On entire functions of fast growth
Authors: Bajpai S. K. , Kapoor G. P. , Juneja O. P. 
Published: July 2010

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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M left-parenthesis r right-parenthesis equals max Underscript StartAbsoluteValue z EndAbsoluteValue equals r Endscripts StartAbsoluteValue f left-parenthesis z right-parenthesis EndAbsoluteValue comma m left-parenthesis r right-parenthesis equals max Underscript n greater-than-or-equal-to 0 Endscripts left-brace StartAbsoluteValue a Subscript n Baseline EndAbsoluteValue r Superscript lamda Super Subscript n Superscript Baseline right-brace">

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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N left-parenthesis r right-parenthesis equals max Underscript n greater-than-or-equal-to 0 Endscripts left-brace right-brace vertical-bar times times times lamda n vertical-bar vertical-bar equals equals of mm left-parenthesis right-parenthesis ran vertical-bar r lamda n period">

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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis asterisk asterisk right-parenthesis lamda equals limit Underscript r right-arrow normal infinity Endscripts sup left-parenthesis log Superscript left-bracket q right-bracket Baseline upper M left-parenthesis r right-parenthesis vertical-bar log r right-parenthesis">

(
∗
∗
)

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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis asterisk asterisk asterisk right-parenthesis upper T equals limit Underscript r right-arrow normal infinity Endscripts sup left-parenthesis log Superscript left-bracket q minus 1 right-bracket Baseline upper M left-parenthesis r right-parenthesis vertical-bar r Superscript lamda Baseline right-parenthesis">

(
∗
∗
∗
)

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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t equals limit Underscript r right-arrow normal infinity Endscripts inf left-parenthesis log Superscript left-bracket q minus 1 right-bracket Baseline upper M left-parenthesis r right-parenthesis vertical-bar r Superscript lamda Baseline right-parenthesis comma q equals 2 comma 3 comma 4 comma midline-horizontal-ellipsis period">

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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row delta 2nd Row mu EndLayout equals limit Underscript r right-arrow normal infinity Endscripts StartLayout 1st Row sup 2nd Row inf EndLayout left-parenthesis log Superscript left-bracket q minus 1 right-bracket Baseline upper N left-parenthesis r right-parenthesis vertical-bar r Superscript lamda Baseline right-parenthesis comma">

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.