Affiliation:
1. Department of Mathematics and Statistics University of North Florida Jacksonville Florida , 32224 U.S.A
2. Deparment of Mathematics AfyonKocatepe University Afyonkarahisar Turkey
Abstract
Abstract
The following notion of bounded index for complex entire functions was presented by Lepson. function f(z) is of bounded index if there exists an integer N independent of z, such that
max
{
l
:
0
≤
l
≤
N
}
|
f
(
l
)
(
z
)
|
l
!
≥
|
f
(
n
)
(
z
)
|
n
!
for all
n
.
$$ \max\limits_{\{l: 0\leq l\leq N\}} \left \{ \frac{|{f^{(l)}(z)}|}{l!}\right \}
\geq \frac{|{f^{(n)}(z)}|}{n!}\quad\text{for all}\,\, n. $$
The main goal of this paper is extend this notion to holomorphic bivariate function. To that end, we obtain the following definition. A holomorphic bivariate function is of bounded index, if there exist two integers M and N such that M and N are the least integers such that
max
{
(
k
,
l
)
:
0
,
0
≤
k
,
l
≤
M
,
N
}
|
f
(
k
,
l
)
(
z
,
w
)
|
k
!
l
!
≥
|
f
(
m
,
n
)
(
z
,
w
)
|
m
!
n
!
for all
m
and
n
.
$$ \max\limits_{\{(k,l): 0,0\leq k, l\leq M, N\}} \left \{ \frac{|{f^{(k,l)}(z,w)}|}{k!\,l!}\right \} \geq \frac{|{f^{(m,n)}(z,w)}|}{m!\,n!}\quad\text{for all}\,\, m \, \text{and}\,\, n. $$
Using this notion we present necessary and sufficient conditions that ensure that a holomorphic bivariate function is of bounded index.
Reference7 articles.
1. Fricke, G. H.: A characterization of functions of bounded index, Indian J. Math. 14 (1972), 207–212.
2. Hamilton, H. J.: Transformations of multiple sequences, Duke Math. J. 2 (1936), 29–60.
3. Hardy, G. H.: Divergent Series, Oxford University Press, 1949.
4. Lepson, B.: Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index. Lecture Notes, 1966, Summer Institute on Entire Functions, Univ. of California, La Jolla, California.
5. Patterson, R. F.: Analogues of some fundamental theorems of summability theory, Int. J. Math. Math. Sci. 23, (2000), 1-9.
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