Letf be analytic on the unit diskD=D(0,1), and assume thatf(0) = 0 and|f(z)| ≤1 for allz ∈D. Then (a)|f(z)| ≤ |z| onD, and (b) |f′(0)| ≤1. Furthermore, if equality
holds in (a) for somez= 0, or if equality holds in (b), thenf is a rotation ofD. That is, there is a constantλwith|λ|= 1 such thatf(z) =λzfor allz∈D.
Proof. Define
g(z) =
f(z)/z ifz∈D\ {0}
f′(0) ifz= 0.
By (2.2.13),gis analytic on D. We claim that|g(z)| ≤1. For if|z|< r <1, part (d) of the maximum principle yields
|g(z)| ≤max{|g(w)|:|w|=r} ≤ 1
rsup{|f(w)|:w∈D} ≤
1
r.
Sincermay be chosen arbitrarily close to 1, we have|g| ≤1 onD, proving both (a) and (b). If equality holds in (a) for somez = 0, or if equality holds in (b), then g assumes its maximum modulus at a point of D, and hence g is a constant λ on D (necessarily
|λ|= 1). Thusf(z) =λzfor allz∈D. ♣
Schwarz’s lemma will be generalized and applied in Chapter 4 (see also Problem 24).
Problems
1. Give an example of a nonconstant analytic functionf on a region Ω such thatf has a limit point of zeros at a point outside of Ω.
2. Verify the statements made in (2.4.5).
3. Consider the four forms of the maximum principle (2.4.12), for continuous rather than analytic functions. What can be said about the relative strengths of the statements? The proof in the text shows that (a) implies (b) implies (c) implies (d), but for example, does (b) imply (a)? (The region Ω is assumed to be one particular fixed open connected set, that is, the statement of the theorem does not have “for all Ω” in it.)
4. (L’Hospital’s rule). Letfandgbe analytic atz0, and not identically zero in any neigh-
borhood ofz0. If limz→z0f(z) = limz→z0g(z) = 0, show thatf(z)/g(z) approaches a limit (possibly∞) asz→z0, and limz→z0f(z)/g(z) = limz→z0f
′(z)/g′(z).
2.4. FURTHER APPLICATIONS 27
6. Letf be continuous on the closed unit diskD, analytic onD, and real-valued on∂D. Prove thatf is constant.
7. Letf(z) = sinz. Find max{|f(z)|:z∈K}where K={x+iy: 0≤x, y≤2π}. 8. (A generalization of part (d) of the maximum principle). SupposeK is compact,f is
continuous onK, andf is analytic onK◦, the interior ofK. Show that
max
z∈K|f(z)|= maxz∈∂K|f(z)|.
Moreover, if |f(z0)| = maxz∈K|f(z)| for some z0 ∈ K◦, then f is constant on the
component of K◦ that containsz
0.
9. Suppose that Ω is a bounded open set (not necessarily connected),f is continuous on Ω and analytic on Ω. Show that max{|f(z)|:z∈Ω}= max{|f(z)|:z∈∂Ω}. 10. Give an example of a nonconstant harmonic functionuonCsuch thatu(z) = 0 for
each real z. Thus the disk that appears in the statement of Theorem 2.4.14 cannot be replaced by just any subset ofChaving a limit point inC.
11. Prove that an open set Ω is connected iff for allf, ganalytic on Ω, the following holds: Iff(z)g(z) = 0 for everyz∈Ω, then eitherf orgis identically zero on Ω. (This says that the ring of analytic functions on Ω is an integral domain iff Ω is connected.) 12. Suppose thatf is analytic onC+={z: Imz >0}and continuous onS=C+∪(0,1).
Assume thatf(x) =x4−2x2 for allx∈(0,1). Show thatf(i) = 3.
13. Letf be an entire function such that|f(z)| ≥1 for allz. Prove thatf is constant. 14. Does there exist an entire function f, not identically zero, for which f(z) = 0 for
everyz in an uncountable set of complex numbers?
15. Explain why knowing that the trigonometric identity sin(α+β) = sinαcosβ + cosαsinβ for allreal αandβ implies that the same identity holds for all complex α
and β.
16. Supposef is an entire function and Im(f(z))≥0 for allz. Prove thatf is constant. (Consider exp(if).)
17. Suppose f and g are analytic and nonzero on D(0,1), and ff′(1(1/n/n)) = gg′(1(1/n/n)), n = 2,3, . . .. Prove thatf /gis constant onD(0,1).
18. Suppose thatf is an entire function,f(0) = 0 and|f(z)−ezsinz|<4 for allz. Find a formula for f(z).
19. Letf andgbe analytic onD=D(0,1) and continuous onD. Assume that Ref(z) = Reg(z) for allz∈∂D. Prove thatf−g is constant.
20. Letf be analytic onD=D(0,1). Prove that either f has a zero inD, or there is a sequence{zn}in D such that|zn| →1 and{f(zn)} is bounded.
21. Letube a nonnegative harmonic function onC. Prove that f is constant.
22. Suppose f is analytic on Ω ⊇ D(0,1), f(0) = i, and |f(z)| > 1 whenever|z| = 1. Prove thatf has a zero inD(0,1).
24. Suppose that f is analytic on D(0,1), with f(0) = 0. Define fn(z) = f(zn) for n = 1,2, . . . , z ∈ D(0,1). Prove that fn is uniformly convergent on compact subsets ofD(0,1). (Use Schwarz’s lemma.)
25. It follows from (2.4.12c) that if f is analytic on D(0,1) and f(zn) → 0 for each sequence{zn} inD(0,1) that converges to a point ofC(0,1), thenf ≡0. Prove the following strengthened version forbounded f. Assume only thatf(zn)→0 for each sequence {zn} that converges to a point in some given arc {eit, α ≤ t ≤ β} where α < β, and deduce that f ≡0. [Hint: Assume without loss of generality thatα= 0. Then for sufficiently large n, the arcsAj ={eit: (j−1)β ≤t≤jβ}, j = 1,2, . . . , n coverC(0,1). Now considerF(z) =f(z)f(eiβz)f(ei2βz)· · ·f(einβz).]
26. (a) Let Ω be a bounded open set and let{fn}be a sequence of functions that are an- alytic on Ω and continuous on the closure Ω. Suppose that{fn}is uniformly Cauchy on the boundary of Ω. Prove that {fn} converges uniformly on Ω. If f is the limit function, what are some properties off?
(b) What complex-valued functions on the unit circle C(0,1) can be uniformly ap- proximated by polynomials in z?