1550251515-Classical_Complex_Analysis__Gonzalez_

20 Chapter^1

Property 4 can be written as

J x^2 + Y^2 :S !xi + IYI

which is equivalent to x^2 + y^2 :::; x^2 + 2JxllYI + y^2 , since both sides of the
inequality are nonnegative real numbers. The last inequality reduces to
0 :S lxllYI, which is obviously true.
For the first inequality in property 5, we have

lxl = H :S Jx^2 +Y^2 = lzl

and similarly for the second. We note that those inequalities may also be
written as

-lzl :S x :S lzl and (1.6-2)

respectively. Hence we have

Rez:::; lzl and Imz:::; lzl (1.6-3)

It is clear that Re z = lzl holds iff z is real and nonnegative. On the

other hand, Imz = lzl holds iff z ,,; iy with y 2:: 0.
Property 6 follows from ·

lz1z21

2

= (z1z2)(z1z2) = (z1z1)(z2z2) = lz11

2

lz21^2

by taking square roots, since both sides of lz1 z2 I = Jz1 I lz2 I are nonnegative.
This extends easily to n factors; i.e., we have

where n is any positive integer.
To prove property 7, we note that
lz1 + z21^2 = (z1 + z2)(z1 + z2)
= lz11
2

• Z1Z2 + Z1Z2 + lz2 l^2

But the conjugate of z 1 z 2 is z 1 z2, so that

z1z2 + z1z2 = 2Re(z1z2):::; 2lz1z2I = 2lz1llz2I

Therefore,

Jz1 + z2 l

2

:S Jz11

2

+ 2Jz1 Jlz2 I+ lz2 l

2

= (Jz1 I+ lz21)

2

and it follows that

(1.6-4)

(1.6-5)
The preceding proof shows that the equality holds in (1.6-5) iff
Re(z1z2) = lz1z2I, which is true iff z 1 z 2 2:: 0. This condition is trivially
satisfied if either z1 = 0 or z2 = 0. Assuming that z 2 ¥-0, the condition