57.6. EXERCISES 1809

Use Jensen’s equation to establish the following inequality.

M (2r)≥ 2n(r)

where n(r) is the number of zeros of f in B(0,r).

2. The version of the Blaschke product presented above is that found in most complexvariable texts. However, there is another one in [89]. Instead of αn−z

1−αnz|αn|αn

you use

αn− z1

αn− z

Prove a version of Theorem 57.5.1 using this modification.

3. The Weierstrass approximation theorem holds for polynomials of n variables on anycompact subset of Rn. Give a multidimensional version of the Müntz-Szasz theoremwhich will generalize the Weierstrass approximation theorem for n dimensions. Youmight just pick a compact subset of Rn in which all components are positive. Youhave to do something like this because otherwise, tλ might not be defined.

4. Show cos(πz) = ∏∞k=1

(1− 4z2

(2k−1)2

).

5. Recall sin(πz) = zπ ∏∞n=1

(1−( z

n

)2). Use this to derive Wallis product,

π

2=

∞

∏k=1

4k2

(2k−1)(2k+1).

6. The order of an entire function, f is defined as

inf{

a≥ 0 : | f (z)| ≤ e|z|a

for all large enough |z|}

If no such a exists, the function is said to be of infinite order. Show the order of anentire function is also equal to

lim supr→∞

ln(ln(M (r)))ln(r)

where M (r)≡max{| f (z)| : |z|= r}.

7. Suppose Ω is a simply connected region and let f be meromorphic on Ω. Supposealso that the set, S ≡ {z ∈Ω : f (z) = c} has a limit point in Ω. Can you concludef (z) = c for all z ∈Ω?

8. This and the next collection of problems are dealing with the gamma function. Showthat ∣∣∣(1+

zn

)e−zn −1

∣∣∣≤ C (z)n2