341
Internat. J. Math. & Math. Sci.
Vol. 9 No. 2 (1986) 341-346
THE MAXIMUM TERM OF A POWER SERIES
G.S. SRIVASTAVA
Department of Mathematics
University of Roorkee
Roorkee, India
and
O.P. JUNEJA
Department of Mathematics
Indian Institute of Technology
Kanpur, India
(Received March 30, 1984)
ABSTRACT.
n
Let
Izl
in the disc
be a power series, representing an analytic function
anZ
0
< R.
f(z)
A characterization for the type of such functions was obtained
by the authors [J. Math. Anal. Appl. 81(1981), I-7] in terms of the maximum term and
It is proved in this paper by means of an example, that a similar relation does
rank.
not hold in general for lower type and sufficient conditions have been obtained for
the validity of the corresponding result for lower type.
Alternative coefficient
characterization for type and lower type have been given and a necessary and sufficient condition for the analytic function
f(z) to be of perfectly regular growth has
been obtained.
Analytic function, order, type, lower type, maximum term, and
FEY WORDS AND PHRASES.
index.
.ntra
AMS SU{iJECT CLASSIFICATION CODE
i.
30B10.
INTRODUCTION.
Let
f(z)
R(0 < R < ).
N(r)
max{%
n>0
n
n
For
n
an z
0
0
m(r)
be an analytic function with radius of convergence
M(r)
R, we set
r
la n Ir
}.
M(r)
Then
max
If(z)
m(r)
and
modulus, maximum term and central index respectively.
ined as
lim sup
r+R
I$+ I$+
M(r)
log x
,
where x Rr/(R-r). When 0
Further for 0 < 0 < =, the type
0 <
N(r)
max
{la n Ir
and
are called the maximum
The order
0
of
f(z)
is de-
(I.I)
can be replaced by m(r) in (I.I) [l,Lemma2].
and lower type t of f(z) are defined as
M(r)
T
lim sup log m(r)
0
r/R inf
0
0,
m(r)
X
T
t
(1.2)
G. S. SRIVASTAVA AND O. P. JUNEJA
342
f(z)
The function
.
side of (I.I).
T <
is said to be of regular growth if limit exists on the left hand
f(z)
Further,
0 < t
is said to be of perfectly regular growth if
Maclane [2] obtained the following formula for the order
lanl
lg+ lg+
sup
log
of
0
f(z):
P
(1.3)
p+l
n
Further, Bajpai, Tanne, and Whittier [3] obtained the following coefficient charac-
T
terization for
and
t:
lim sup [log+
n
inf
(p+l)
B
where
p+I
(R/p)
n
(i)
conditions that
p.
(lanl
B
R
n)]
T
(1.4)
t
Pn
The expression for the lower type
Xn+l
function of n.
-
loglan/an+ll
n+l n
(n)
(ii)
and
holds under the
t
is a non decreasing
In a recent paper of [4], we obtained a formula for the type
M(r)
N(r).
and
Thus we have
(p+l---)
p+l
pT
(-)
lim sup
N(r)
x
r/R
x
m(r)
{I+ oN(r)
T
in terms of
(p+l)
(1.5)
In this paper, we show by means of an example that a corresponding result for lower
type
We shall obtain the corresponding result for lower
does not always hold.
t
In the process, we also obtain formulae corresponding
type under certain conditions.
to (1.4) without any additional conditions on
%
n
or
a’s
n
Izl
< R
thus improving upon
(1.4).
WE PROVE THE FOLLOWING;
2.
THEOREM I.
T
type
f
n
f(z)
Let
and lower type
{r
and
k}
0
n
a z
n
be analytic in
{n
Let
t
denote the sequence of principal indices of
N(r), i.e.,
the sequence of jump points of
N(r)
n
+lim
sup
for
k
r
r <
k
1,2,3
k
rk+
n
Then
k
lim inf
k
where
k}
and of order
x
R
k
(R--k)[
IOg+
log m(r k)
({ank
Rp+lk)
(p+I)
(2.1)
(2.2)
t
(Xk)P
rk/(R-rk)
Before providing the above theorem, we give an example to show that (1.5) does
not always hold for the lower type.
EXAMPLE. Let us consider the function
seen that
m(r)
e
2
f(z) is analytic in
n
r
4
Izl
f(z)
n
0
< I, of order
for
-I/3(2
n-l)
< r < e -I/3(2
2n
z
p =I.
n
e
e
n)
4n
Then it can be easily
Further,
N(r)
4n
343
MAXIMUM TERM OF A POWER SERIES
Hence, from (2.1) and (2.2) above, we have
(for R
and p
I)
N(r)
]p+l
x log m(r)
+ 7(r)
x-
lim
r.+R
I/4, t
T
>
2/9.
0.+I
(@_
P
8
g
However, in this case
(R)
Hence we see that although limit exists on the right hand side of (1.5), the equality
fails to hold for lower type t
The above inequality can be generalized as follows:
Let
THEOREM 2.
f(z)
(0+I)
be defined as before.
p+l
(pt)
R
< lim inf
Then
N(r)
r->R
In order to get reverse inequality for
THEOREM 3.
{r
f(z)
Let
n
(0+I)
0
where
x
(2.3)
we prove
t
be defined as in Theorem I.
R
tending to
(p+l)
+-xr log re(r)
Then for any increasing sequence
we have
p+l
0t
(--)
N(rn_ I)
>
lira inf
(x)
l
x
+
p+l
log m(r
n
rn
n
p+l
n
(2.4)
N(rn)
R r /(R-r ).
n
n
n
The following corollaries are immediate consequences of Theorem 3.
COROLLARY I.
We have
()o+I
p+l
ot
()
N(rn-I
max
{r
lim inf
p+l
n+
x
n
n
{I +
__n_nX
r
log m
N(r
n
(rn
p+l
(2.5)
n
where the maximum on the right hand side is taken over all increasing sequences
such that
r
R
n
COROLLARY 2.
as
n
p+I
()
{max
{r
lim inf
n-=
n
N(rn_
N(rn
{lim inf N(r)
rR
x
n
=.
We have
(p+l)
P
where
{r
Rr/(r-r)
and
{r
m(r)
+ 7x logN(r)"
0+I
(2 6)
is as defined above.
n
Lastly we state two theorems.
THEOREM 4.
If
f(z)
satisfies the condition of Theorem
ing sequence of positive real numbers
N(r
N(r
n-1
{r
n
then
p+l
pt
(-)
THEOREM 5.
If
f(z)
lim inf
r/R
n
such that
as n-
N(r)
x
and there exists an increasr
n
lim
r+R
N(r)
as
=,
n
and
(2 7)
+ x lo(r)
N(r)
satisfies the conditions of Theorem
growth, then
R
p+l
(2. 8)
and is of perfectly regular
m(r)
+ x logN(r)
(2 9)
exists.
Conversely, if limit in (2.9) exists and equals
A
and
N(r)
satisfies the con-
G.S. SRIVASTAVA and O. P. JUNEJA
344
dition (2.7), then
f(z)
is of perfectly regular growth and
3.
for
k
r
_<
_<
k
r
r <
Qk(r)
Further, for fixed
case
r
a non
k
r
k
k
<
k
for
r
P(r) decreases for
P(r)
r
for
r
Qk(r)
Since
(3.1)
R-r
R
log (R/r)
as
r/R
lank l-0/(0+l)nk
RI/(0 +I)
it
r
P(r)
r
k
increases for
while if
rk+l,
r
k
(3.2)
has an absoute maxima for
then the function
rk+
I-0
for
the function
decreasin function of
r
r
+ n k log r)
0
r*
k
r
< r
[Rr/(R-r) ]0
is a differentiable function of
I-0
R
(R_-)
()n k -0(loglank
can be easily seen that
k
P(r)
+ n k log r
Differentiating, we have
rk+
If
lank
P(r)
Then
[Rr/(R-r) ]0
It is clear that
rk+
Qk(r)
where
k
(2.10)
log
log m(r)
P(r)
Let us set
P’(r)
r
(-).
PROOFS OF THEOREMS.
THEOREM I.
r
AR
p
()
T
t
rk+ I.
r
k
r
e
k
r
<_ r
rk+ I,
r
<- r
k
only.
k*
then
In
P(r)
In every case, we have for
rk+l,
P(r) e
e(rk), or
P(rk+l).
P(r) e
(3.3)
(3.4)
Thus (3.3) and (3.4) give
lim inf P(r)
rR
lim inf
k-
P(rk)
(3.5)
lim inf
P(rk).
(3.6)
But we obviously have
lim inf P(r)
Thus (3.5) and (3.6) lead to (2.2).
To prove (2.1). we note that for
r
k
lank
log
P(r)
[Rr/
since for fixed
P(r)
k
r <
rk+l,
+ n k log r
k
(R-r )
k
has an absolute maxima at
r
r
k.
The above inequality,
after simple calculation leads to
T
lim sup P(r)
r+R
< lira sup
(--)
0
log(l an
k+
R
nk)
k
0+
The reverse inequality is trivial since
T
lim sup P(r) >- Jim sup
k-
rR
P(r)
i+I
(.7)
is
345
MAXIMUM TERM OF A POWER SERIES
(,--)
l.im sup
k-
[’
(3.8)
]
0+I
Thus we gel (2.1) on combining (3.7) and (3.8).
This proves Theorem
I.
The result (2.1) is an improvement over the corresponding resul (1.4) of
REMARK.
Bajpai e.al. [3].
I is easy o verify ha for
THEOR 2.
and
O,
u
(1) p+l"
()
pu
0 > 0
The inequaliy (2.3) now follows on making proper substitutions for
and proceeding
To prove (2.4), we pu
TBEOR 3.
N(rn
lira inf
x
log m(r
r
N(r
+
n
Then for
u
0 <
N(r
log m(r
R-r
n-I
[-r0+l
n
q
n
no(E)
n >
q, we have for
E
n
0+I
n
+
N(rn
n ] 0+I
This leads to
a-r
a
I (p/
log m(r
n)
>
[rn{N(-rn_i)
n]N(rn)"
(3.9)
From the relation [5,p.196]
log M(r)
log m(r
o)
fr N(t)d t
+
o < r
ro
O
<r R,
(3.1o)
we have
log m(r
log re(r)
n)
+
r
n
N.(t)
t
dt
-> log m(r + N(rn) log (r/r)
n
n
log m(r n) + N(rn){lg(R/rn
R-r
Using (3.9) and the fact that
log
(R)
as
r-R, we get
1/(p/l)
log m(r) >
Let
{s
n
R-r
R
N(rn)[rn{N(rn_1)}
be a sequence defined by
s
n
It is clear that
x
log(R/r)}.
R
O
sR(q-E)
.I/(p+l) ]
n > no
exp[--[,
as
s /R
n
n+.
If
r
s
satisfies
n
< r <
Sn+l’
Rr/(R-r),
lOgxpro(r)-
>
N(rn)
.R-Sn+ l]p
tRn+
]P
N(rn) [R-Sn+.
R Sn+
(p+l)r
n
log(Rls n)
oR
(p+l)r (R-s
n
pR2
n)
R-s
R -s n
R
n
then for
G.S. SRIVASTAVA and O. P. JUNEJA
346
Proceeding to limits as
O
0+I R(q-_ (R-s n)
0
o+1 (q-c)R
R, we get
r
0
(0--f)
which yields (2.4).
0+1
Ot
&
(-)
q
The inequality obviously holds if
As already mentioned, Corollaries
q
0.
This proves Theorem 3.
and 2 are direct consequences of Theorem 3.
Theorem 4 follows on combining (2.3) with (2.6) under the condition (2.7).
Hence we
omit the proofs.
Lastly we prove
THEOREM 5.
f(z)
Let us suppose that
is of perfectly regular growth, i.e.
T
t.
Then from (1.5) and (2.3), we have
lira sup
r/R
_<
N(r)
m(r)
+ 7x logN(r)
x--
-.-N(r)
lira inf
r/R
x
l+
7
x
p+l
(o+l.P+l
_) (0__)
lo$ re(r) 0+1
--
N(r)’
that is, limit exists in (2.9).
Conversely, let
lim
N(r)
+
r+R x
x log m(r)
r
N(r)
0+1
and suppose that (2.7) is satisfied for some sequence
(1.5), we have
(__) O+
i.e.
T
t
0+I AR
(0+I---) (-).
ThiF proves Theorem 5.
Ot
(-)
ot
(o+1. o+1 (-)
=A
{r }.
n
Then from (2.8) and
A,
REFERENCES
I.
SONS, L.R.
306.
2.
MacLANE, G.R. Asymptotic values of holomorphic functions, Rice University Studies,
Houston,(Texas), 1963.
3.
BAJPAI, S.K., TANNE, J., and WHITTIER, D. A decompostion theorem for an analytic
function, J. Math. Anal. Ap_pl. 48(1974), 736-743.
4.
SRIVASTAVA, G.S. and JUNEJA, O.P. The maximum term of a power series, J. Math.
Anal. Appl. 81(I) (1981), I-7.
5.
VALIRON, G.
Regularity of growth and gaps, J. Math. Anal. Appl.
Fonction
2__4(2)(1968),
296-
Analytiques, Press Unlversltalres de France, Parls,1964.