Heine–Borel theorem.
By attainment of bounds, $\abs{g}$ has a maximum on $K_r$ at some $z^*$ .
If $\abs{z^*} < r$ , then $\abs{g}$ has a local maximum at $z^*$ in the domain $\mathbb{D}$ . By the maximum modulus principle, $g$ is constant, so the maximum is also attained on $\abs{z} = r$ .
So in all cases the maximum of $\abs{g}$ on $K_r$ is attained at a point $z'$ with $\abs{z'} = r$ . There $\abs{g(z')} = \abs{f(z')}/r < 1/r$ , since $f$ maps into $\mathbb{D}$ . This proves the claim.
Fix $z \in \mathbb{D}$ . Step 7 holds for every $r$ with $\abs{z} \le r < 1$ . Letting $r \to 1$ gives $\abs{g(z)} \le 1$ .
For $z \ne 0$ , step 11 gives $\abs{f(z)} \le \abs{z}$ , and this also holds at $z = 0$ . At $z = 0$ it gives $\abs{f'(0)} \le 1$ .
Now assume $\abs{f(z_0)} = \abs{z_0}$ for some $z_0 \ne 0$ , or $\abs{f'(0)} = 1$ . Then $\abs{g(z_0)} = 1$ or $\abs{g(0)} = 1$ .
By step 11, $\abs{g} \le 1$ on $\mathbb{D}$ . So $\abs{g}$ has a local maximum at an interior point of $\mathbb{D}$ .
By the maximum modulus principle, $g$ is a constant $a$ . By step 13, $\abs{a} = 1$ .
Then $f(z) = z g(z) = a z$ for $z \ne 0$ , and also for $z = 0$ since $f(0) = 0$ .

Idea

Dividing by $z$ removes the zero at the origin and gives a holomorphic $g$ with $g(0) = f'(0)$ . On the circle $\abs{z} = r$ we have $\abs{g} < 1/r$ , and the maximum modulus principle carries this bound inside. Letting $r \to 1$ gives $\abs{g} \le 1$ , and equality at one interior point forces $g$ to be a constant of modulus one.

Reading path

To understand this page from the ground up, read these 482 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the derivative of a map into R^n by components, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the mean value property via a holomorphic function with real part u, Proof that open sets are the sets that are neighbourhoods of all their points, Proof of the diameter characterisation of Cauchy sequences, Proof of the commutativity of addition by induction, Proof of the Lebesgue number lemma by a finite cover of balls, Proof of the epsilon-delta form of continuity, Proof of the Laplacian as a sum of second partial derivatives, Proof of the triangle inequality for norms, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that a complex number times its conjugate is the square of its modulus, Proof of the Cauchy–Schwarz inequality, Proof of the Heine–Borel theorem by sequential compactness, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule in several variables, Proof of the chain rule with an auxiliary difference quotient, Proof of the reverse triangle inequality, Proof of the derivative as a little-o approximation, Proof of the equality of complex numbers, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that a linear map between normed spaces is continuous if and only if it is bounded, Proof that a function is bijective iff it has a two-sided inverse, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of continuity via closed sets, Proof of the double negation law by a truth table, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof that compact subsets of a metric space are bounded, Proof that compactness and sequential compactness agree in metric spaces, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the uniqueness of limits, Proof that holomorphic functions are infinitely differentiable, Proof that Lipschitz functions are uniformly continuous, Proof that the integral of a continuous function is an antiderivative, Proof that the coordinate map is a linear isomorphism, Proof of strong induction from ordinary induction, Proof that linear maps on finite-dimensional normed spaces are continuous, Proof of the limit of a product of sequences, Proof of Cauchy's integral formula by a primitive of the difference quotient, Proof of the linearity of the derivative, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the derivative of a continuous linear map, Proof of the Cauchy–Riemann equations, Proof of the existence of square roots of nonnegative reals, Proof of the chain rule for a composition with a curve, Proof that the continuous image of a connected set is connected, Proof of the linearity of differentiation, Proof of the mean value theorem for integrals, Proof that a continuous image of a compact set is compact, Proof that holomorphic functions on a disc have primitives, Proof that compact subsets of a Hausdorff space are closed, Proof that the composition of continuous maps is continuous, Proof of epsilon-delta continuity at a point in metric spaces, Proof of Riemann's criterion for integrability, Proof of the sequential characterization of limits, Proof of the description of the open sets of a metric space, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof that a zero derivative on a connected open set implies constant, Proof of the first fundamental theorem of calculus, Proof of Schwarz's theorem for C^2 functions by the mean value theorem, Proof that compactness of a subset can be tested with open sets of the ambient space, Proof of the comparison test, Proof of the existence of the infimum by negation, Proof of the equivalence of norms in finite dimension by compactness of the unit sphere, Proof that path-connected spaces are connected, Proof that every spanning set contains a basis, by Zorn's lemma, Proof of the estimation lemma, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof that connected subsets of the real line are intervals, Proof that closed sets contain their limits, Proof of the linearity of the integral, Proof of integration by substitution, Proof of Zorn's lemma, Proof that metric spaces are Hausdorff, Proof that the derivative gives the directional derivatives, Proof of the second fundamental theorem of calculus, Proof of the intermediate value theorem by the supremum, Proof that differentiable functions are continuous, Proof of the sequence characterisation of uniform continuity, Proof of the epsilon-delta form of the limit of a function, Proof of the isolated zeros theorem by power series, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the fundamental theorem of contour integration, Proof of Goursat's theorem by repeated subdivision, Proof of the Archimedean property in an Archimedean ordered field, Proof that a continuous real function on a compact set attains its bounds, Proof of the derivative of a map into a product, Proof of the identity theorem by connectedness, Proof of the maximum modulus principle by Cauchy's integral formula, Proof that a composition of continuous functions is continuous, Proof of Cantor's intersection theorem, Proof of the additivity of the integral over intervals, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that holomorphic functions are analytic, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that absolutely convergent series in a Banach space converge, Proof of the symmetry of second derivatives by a second difference, Proof that open balls are open, Proof that power series are holomorphic in the disc of convergence, Proof of the monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof that Lipschitz maps are uniformly continuous, Proof that C^k functions are characterised by continuous derivatives, Proof that continuous partial derivatives imply differentiability, Proof that a set is open if and only if its complement is closed, Proof of the fundamental theorem of calculus, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof that the real part of a holomorphic function is harmonic, Proof of the mean value theorem from Rolle's theorem, Proof that sums and products of continuous functions are continuous, Proof of continuity via open sets in metric spaces, Proof that continuous functions are Riemann integrable, Proof of the mean value inequality, Proof of the constant multiple rule.

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Differentiable functions are continuous statement
Proof of the sequence characterisation of uniform continuity proof
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Limit of a sum of sequences statement
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Limits preserve non-strict inequalities statement
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Proof that there is a natural number n with 1/n below any positive epsilon proof
Proof that Darboux and Riemann integrals agree proof
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Continuity via open sets statement
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Proof of the limit of a product of sequences proof
Epsilon-delta uniform continuity statement
Monotonicity of the integral statement
Darboux and Riemann integrals agree statement
There is a natural number n with 1/n below any positive epsilon statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
Triangle inequality for norms statement
Proof that a decreasing sequence bounded below converges proof
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Proof of the comparison test proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Cauchy–Schwarz inequality statement
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Proof of the chain rule with an auxiliary difference quotient proof
Proof of the linearity of differentiation proof
Proof that sums and products of continuous functions are continuous proof
Proof of the additivity of the integral over intervals proof
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Proof of the derivative of a map into a product proof
Product of continuous functions is continuous statement
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Proof of the linearity of the derivative proof
Additivity of the integral over intervals statement
Proof of the Heine–Borel theorem by sequential compactness proof
Operator norm definition
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Linearity of differentiation statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Derivative of a map into a product statement
Linearity of the derivative statement
Proof that power series are holomorphic in the disc of convergence proof
Heine–Borel theorem statement
Continuous functions on a closed bounded interval are uniformly continuous statement
Boundedness theorem statement
Proof of the constant multiple rule proof
Proof of the first fundamental theorem of calculus proof
Power series are holomorphic in the disc of convergence statement
Derivative of a map into R^n by components statement
Constant multiple rule statement
Proof that a continuous real function on a compact set attains its bounds proof
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Extreme value theorem statement
Continuous real function on a compact set attains its bounds statement
Proof of Fermat's theorem on stationary points proof
Proof that continuous functions are Riemann integrable proof
Proof of the equivalence of norms in finite dimension by compactness of the unit sphere proof
Fermat's theorem on stationary points statement
Continuous functions are Riemann integrable statement
Equivalence of norms in finite dimension statement
Proof of Rolle's theorem proof
Rolle's theorem statement
Proof of the estimation lemma proof
Proof that linear maps on finite-dimensional normed spaces are continuous proof
Proof of the mean value theorem for integrals proof
Estimation lemma statement
Linear maps on finite-dimensional spaces are continuous statement
Proof of the mean value theorem from Rolle's theorem proof
Mean value theorem for integrals statement
Mean value theorem statement
First fundamental theorem of calculus statement
Proof of the chain rule in several variables proof
Proof that continuous partial derivatives imply differentiability proof
Chain rule in several variables statement
Proof that the integral of a continuous function is an antiderivative proof
Proof of the second fundamental theorem of calculus proof
The integral of a continuous function is an antiderivative statement
Continuous partial derivatives imply differentiability statement
Proof of the chain rule for a composition with a curve proof
Second fundamental theorem of calculus statement
Proof of the fundamental theorem of calculus proof
Chain rule for a composition with a curve statement
Proof that C^k functions are characterised by continuous derivatives proof
Proof of integration by substitution proof
C^k function characterised by continuous derivatives statement
Integration by substitution statement
Fundamental theorem of calculus statement
Proof of the mean value inequality proof
Proof of the fundamental theorem of contour integration proof
Proof of Schwarz's theorem for C^2 functions by the mean value theorem proof
Vector field definition
Mean value inequality statement
Fundamental theorem of contour integration statement
Divergence definition
Proof that a zero derivative on a connected open set implies constant proof
Proof of the symmetry of second derivatives by a second difference proof
Laplacian definition
Proof of Goursat's theorem by repeated subdivision proof
Symmetry of second derivatives statement
Zero derivative on a connected open set implies constant statement
Goursat's theorem statement
Schwarz's theorem for C^2 functions statement
Proof of the Laplacian as a sum of second partial derivatives proof
Proof of Cauchy's integral formula by a primitive of the difference quotient proof
Proof that holomorphic functions on a disc have primitives proof
Laplacian as a sum of second partial derivatives statement
Cauchy's integral formula statement
Holomorphic functions on a disc have primitives statement
Harmonic function definition
Proof of the mean value property via a holomorphic function with real part u proof
Proof that holomorphic functions are analytic proof
Holomorphic functions are analytic statement
Proof of the isolated zeros theorem by power series proof
Proof that holomorphic functions are infinitely differentiable proof
Holomorphic functions are infinitely differentiable statement
Order of a zero definition
Proof that the real part of a holomorphic function is harmonic proof
Isolated zeros theorem statement
The real part of a holomorphic function is harmonic statement
Proof of the identity theorem by connectedness proof
Mean value property statement
Identity theorem statement
Proof of the maximum modulus principle by Cauchy's integral formula proof
Maximum modulus principle statement
Proof of the Schwarz lemma by the maximum modulus principle proof

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8 pages build on this one.

Tag	`5D4H3BBXSS`
URI	`m2://5D4H3BBXSS`
Kind	proof
Writing	W0 stub
MSC	30C80
Proves	Schwarz lemma
Uses	Holomorphic function, Holomorphic functions are analytic, Disc of convergence, Power series are holomorphic in the disc of convergence, Heine–Borel theorem, Continuous real function on a compact set attains its bounds, Maximum modulus principle
Changed	4 November 2026

Proof of the Schwarz lemma by the maximum modulus principle

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