C OMPOSITIO M ATHEMATICA
R. S. L. S RIVASTAVA O. P. J UNEJA
On proximate type of entire functions
Compositio Mathematica, tome 18, n
o1-2 (1967), p. 7-12
<http://www.numdam.org/item?id=CM_1967__18_1-2_7_0>
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7
by
R. S. L. Srivastava and O. P.
Juneja
1
In
studying
thegrowth
of an entire functionf(z)
of finite order p, use is made of acomparison
functionp(r)
called theproximate
order
[1,
p.54]
off(z),
which possesses thefollowing properties:
i) 03C1(r)
isreal,
continuous andpiecewise
differentiable forr>l,
iii) 03C1’(r)r log
r ~ 0 as r - oo, where03C1’(r)
is either theright
orleft hand derivative at
points
wherethey
aredifferent,
where
It is evident that
p(r)
has been linked with the order p andlog M(r)
togive
information about thegrowth
off(z).
Besidesthe order and the lower order there are two other constants,
viz.,
thetype T
and the lowertype t
off(z)
whichgive
a moreprecise
information about thegrowth
thangiven by
the orderand lower order. These are determined as
Since the
proximate
orderp(r)
is not linked with thetype
off(x)
it becomes natural to search for anothercomparison function, T(r),
say, which should take into account thetype
of the functionand be
closely
linked with its maximum modulusM(r).
Inanalogy
with the
proximate
order we call this functionT(r)
as aproximate type
of the entire functionf(z).
In this paper we first define
proximate type
of an entire function8
and then prove its existence on lines similar to those of Shah
[2]
for the case of
proximate
order. The idea is further extendedby defining
a lowerproximate type. Finally,
we demonstrate that r-Plog M(r)
is aproximate type
for a certain class of entirefunctions.
2
DEFINITION. A
function T(r)
is said to be aproximate type of
an entire
function f(z) of
order p(0
p~)
andfinite type
Tif it
has thefollowing properties:
(2.1) T(r) is real,
continuous andpiecewise di f f erentiable for
(2.3) rT’(r)
- 0 as r ~ oo, whereT’(r)
is either theright
or theleft
hand derivative atpoints
wherethey
aredifferent,
LEMMA.
exp {r03C1T(r)} is
anincreasing function of
rfor
r > ro.By (2.1), (2.2)
and(2.3)
we havefor r > ro, so the result follows.
THEOREM 1. For every entire
function f(z) of
order p(0
p~)
and
finite type
T there exists aproximate type T(r).
PROOF. Let
S(r)
= r-Plog M(r).
Then two cases arise. Either(A) S(r)
> T for a sequence of values of rtending
toinfinity,
or
(B) S(r) S
T for alllarge
r. In case(A),
we defineQ(r)
=maxx~r{S(x)}.
SinceS(x)
iscontinuous,
limsupx~~S(x)
= Tand
S(x)
> T for a sequence of values of xtending
toinfinity, Q(r)
exists and is anon-increasing
function of r.Let ri be a number such that ri > ee and
Q(rl)
=S(ri).
Suchvalues will exist for a sequence of values of r
tending
toinfinity.
Next,
supposeT(r1)
=Q(r1)
and let t1 be the smallestinteger
not less than
1+r1
such thatQ (rl )
>Q (tl )
and setT(r)
=T(ri )
=Q(rl) for ri
C r S tl. Define ul, as followsbut
Let r2 be the smallest value of r for which r2 > ul and
Q(r2)
=S(r2). If r2
> ul then letT(r)
=Q(r)
foru1 ~
r r2 .Since
Q(r)
is constant foru1~r~r2,
thereforeT(r)
is constantfor
u1~r~
r2. Werepeat
theargument
and obtain thatT(r)
is differentiable in
adjacent
intervals.Further, T’(r)
= 0, or(-1/r log r)
andT(r)~Q(r)~S(r)
for allr~r1. Further, T(r)
=S(r)
for aninfinity
of values of r = ri, r2, ...,;T(r)
isnon-increasing
andlimr....oo Q(r)
= T.Hence,
and since
M(r)
= exp{r03C1S(r)}
= exp{r03C1T(r)}
for aninfinity
ofr,
M(r)
exp{r03C1T(r)}
for theremaining
r, thereforeCase
(B).
LetS(r)
T for alllarge
r. Here there are two pos- sibilitiesfor at least a sequence of values of r
tending
toinfinity;
for all
large
values of r.In case
(B.1 )
we takeT(r)
= T for all values of r.In case
(B.2)
letP(r)
= maxX~x~r{S(x)}
where X > ee is such thatS(x)
T for x > X.P(r) is non-decreasing.
Take asuitably large
value of r, > X and letwhere si ri is such that
P(sl)
=T(si).
IfP(s1) ~ S(81),
then we take
T(r)
=P(r) upto
the nearestpoint
t1 sl, at whichP(tl)
=S(ti). T(r)
is then constantfor t1
~r~ sl. IfP(81)
=S(s1),
thenlet t1
= sl.Choose r2
> risuitably large
and letT(r2)
=T,
10
where s2 (r2)
is such thatP(82)
=T(82)-
IfP(S2) =1= S(à
then let
T(r)=P(r)
fort2~r~s2
wheret2(s2)
is t]point
nearestto s2
at whichP(ta)
=S(t2).
If
P (82)
=S(s2),
thenlet t2
= s2. For rt2
letwhere
u1( t2)
is thepoint
of intersection of y = T withLet
T(r)
= Tfor r1 ~
r ui. It isalways possible
to chooser2 so
large that rl
ul. Werepeat
theprocedure
and note thatand
T(r)
=S(r)
for r = t1, t2,t3,....
Henceand
REMARK: It is
possible
to have a(smaller)
class of functionsT(r) satisfying
the conditions(2.1)
to(2.4)
and the relationThe
only change required
inproving
the existence of such functions is to take curves of the forminstead
of y
=A ::f:l2T
in our construction forT(r).
3
Let
f(z)
be anintegral
function of orderp(0
poo),
finitetype
T and lowertype
t. We consider the class of functionst(r) satisfying
thefollowing
conditions:(3.1) t(r)
is anon-negative
continuous function of r for r > ro,(3.2) t(r)
is differentiable for r > roexcept
at isolatedpoints
at which
t’(r-0)
andt’(r+0) exist,
These functions are defined in the same way,
except
for(3.4)
and
(3.5),
as theproximate type
definedin §
2. We callt(r)
alower
proximate type
for the functionf(z).
The existence of such functions can beproved
in the same way asproved
forT(r)
andso we omit the
proof.
4
In this section we construct a
proximate type
for a class ofentire functions. It is known
[3,
p.27]
thatwhere
W(r)
is apositive, indefinitely increasing
function. Hencedifferentiating
weget M"(r)IM(r)
=W(r)lr
whereM’(r)
is thederivative of
M(r)
which exists for almost all values of r.PROOF. For any e > 0 and r
> rô
=r’0(03B5),
we have from(4.2)
So,
for r > max(ro, r’0),
we haveIntegrating
the aboveinequalities
between suitable limits andthen
dividing by
rP andproceeding
to limits weget
the resultin
(4.3).
We are now in a
position
to prove thefollowing:
THEOREM 2. Let
f(z)
be anintegral function o f
order p(0
p~)
and
type
T(0
T~)
and letM(r)
=max|z|=r If(z)1
andW(r)
be
given by (4.1). Il limr~~ (W(r»IrP
exists then(log M(r))/r03C1
is a
proximate type of f(z).
12
PROOF. Let
Since
log M(r)
is areal, continuous, increasing
functionof r,
which is differentiable in
adjacent intervals,
it follows thatT(r)
satisfies
(2.1).
Sincelimr~~r-03C1W(r) exists, (4.3)
shows thatlimr~~
r-03C1log M(r)
also exists and soT(r)
~ T as r ~ 00. FurtherT(r)
ispiecewise
differentiable and it hasright
and left handderivatives where
they
aredifferent,
soor,
Thus, T(r)
satisfies the condition(2.3)
also.Finally
follows from
(4.4).
Hence the theorem is established.REFERENCES
CARTWRIGHT, M. L.
[1] Integral Functions, Cambridge Tract, C.U.P., 1956.
SHAH, S. M.
[2] On proximate orders of integral functions. Bull. Amer. Math. Soc. 52, (1946),
3262014328.
VALIRON, G.
[3] Lectures on the general theory of integral functions. Chel. Publ. Co. New York, (1949).
(Oblatum 24-10-65)