1550251515-Classical_Complex_Analysis__Gonzalez_

24 Chapter^1

Corollary 1.1 (Cauchy-Schwarz Inequality). If {zi, ... ,zn} and {w 1 , ... ,

wn} are two sets of complex numbers (n ;::: 1), then

n^2 n n

I>kw1, SL lzkl

2

L lwkl

(^2) (1.6-15)
k=l k=l k=l
where the equality sign holds iff the Zk are proportional to the Wk.
Proof This follows at once from (1.6-14) since the right-hand side of this
equation is a nonnegative number. We have equality only when that right-
hand side vanishes, i.e., iff
ZiWj - ZjWi = 0 (1.6-16)
which is equivalent to z; = rw;, where r :f 0 is a constant of proportionality.
Of course, condition (1.6-16) is trivially true if all Zk or all Wk are zeros.

Exercises 1.2

1. Find the absolute value of each of the following.
   (a) 15-Si (b) 3+i
   (c) 4/s-^3 / 5 i (d) (a^2 - b^2 ) + 2abi


2. Prove that z1 z 2 = 0 iff z 1 = 0 or z 2 = 0 by means of the properties of
   the modulus of a complex number.

3. If lz1 I :f lz2 I show that

I

w I 1w1

z1 + z2 S llz1I - lz2ll

4. (a) If z = x + iy, prove that lxl + IYI S V2°1zl.

(b) If z1 = X1 + iy1 and z 2 = x 2 + iy 2 , prove that

I

X1 Y1 I = Z1Z2 ~ Z1Z2

x2 Y2 2z

5. A complex number u such that lul = 1 is called unimodular. Show that
   the following complex numbers are unimodular.
   (a)
   1

;

1

(z:f:O) (b) cosB+isinB

(c) l+k~ (d) :-~ (z:f w)

1-ki z-w

6. Show that if u and v are unimodular, the numbers u, 1/u, uv, u/v,
   and un are also unimodular.


7. If lul = 1 and u :f -1, prove that u can be expressed in the form

u = (l+ki)/(1-ki), where k is a real number. Deduce that unimodular