Let
\[
(
∗
)
f
(
z
)
=
∑
n
=
0
∞
a
n
z
λ
n
(\ast )\quad f(z) = \sum \limits _{n = 0}^\infty {{a_n}{z^{{\lambda _n}}}}
\]
be a transcendental entire function. Set
\[
M
(
r
)
=
max
|
z
|
=
r
|
f
(
z
)
|
,
m
(
r
)
=
max
n
≥
0
{
|
a
n
|
r
λ
n
}
M(r) = \max _{|z| = r} |f(z)|,\; m(r) = \max _{n \geq 0} \{ |{a_n}|{r^{{\lambda _n}}}\}
\]
and
\[
N
(
r
)
=
max
n
≥
0
{
λ
n
|
m
(
r
)
=
|
a
n
|
r
λ
n
}
.
N(r) = \max _{n \geq 0} \{ {\lambda _n}|m(r) = |{a_n}|{r^{{\lambda _n}}}\} .
\]
Sato introduced the notion of growth constants, referred in the present paper as
S
q
{S_q}
-order
λ
\lambda
and
S
q
{S_q}
-type
T
T
, which are generalizations of concepts of classical order and type by defining
\[
(
∗
∗
)
λ
=
lim
r
→
∞
sup
(
log
[
q
]
M
(
r
)
|
log
r
)
(\ast \ast )\quad \lambda = \lim _{r \to \infty } \sup ({\log ^{[q]}}M(r)|\log r)
\]
and if
0
>
λ
>
∞
0 > \lambda > \infty
, then
\[
(
∗
∗
∗
)
T
=
lim
r
→
∞
sup
(
log
[
q
−
1
]
M
(
r
)
|
r
λ
)
(\ast \ast \ast )\quad T = \lim _{r \to \infty } \sup ({\log ^{[q - 1]}}M(r)|{r^\lambda })
\]
for
q
=
2
,
3
,
4
,
⋯
q = 2,3,4, \cdots
where
log
[
0
]
x
=
x
{\log ^{[0]}}x = x
and
log
[
q
]
x
=
log
(
log
[
q
−
1
]
x
)
{\log ^{[q]}}x = \log ({\log ^{[q - 1]}}x)
. Sato has also obtained the coefficient equivalents of
(
∗
∗
)
(\ast \ast )
and
(
∗
∗
∗
)
(\ast \ast \ast )
for the entire function
(
∗
)
(\ast )
when
λ
n
=
n
{\lambda _n} = n
. It is noted that Sato’s coefficient equivalents of
λ
\lambda
, and
T
T
also hold true for
(
∗
)
(\ast )
if
n
n
’s are replaced by
λ
n
{\lambda _n}
’s in his coefficient equivalents. Analogous to
(
∗
∗
)
(\ast \ast )
and
(
∗
∗
∗
)
(\ast \ast \ast )
lower
S
q
{S_q}
-order
v
v
and lower
S
q
{S_q}
-type
t
t
for entire function
f
(
z
)
f(z)
are introduced here by defining
\[
v
=
lim
r
→
∞
inf
(
log
[
q
]
M
(
r
)
|
log
r
)
v = \lim _{r \to \infty } \inf ({\log ^{[q]}}M(r)|\log r)
\]
and if
0
>
λ
>
∞
0 > \lambda > \infty
then
\[
t
=
lim
r
→
∞
inf
(
log
[
q
−
1
]
M
(
r
)
|
r
λ
)
,
q
=
2
,
3
,
4
,
⋯
.
t = \lim _{r \to \infty } \inf ({\log ^{[q - 1]}}M(r)|{r^\lambda }),\quad q = 2,3,4, \cdots .
\]
For the case
q
=
2
q = 2
, these notions are due to Whittakar and Shah respectively. For the constant
v
v
, two complete coefficient characterizations have been found which generalize the earlier known results. For
t
t
coefficient characterization only for those entire functions for which the consecutive principal indices are asymptotic is obtained. Determination of a complete coefficient characterization of
t
t
remains an open problem. Further
S
q
{S_q}
-growth and lower
S
q
{S_q}
-growth numbers for entire function
f
(
z
)
f(z)
we defined
\[
δ
μ
=
lim
r
→
∞
sup
inf
(
log
[
q
−
1
]
N
(
r
)
|
r
λ
)
,
\begin {array}{*{20}{c}} \delta \\ \mu \\ \end {array} = \lim _{r \to \infty } \begin {array}{*{20}{c}} {\sup } \\ {\inf } \\ \end {array} ({\log ^{[q - 1]}}N(r)|{r^\lambda }),
\]
for
q
=
2
,
3
,
4
,
⋯
q = 2,3,4, \cdots
and
0
>
λ
>
∞
0 > \lambda > \infty
. Earlier results of Juneja giving the coefficients characterization of
δ
\delta
and
μ
\mu
are extended and generalized. A new decomposition theorem for entire functions of
S
q
{S_q}
-regular growth but not of perfectly
S
q
{S_q}
-regular growth has been found.