ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXIV (1984)
G . S . Sr i v a s t a v a
(Roorkee, India) and O. P.
Ju n e j a(Kanpur, India)
On entire functions of slow growth
1. Let / (z) = Y j ar£n-> where z = reie, be an entire function and let M(r), n= 0
p(r) and v(r) denote the maximum modulus, maximum term and the rank of the maximum term of /(z ) for |z| = r, that is, M(r) = max |/(z)|,
|z| =
rp{r) = max {|a„|r"} and v(r) = max [n: p(r) = \an\ rn}. Thus, if for r > 0, о
p(r) — |a jr " = |an+ 1 |r "+1 = ... = ja„+k|r " +,£, then the rank v(r) is defined to be n + k. It is clear that v(r) is an unbounded, non-decreasing function of r.
The values assumed by v(r) are called the principal indices of /(z).
As usual, we define the order q and type T of / (z) as ( 1 . 1 )
and
lim sup Г-* 00
log log M(r)
log r Q, 0 < Q ^ 00 ,
( 1 . 2 ) lim sup
Г - * X
log M(r)
= T, where 0 < g < oo.
The function/(z) is said to be of regular growth if limit exists on the left- hand side of (1.1). Similarly, if 0 < T < oo and limit exists on the left-hand side of (1.2), then/ (z) is said to be of perfectly regular growth. Further,/(z) is said to be of very regular growth if two finite and positive constants В and D can be found such that
(1.3) В < lim inf
r~* cc
log M(r)
lim sup
Г - * oo
log M{r) D.
Valiron [5] derived the necessary and sufficient conditions on the coefficients a„’s so that a function / (z) may have regular, perfectly regular or very regular growth. Thus we have ([5], Theorem 15):
T
h e o r e mA. The necessary and sufficient condition that a function of
order q be of regular growth is that the coefficients an's satisfy the inequality
(1.4) |a j 1/n < n~l/(e+e),
£ being arbitrarily small and positivé, for all sufficiently large values of n, and that there is an infinite sequence of numbers (np) such that
(1.5) lim
p~*O0
log np+1 log np = 1, for which
(1.6) \a„J1/np > (np) 1/(<? Bp), where sp ->0 as p-»oo.
Similar conditions also exist for functions of perfectly regular and very regular growth ([5], p. 44-46).
It is clear that for an entire function of order g = 0, conditions (1.4) and (1.6) are no more relevant. For such functions, said to be of slow growth, the concept of logarithmic order g* is used, where p* is defined by
(1.7) lim sup
r o o
log log M(r)
log log r 1 < g* < oo.
For 1 < g* < oo, the logarithmic type T* is defined as
n Q4 v log г* ^
(1.8) lim sup —---— = T *, 0 ^ 7 * ^ x . r - * (log r f
We also define the lower logarithmic order A* and the lower logarithmic type t* by
(1.9) lim inf
Г-* oo
log log M(r)
log log r A*, 1 ^ A* ^ g* ^ oo, and
(1.10) lim inf ^ ~ = t*, 0 ^ t* ^ T* ^ oo.
r^ x (log r)e
An entire function / (z) is said to be of regular logarithmic growth if and only if A* — g*, and of perfectly regular logarithmic growth if and only if 0 < t* = T* < oo.
In the present paper, we shall find necessary and sufficient conditions for an entire function to be of regular logarithmic growth etc. To avoid trivial cases, we shall consider only transcendental entire functions.
The following result was proved by Valiron ([5], Theorem 11).
( 1 . 11 ) p(r) < M{r) < p{r) 1 + 2v < r +
v(r)
We also have the following relation between p(r) and v(r) ([5], p. 31):
( 1 . 12 ) log Mr) = log Mr0) + v(*)
X dx.
r0
Now, from (1.1), (1.11) and (1.12), it is easy to prove
T
h e o r e mB. For entire functions of finite order
(1.13) lim
r~* ao
log M(r) log Mr) or, in the usual notation,
(1.14) log M{r) ~ log Mr). N
In view of the asymptotic relation (1.14), it is easily seen that M(r) can be replaced by p(r) in (1.7) to (1.10).
2. We now prove
00
T
h e o r e m. 1. Let f ( z ) = £ anzn be an entire function of logarithmic order
n= 0
Q* and lower logarithmic order к*, 1 < k* < q * < oo. Then ( 2 . 1 ) \an\1,n < exp {— n1/(e*~ 1 +£))}
for all sufficiently large values of n, where e > 0 is arbitrary. Further, if (np) denotes the sequence of principal indices of f (z), then
( 2 . 2 ) and
lim sup p-+ 00
log np+ ! log np
e * - i
(2.3) ,l/«r
> exp { - ( n p) 1 ~epb h 0 as p 00 .
P ro o f. We shall use the Newton’s polygon method as adopted in [5].
We put
9n = -lo g \an\.
Since f(z ) is entire, we have
(2.4) lim ~ = oo.
It~* 00 tl
Taking the axes of coordinates OX and 0У, we plot the points A n of
coordinates (n, gn). Thus we construct a Newton’s polygon n{f) with some of
the points /l„’s as its vertices and the rest of lying on or above it. The
rank and the maximum term of / (z) are given corresponding to those /l„’s
which are vertices of n (f). For details, we shall refer to ([5], p. 28 -31). Now we consider another dominant function
X
(2.5) W(r) = £ e~°nrn,
n = 0
where Gn is the ordinate of the point of abscissa n on the polygon n ( f ). It is easily seen that f ( z ) and W(r) have the same maximum terms and ranks for all r. Thus logarithmic order and lower logarithmic order of f (z) and W(r) are also the same. Further, the ratio e n/e "+1 is a non-decreasing function of n. Thus, from [4] we get for 1 < A* ^ q * < oo,
( 2 . 6 ) lim sup
O D
log n
log (GJn) 1,
(2.7) lim inf
n~* oo
log n
log (GJn) A * - l . Hence, given e > 0, we have for all n > n0(e),
A* — 1 —
F. <log n
log (GJn) < £* — 1 +e,
Fig. 1
or,
( 2 . 8 ) n(e* + e)l(Q*- 1+fi) < Q < и (Л*-г)/(Л*- l-£)_
From the construction of тг(/), it is clear that G„ < gn for all n. Hence (2.1) follows from the left-hand inequality of (2.8). Now, we see that for all n> n0, the polygon n ( f) lies between the two curves
( f l - y = x U * - e ) / { À * ~ 1 - E ) .
y
_ x (q* + e)/(q* ~ 1 + £ )Supposing that A„ and An< are two consecutive vertices of n{f), the chord A„A„' lies between the two curves Ct and C2. The difference n' — n is maximum when A„ and A„> are points of intersection with C 2 of a tangent to Cj. Thus, let F be a point of C1 and let X t , X 2 be the abscissas of points of intersection with C 2 of the tangent at F. Then X x and X 2 are the roots of the equation
JC(e*+«)/(e * -1 +«)_ ^(д*-в)/(я*- 1 - e ) W - Z ) Х ш *~ 1 ~£), that is
(2.9) л/(е*~ i +£). À* W w * - ! - !
— 1—8/
The left-hand side of (2.9) is positive for the two values
X ( X*-8 ' (e*~1+e)
1 -£)
= °-
X = A * -e ’
1 +e)/(A*- l-£) and is negative for x = X . Hence we must have
X
A*-£ < X i < x 2 < Л * - £
Л * - 1 - в
(e*~ 1 +E)
Hence we get
log * 2 log X,
Q* — 1 + £
I * - l — -£ (l+ o (l)), or
(
2
.10
)log rt'
log n
Q* — 1 + £
Л* — 1 —£ ( 1 + 0 ( 1 )).
Thus we see that the coefficients \a„\ are equal to the numbers e G" for the sequence |/ip] of principal indices of f(z ) satisfying (2.10). Since £ > 0 is arbitrarily small, we have
lim sup p- X
log Kp+l log «p
1
(
2
.11
)Г
and
(2.12) K p|1/np> e x p [ - ( « p)1/(A*~1_8p)], as oo.
This completes the proof of Theorem 1.
With the help of Theorem 1, we now prove 00
T
h e o r e m2. Let f(z ) = £ a„zn be an entire function of logarithmic order
n = 0
g*, 1 < g* < oo. Then f (z) is of regular logarithmic growth if and only if the following conditions are satisfied:
(i) For arbitrarily small e > 0 and all sufficiently large values of n, (2.13) \an\1/n < exp [ — n1/(e*-1+£)].
(ii) I f {np} denotes the sequence of principal indices of f (z), then
(2.14) lim
p - * oo
log np+ 1 log np = 1 , and
(2.15) \a„p\ilnp > exp[ —(»p)1/(e 1 £p)], ep -» 0 as p o o .
P ro o f. First, let /(z ) be of regular logarithmic growth and logarithmic order g*. Then, from (2.1), (2.2) and (2.3), we immediately have (i) and (ii).
Conversely, suppose / (z) satisfies conditions (i) and (ii). We consider an auxiliary function
00 F(z)= £ a „ / p.
p— о
Then, it is obvious that /(z ) and F(z) have the same sequence of principal indices and have the same maximum term for each \z\ = r. Hence they have the same logarithmic order and lower logarithmic order etc. Now from (2.13) and (2.15), we have
r 1V7© rtP * t
lim ---- --- , - t -, = o* - 1.
log [log |fl„p| 11 p]
From Theorems 1 and 2 of [1], it immediately follows that F(z) is of regular logarithmic growth and logarithmic order g*. The same is therefore true for /(z ) also.
This proves Theorem 2.
Now, let us suppose that / (z) is of logarithmic order g* (1 < g* < oo)
and of logarithmic type T* and lower logarithmic type t* (0 < t* ^ T*
< oo). Then, considering again the function W(r) defined by (2.5), we have, following Rahman [3],
(2.16) lim sup
Q O
П
Г 1(e*-D
_{e* - 1 ) n _
Q*T*,
(2.17) lim inf
n-*oo
n
Q*Gn 1(e*-D _te*-l)w_
Q*t*.
We now consider the curves V - i C'i: У = C^: y =
xe*l{e*~ i) g~ / (g*t* — e)1/<e* 1)5 Q * - l \
{g*T*+e)lKe*~l)
where e > 0 is arbitrarily small. Then the polygon n ( f) must lie between the curves Ci and C'2, beyond a certain value of n. Further proceeding as earlier, we can prove the following
00
T heorem 3. Let f(z ) = ]T anzn be of logarithmic order q * (1 < q * < oo),
n= 0
of logarithmic type T* and lower logarithmic type t* (0 < t* ^ T* < oo).
Then, given arbitrarily small e > 0, we have for all sufficiently large values of n,
(2.18) \a„\lln < exp (g* — 1 ) n1/(<?*
q *( q * Т*-Ье)1/(е*_1)_
Further, if {np} denotes the sequence of principal indices of f (z), then (2.19)
and
г
ПР+ ilim sup ——
(2.20) \ a f !> > exp
g* (g*t* —£p)1/(e*~ t;p —> 0 as p-> oo,
x t and x 2 being the smallest and largest roots of the equation
( 2 . 2 1 ) (e* _ i ) x«, _ e*x<c*-4+ i l = o.
The following theorem gives necessary and sufficient conditions for an entire function / (z) to be of perfectly regular logarithmic growth.
3C
T heorem 4. Let f(z ) = ]T anzn be an entire function of logarithmic order n — 0
g*, 1 < q * < X . A necessary and sufficient condition that f (z) be of perfectly regular logarithmic growth and logarithmic type T* is that (2.18) holds and that for the sequence of principal indices {np) of f(z), we have
(2.22) l i m ^ = l ,
p- * tip and
(2.23) , 1
mP > exp ,
Q* {Q* T* — £р)1/{в*~ Л J &p -> 0 as p -> x .
P ro o f. The proof follows on the lines of Theorem 2. First, let/(z) be of perfectly regular logarithmic growth and logarithmic type T*. Then it is clear that equation (2.21) has a unique root at x = 1. Hence we immediately have (2.22) and (2.23) from (2.19) and (2.20).
Conversely, suppose / (z), satisfies (2.18), (2.22) and (2.23). Considering again the function F (z) as defined earlier we have from (2.18) and (2.23) that
p-» G lim C[log к г 1'*]
l/» n i d ? * - 1)Q* \ (l 1)
Further, since (2.22) is satisfied, we have (see [2], Theorem 4) lim inf
p
(X
ПР
