On the Growth and Polynomial Coefficients of
Entire Series
Huzoor H. Khan1, Rifaqat Ali2
1Department of Mathematics, Aligarh Muslim University, Aligarh, India 2Department of Applied Mathematics, Aligarh Muslim University, Aligarh, India
E-mail: huzoorkhan@yahoo.com, rifaqat.ali1@gmail.com Received July 9, 2011; revised August 6, 2011; accepted August 13, 2011
Abstract
In this paper we have generalized some results of Rahman [1] by considering the maximum of f z( ) over a
certain lemniscate instead of considering the maximum of f z( ) , for z r and obtain the analogous
re-sults for the entire function
11
( ) k k
k
f z p z q z
where q z
is a polynomial of degree m and
kp z is of degree m
1. Moreover, we have obtained some inequalities on the lover order, type and lowertype in terms of polynomial coefficients.
Keywords: Lemniscate, Lower Order, Lower Type, Slowly Changing Function, Polynomial Coefficients and Entire Functions.
1. Introduction
Let
0
( ) n
n n
f z a
zbe a nonconstant entire function and assume that an≠0 for n = 1, 2, 3,
For classifying entire functions by their growth, the concept of order was introduced. If the order is a (finite) positive number, then the concept of type permits a subclassification. For the class of order = 0 and = no subclassification is possible. For exam-ple all entire functions that grow at least as fast as exp (exp (z)) have to be kept in one class. For this reason, numerous attempts have been made to refine the concept of order and type. Boas [2] define the order (0 ≤≤) and the type T (0 ≤T≤) as follows:
1
log log , log
lim sup limsup
log log| |
r n n
M r f n n
r a
(1.1)
/log , 1
lim sup limsup | | n
n
r n
M r f
T n a
e r
|
(1.2)
where
| |
, max |
z r
M r f f z
,
Rahman [1] studied the type by taking the function
, in place of
11 log a
r r
log
akkr r
and shown
that
1 1
1
1
/ 1
1
log ,
lim sup ( )
log ,... log
| | limsup
log ,... log
k
k
a a
r
k n a
n
a a
n
k
M r f
T
r r r
n a
e n n
(1.3)
In this paper, we show that instead of considering the maximum of |f(z)|, for |z| = r, we can consider the maxi-mum of f(z) over a certain lemniscate and obtained analogous results for entire function
f(z) =
1,1
( ) k
k k
p z q z
H. H. KHAN ET AL. 1125
Rice [5] has extended the results (1.1) and (1.2) for the lemniscate R.: |q(z)| = R, i.e.,
1log log , log
lim sup / limsup ,
log log || ( ) ||
R
R n
n a
M f n n
m
R p z
(1.4)
//
log ,
lim sup R limsup ( ) mn,
n
m a
R n
n
M f m
T n p z
e R
(1.5)
where
M(R, f) = | ( ) || R max | ( )
R
z
f z f z ,
R
is the boundary of the lemniscate.
Analogous to (1.4) the lower order if f(z) can be de-fined as
,
log log lim inf
log
R R
M f
m R
2. Definitions and Auxiliary Results
Definition 2.1. Slowly changing function (r) is defined as:
1) (r) is positive, continuous and tends to as r,
2)
lim 1
r
kr r
, for every fixed k > 0,
3)
1/
0
x x
x
, bounded as x .
Now we prove
Lemma 2.1.
11
( )[ ( )]k k
k
f z p z q z
is an entire function of order > 0 and type T with re-spect to order , if and only if,
/ 1/
log ,
lim sup R .
m m
R
M f
T R R
Proof. Set S = R1/m (1+ O(1)), so that from the
esti-mate Rice [5], as R for
z R,|z| = S, we have
1/|| || 2π m 1 1
R R O
/ 1/
log , log , log ,
lim sup lim sup lim sup .
(1 (1) (1 (1)
R m m
R S S
M f M S f M S f
T S S
R R S O S O
Lemma 2.2.
11
( )[ ( )]k k
k
f z p z q z
is an entire function of order > o and lower type T, if and only if
/ 1/
log ,
lim inf R .
m m
R
M f
t R R
Proof. Proof can be done in a similar manner as Lemma 2.1.
3. Main Results
First we prove the inequality for lower order in terms of polynomial coefficients.
Theorem 3.1. Let be fixed and
11
( )[ ( )]k k
k
f z p z q z
be an entire function of order > 0 and lower order ,
then,
1log lim inf
log || ( ) ||
k
k a
k k m
p z
.
Proof. Let
1log
lim inf ,
log || ( ) ||
k
k a
k k
p z
so
log
log || k( ) || a ,
k k p z
for k > K().
Using the relation Rice [5]
,
|| || ( )
|| ( ) || || ( ) || ,
2π R
R R
k a k
M f
p z Q z a R
R
(3.1)
have Q(z) is a polynomial of degree m 1, independent of K and R, we get
|| ||
2π log
log ( , ) log || ( ) || log log || ( ) ||
|| || || ( ) || 2π R
R
k
R
R k
R
R k k
M f p z k R Q z
Q z
Choose R = ek1/ , since < R, for sufficiently large k, we obtain
1
1/ log ( , ) log log ( ) log ( )
2π R 2π R
R R
R
M f k Rk Q z k Q z
or
log , log ( )
2π R
R R
M f R e Q z
or
log log , log log 1 log ( )
2π R
R R
M f R e e R Q z
In view of a result of Rice [5]
1/ || ||
1 (1) 2π
m
R R O
,
we get
log log ,
(1) log
R
M f
O
R
.
Proceeding to limit as R, we get
m .
Hence the proof is complete.
Theorem 3.2. Let be fixed. The necessary and suf-ficient condition that
11
( )[ ( )]k k
k
f z p z q z
is an entire function of type T (0 ≤T< ) with respect to order (0 ≤ < ), is
/log ,
lim sup limsup || ( ) ||
( ) ( )
( )
mk k
r k
M r f T m k p
e k
r r
z
(3.2)Proof. Let
/lim sup ( )
( )
mk k
k
k
p z
k
.
Then for a given > 0, we get
/
( )
|| ( ) || ( )
mk
k z k k
.
For an infinite sequence of values of k, so that from (3.1), we have
/
/ ( )
2π ( )
( , )
( )
R
mk m
R
R
R k
k
M f
Q z
( 3.3)
Choosing a sequence of values of R such that
m ke
R k
.
For these R the right hand side of (3.3) attains its maximum value, that is,
/ 2π( ) ( , )
( )
R
mk R
R
e
M f
Q z
,
or
log ( , ) log 2π log ( )
R
R R
mk
M f Q z
or
/ 1/ / 1/ 1/
log ( , ) (1) ( ) (1)
( ) ( )
( ) ( R
m m m m
M f m k O m k O
e
R R R R ke
k
Applying the limits as R, we get
( ) 1
( )
m T
e
(3.4)
In order to prove reverse inequality, from (3.2) we have
/( ) ,
( )
mk k
k
p z k
for sufficiently large k.
H. H. KHAN ET AL. 1127
( ( )
m e
)
(3.6)
Since is arbitrary, combining (3.4) and (3.6) we get
( )
m e
Now we have to prove that T is a type of f (z) with
respect to order , when
( )
m e
In fact we can assume that the inequality (3.5) holds for all k as we can always add a polynomial to f(z) with-out afficting its order. Thus
/
1 1 2 2
1 1 1 1
( )
( ) | ( )[ ( )] | | ( ) | || ( ) || ( )
mk p
k k k m m
k k k
k k k K
k
f z p z q z P k R p z R R k
k
RFor k = k1, given by k1 = ( + ) (k1) R/meO(1)−1, we find that
/ ( ) ( )k mk
k
Rk is maximum and the maximum value is
/ (1) 11 exp 1 O(1) m ( )k Rm O
e
Now choosing K1such that K1< k1 < K1 + 1. Then
1
1
/ /
2 2
1 1 1
2 / (1) 1 2 *
1 1
( ) ( ) ( )
exp 1 (1) ( )
mk K mk mk
m k m k
k k k K
m m O m
R k R R k R k R
k k k
m
K R O k R e R N
/ k
where N* is the sum of the convergent series
1
/
1
( )
mk k k K
k R
k
.So
M(R, f) =
R
z
f
(
)
||
||
2
/
1 exp 1 (1) 1 1 (1)
m m m
K R O k R e O
O(1) 1
/ (1) 1 2
/ (1) 1
1 1
( )k Rm Oe Rm exp 1 O(1) m ( )k Rm Oe 1 O(1)
which gives
/ (1) 1 1
, (1) 1
1/
/ 1/ / 1/
1 1
1 (1) ( )
log ( )
(1) 1 (1) (1).
( ) ( )
m O
R O
m m m m
m
O k R e
M f m k
O O e O
e
R R R R k e
k
In view of Lemma 2.1 and condition (3) of Definition 2.1, we get
( ) 1
. ( )
m T
e
Similarly we can prove
( ) 1
( )
m T
e
,
by taking
/( ) ,
( )
mk k
k p z
k
for an infinite sequence if values of k and choosing
a corresponding sequence of values of R such that
/
( ) m
e k
R
k
with (3.1). Hence the proof is complete.
Theorem 3.3. Let be fixed. If
11
( )[ ( )]k k
k
f z p z q z
is an entire function of order > 0 and lower type ,t
where
log ( , )
liminf ,
( ) r
M r f t
r r
Then
/liminf ( )
( ) ( )
mk k
k
m k
p z t
e k
(3.7)
Proof. The proof of this theorem follows on the lines of the necessary part of Theorem 3.2, by writing
/' liminf ( )
( )
mk k
k
k
p z
k
and for a given > 0,
/
( ) mk ( ' )
k
k p z
k
for k>K(),
so that we may choose a sequence of values of R satisfy-ing
/ .
( ) ( ' )
m k e
R
k
Now using the relation (3.1) and Lemma 2.2, we get the required result. Hence the proof is complete.
Remark 3.1. Theorem 3.2 is the generalization of the result (1.3) by Rahman [1].
Remark 3.2. Rahman’s theorem [1], if
1
1
log ,
( ) lim inf
(log ) ,....(log ) k
r k
M r f T
k r r r
then
1
1
1
/ 1
( ) lim inf | |
(log ) ,....(log ) k
n n
n k
n
T a
e n n
is a special case of Theorem 3.3, if we take
and consider the circle |z| =
r,
instead if the lemniscate |
q(z)| = R.1
1
log a ,... log ak
k
r r r
4. References
[1] Q. I. Rahman, “On the Coefficients of an Entire Series of Finite Order,” Math Student, Vol. 25, 1957, pp. 113-121. [2] R. P. Boas, “Entire Functions,” Academic Press, New
York, 1954, pp. 9-11.
[3] J. L. Walsh, “Interpolation and Approximation,” Ameri-can Mathematical Society, Colloquim Publications, Provi-dence, Vol. 20, 1960, p. 56.
[4] P. Borwein, “The Arc Length of the Lemniscate{|p(z)| = 1},” Proceedings of the AMS—American Mathematical Society, Vol. 123, 1995, pp. 797-799.
[5] J. R. Rice, “A Characterization of Entire Functions in Terms of Degree of Convergence,” Bulletin of the Ameri-can Mathematical Society, Vol. 76, 1970, p. 129. doi:10.1090/S0002-9904-1970-12396-5
[6] J. R. Rice, “The Degree of Convergence for Entire Func-tions,”Duke Mathematical Journal, Vol. 38, No. 3, 1971, pp. 429-440. doi:10.1215/S0012-7094-71-03852-X [7] O. P. Juneja, “On the Coefficients of Entire Series,”
Journal of Mathematical Analysis, Vol. 24, 1971, pp. 395-401.
[8] O. P. Juneja and G. P. Kapoor, “Polynomial Coefficients of Entire Series,” Yokohama Mathematical Journal, Vol. 22, 1974, pp. 125-133.
[9] D. Kumar, “Approximation Error and Generalized Orders of an Entire Function,” Tamsui Oxford Journal of Mathematical Sciences, Vol. 25, No. 2, 2009, pp. 225- 235.
[10] D. Kumar and H. Kaur, “Lp—Approximation Error and
