J oî oî

ANNALES SOCIETATIS MATHEMATICAL POLONAE Series I: COMMENTATIONES MATHEMATICAE X V II (1973) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE X V II (1973)

Z. J

and J. K

(Szczecin)

O n orders oî growth oî the entire functions represented b y the Dirichlet series

Let us assume that the Dirichlet series

00

(!) f{s) = ^ a nes*n (s = a + it;

< l x < Л

< ...) ft— i

represents an entire function, i.e. is absolutely convergent in the whole plane. Then, for c > 0 the following functions:

Ж (a) = max \f{o + it)\,

— oo<t<oo

ju{a) — max \an\e*

1

( 2 )

C{<f) = X 1^1

T

/

Ip(a) = lim J \f(a + it)\pdtj (

< p < oo) are well defined.

Using these functions one can define the orders of growth of a function / in the following manner:

( 3 )

q

= lim sup

q

= lim sup

Л = lim sup

qp

= lim sup

loglogilf(o')

log log /л {a)

log log

(d)

loglogJp(d)

(Eitt [3]),

(Sugimura [4]),

[2 ],

[ 1 ] ,

7 — R o e z n i k i P T M — P r a c e M a t e m a t y c z n e X V I I

In the present paper we shall generalize certain results from [1] and [2].

Namely we shall introduce the function:

№ < W = ( j ^ l a S e ^ Y ”,

n= I

and, similarly as in [1] the functions:

1 0

= — — J K(x)P(x)dx,

(5) a

Np{a) = exp JK(°c)togP(x)dxJ,

which are more general than those investigated in [1], since P(x) can be With the

„ _. logK(x)

any of functions (2) and, moreover, limsup --- < oo.

formulae analogous to (3) we shall define orders of growth and shall study their connections with orders (3).

T

1. Let us denote jun = — log\an\jkn (if an = 0, then цп — oo)- For every entire function f defined by formula (1) series (4) representing the function Jp(a) is convergent for 1 < p < oo. I f there exists a constant

C O

00

L such that (a) У e~Ll*n < oo or a constant M such that (b) У е~Шп < oo,

fi—

then formula (4) represents the function Jp(a) for 0 < p < oo.

P roof. First assertion is obvious. Indeed, we have, for 1 00

У (\an\peapXn) < G (pa) for sufficiently large N, since liman = 0. Put-

n = N

ting An = loglogn and an = n “ (a > 1) we see that the series У ane<sXn

oo n —1

= У n~a(logn)a is convergent for every <r, while series (4) is divergent for p < 1/a. From the definition of the sequence {yn} we have:

00

(*) У \an\eaXn — У е аХп~^пХп — У е1п^а~^п\

П=1

It follows then that the condition lim pn = oo is necessary for the con-

f l —X X )

vergence of series (*) for every or.

Taking into account this condition we obtain

F K l'e ’”*'

n = N

ep*n(a-t*n) < eP*Na. e-P*N(*n < oo

n = N

n=N

Orders of growth of entire functions

when (a) holds. On the other hand

OO 0 0

JT* \an\pepaXn < ^ e~MXn since y n — a > Mfp for n > N .

n = N n= 1

B e mark. Similarly as in XJogn

(a') limsup

[

2

implies (a)

we demonstrate that the condition while (b') logw

_____ x„, ___ x._ , limsup

--- -logtf^ n-xx)

is equivalent to (b). Hence, the function Jp(a) is defined by series (4) for

< p <

when one of conditions (a') or (V) is satisfied.

T im OREM 2. For every p

0 and for every a > 0

p >

=> Jp (<y) ^ C ( o ) , p <

=> Jp (a) > (7(cr),

> /*(<*)•

Proof.

( K \ e a**)p]llP> [max(|un|ea^)p]1/iJ = /л{a);

J

I 1 r \ 1,p

I— J \f {a + it)\p dt\ < M(o). For p = 1 we have Jp{o) = C(o) for p =

, from the Parseval’s identity it follows that J 2 (o) = I A a) < M(o) < (7(cr).

For arbitrary p >

we obtain the desired inequalities putting qk

= {\ак\еа*к)р‘, a — 1 /p into the inequalities:

> 2 <ll for « >

( i » ° ( & >

, * =

,

, ...)•

< 2

' Я .1 for

< a <

fc=i

Now, from the Hadamard formula for the coefficients of the Dirichlet series :

T

an = lim ^ f f{(f + it)e{~a~ü)indt

1

T

we have that \ап\еаЛп < lim —— f \f(a + it)\ dt = I i(d) for n =

,

, ...

2 ’—x x ) 2T _ у

Taking supremum on the left-hand side of the latter inequality we obtain

А*И < 1 г(о).

T

3. For arbitrary real a > 0, > 0, «5 > 0, e > 0 satisfying condition ôs = a(3, the following inequalities are true:

(a) J ™ î | < I Ï +. - J j +4 . U 2 = l ) , (b)

Proof. Prom Holder’s inequlity

T T T

(*) j |/(<r + «)|“+', dt < ( J |/(<r+ й)Г<й)1/3’- ( / |/(<Н-й)|*<в)1Л',

_2> - У - У

where l Jp- \ - l j q = 1. Dividing by 2T = (2T)llp(2T)l/q we obtain in the limit as T tends to oo;

/* \ t«+P < Ta

V*/ 5=51 -*-«Ф J-PQ’

Denoting now ap = a + e, pq = ft + à and making use of the condition 1/p + l j q = 1 we prove (a). For the function Jp we have:

O O oo

n=l n=l

that is (b) holds.

T

4. Pe£ ws assume that:

OO

(a) f(a) = £ anea*n and an Ф 0 for infinitely many n’s,

71=1

(b) P(x) — J p (x) or P(x) is any of functions (2), (c) m*P and N p are defined by formulae (6),

(d) K( x) is an arbitrary non-negative non-deer easing function such logK(x)

that lim sup ---< oo.

x~xx> X

Then, for any non-negative number <5:,

(a7) logP(<r- à) { Ô-~ < l o g g e r ) < fflogP(a) K{ o)

and

(b#) £logP(<r — (5) < \ogm*P {o) < 21ogP(cr) {a— ô > <r0; p > 1).

Proof. From (b) and (d) we have immediately:

«7 a

logP(<r-<5)d£(<r-<5) < J l ogP( x) K( x) dx ^ J \ ogP{ x) K( x) dx

o-ô 0

^ ol ogP (a) К (a).

Orders of growth of entire functions

Dividing by K{a) we get (a). Similarly,

logP(<r-a) 1 log Ô + log К {(г - ô) - lo gK(a) I logP(<r-<5) J

< log m,p (a) < logP(<y) | l b g g ~|

l0gP(<7)J In view of (a) we have, for every k, P ( a — <5) > Ceka for a > cr 0 (k) bnt logPT(cr) < A c hence the term in square brackets tends to 1 when o' -> oo.

Thus for a > o

(b') holds.

Th e o e e m 5.

For every p >

log Jp (a) is a convex function.

o o

Proof. Let us denote f(cr) = \an\pepaXn. It suffices to show that

n = 1

/ ! ( X r S ~ /K)/(<7a) ^

for any non-negative numbers <r1) cr2. De­

signating the left-hand side of this inequality by L and denoting pax = xx;

p a 2 = x2; \an\p — an we have:

oo oo oo

L = ( JT aneAn(æi+x

)/

J

— ( jj? anex^ ( anex^

7 1 = 1 7 1 = 1 7 1 = 1

— S

a a e(*n+*m)(Xi+x

— V

a a eXl*n*X2Am = V a a a

— / t Un Um V U"m U'n ° x t

m ,n m ,n m t n

2 а П а т п

(

“H

where amn = ^ n+'*m^a:i+a;2^2— ^Рл+яг*»».

For fixed n and m let us further denote ххЛп + х 2 Лт = и; ххЛт + х 2 Лп (

eu -±-ev \

e (u + v) / 2

---— j o since

is

convex func­

tion. Now, an > 0 and am ^ 0 imply L < 0.

C

. 1. ( 0 < g < p ) => Д [I 3(or) < 7^(сг), 7 в(<7)> Inde- aX)

ed, in view of Theorem 3 (a) (i = б = 0 yield the first inequality. From Theorems

and 3 (b) we have Jl%p{cr) ^ Ja{a)m J i { 0 )) hence the other inequality.

2

. Let us denote

* = limsup

G—►OO

loglogjpjo)

a and let

, £*, {?**,

be defined by formulae (3). Then 0 < p < q => (e3< Qp, Q *q^Qp)i A Qp^' Q* y Qp

p> о

< Qi p > 1 => {

I < Q**, Qp > Q*)’, p < 1 => Qp > Q**- All these inequalities

follow immediately from Corollary 1 and Theorem 2.

3. If \ogfjb{a) ~ \ogC{a) and p ^ 1, then logJp {a) ~ log^(cr). If loggia) ~ l o gM(a), then for p > 0, \ogIp {a) ~\ ogM( a) . First of these asymptotic equalities follows easily from Theorem 2, since for p > 1 fi(cr) < J p{<y) < 0(or). The latter results from Theorems 3 and 2. Indeed, putting in (a) of Theorem 3 a-b/9 = 1 and by yirtue of 2 we get: ^(cr)

< Il И 1а+б(0')-^1 а(0')* bogarithming this, we obtain in view of 2:

г > logIq+e(g) > ± 1 Г log/* И Л

^ logJf(o-) "" a [logif(o-) J ’ Thus the proof is completed.

4. If (*) limsup

П—>oo

lo gn

<

then, for p > 0,

holdà, then by Theorem 1 functions Jp(a) are defined by (4), exist for p > 0 and

* is also defined. Moreover, =

= . From Theorems 2 and 3 it follows that for h > 0 :

\an\pep{a+h)X*e~pMn yp^ ц(а + h ) ( ^ e - pMn)llp= A/л {a + h) .

n=1 n=l

Clearly for every h > 0

log log(cr) loglog((T + h) lim sup--- = lim sup---

a—xx> @ o—>oo O’

from what it follows that e* = —

. (It is well known ([5], p. 73) that log 71

lim sup---< oo implies \ogfi{o) ~ log-Щаг). Thus, by Corollary 3, log/Ln

log Ip ~ lo gM for p > 0, hence

= g.)

5. Let P(<r) = Jp{<y) or P(<r) — Ip{a). Then:

a

(+•) logP(or) = 1 OgP(o'0) ~h j '<pP(x)dx, ao

where <pP{x) is a non-decreasing function. Moreover,

/+4 .. loglogP(o') log

?p(<r)

(X) Q

= lim sup--- = lim sup---

G— >O0 О (j— xx) a

O O C O

- i n f b > 0 : / < со) - i n f (л > 0 : / < ос}.

(+) follows from Theorem 5 and ([1]), p. 92), while (J) is a con­

sequence of Lemma 4 in [2].

6. Let the functions mp(cr) and Лр(о) be defined by (5) and satisfy

the assertions of Theorem 4,

be as in Corollary 5 the order of growth

Orders of growth of entire functions

assigned to the appropriate function P(cr). Then from Theorem 4 we have immediately

lim sup

о — X X )

loglogmp(cr) a

loglogNp(a) lim sup---

a— юо &

References

[1] J a in , P a w a n K u m a r , Growth o f the m ean values o f an entire fu n c tio n repre­

sented by D irichlet series, Math. Nachr. 44 (1970), p. 91-97.

[2] J . K o p e é , Inverse monotone fu n c tio n s and their applications to the theory o f entire functio n s, Prace Mat. 10 (1967), p. 175-187.

[3] J. F . R i t t , On certain p o in ts in the theory o f Dirichlet series, Amer. J. Math. 50 (1928), p. 73-86.

[4] K. S u g im u r a , Übertragung einiger Sàtze aus der Theorie der ganzen FunJctionen a u f Dirichletsche Reihen, Math. Z. 29 (1928-29), p. 264-277.

[5] Y u C h ia - Y u n g , S u r les droites de Borel de certaines fonctions entières, Ann. Sci.

Ecole Norm. Sup. (3) 68 (1961), p. 65-104.

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