TITLE MARCH 2012 - Pakistan Academy of Sciences

42 Rabha W. Ibrahim
G( r, ks; z) = a( r k s )
b z a k b z
2 n 2
| | = (1 ) | | 1,
when r = s =1, z U.
Hence by
Theorem 2.1, we have : If a 0.5, b 0 and
f : U X is a holomorphic vector-valued
function defined in U , with f (0) = ,
then
a( f ( z) zf ( z) )
b z
2
| | < 1 f ( z) < 1.
Consequently, I G( f ( z), zf ( z); z) 0,
(
z)
for every z U.
Consider the function
G : X
2 Y
by
s
G(
r,
s;
z)
= r ,
(
z)
with G ( ,
)
= .
Now for r = s =1,
we have
k
G( r, ks; z) =|1 | 1,
()
z
k 1
and thus G G( X,
Y).
If f : U X is a
holomorphic vector-valued function defined in
U , with f (0) = ,
then
zf ( z)
f( z) < 1
( z)
f( z) < 1.
Hence, according to Theorem 2.2, f has the
generalized Hyers-Ulam stability.