On the lower order <span class="mathjax-formula">$(R)$</span> of an entire Dirichlet series

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A NNALES DE L ’ ^INSTITUT F ^OURIER

P. K. J AIN

D. R. J AIN

On the lower order (R) of an entire Dirichlet series

Annales de l’institut Fourier, tome 24, n^o1 (1974), p. 123-129

<http://www.numdam.org/item?id=AIF_1974__24_1_123_0>

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24,1 (1974), 123-129

ON THE LOWER ORDER (R) OF AN ENTIRE DIRICHLET SERIES

by P.K. JAIN (¹) and D.R. JAIN

1. Introduction.

For an entire Dirichlet series

As) == i, ^\ (5 = a 4- it, \ > 0, \ -> oo with n) (1.1)

n = 1

the lower order (R) X is defined as :

lim i n fl o g l o g M ( q )= X , ( 0 < X < o ° ) , (1.2)

a —> 00 (J

where M(a) = sup { | /(a 4- it) \ : — °° < t < oo}.

Improving upon a result of Rahman [6], Juneja and Singh [4]

have, very recently, proved the following theorem :

THEOREM A. — Let f(s) be an entire Dirichlet series given by (1.1) of lower order (R)X (0 < X < oo). Then

\ log\,_i

lim inf—————— < X . (1.3)

" -> °° log | ^ | Further, if

log n

lim sup ——— < °° , (1.4)

n-^w \

and

(1) The research work of this author has been supported partially by the University Grants Commission (India).

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124 P.K. JAIN AND D.R. JAIN

_ log K - i / ^ l

^⁼ ^————^———^{( L 5 )}

\i - \-l

forms a non-decreasing function of n for n > HQ , then X logX ^

lim inf-f———n—— = X . (1.6)

"^°° log |^|

In this paper, we obtain the estimations for the lower order (R)X in terms of the sequences { \,} a n d { ^ } which hold for every entire Dirichlet series, and one of our estimations includes (1.6) and the result of Rahman [6] as the special cases. In fact, we prove :

00 „ Tk

THEOREM. — Let f(s) = V a^e n be an entire Dirichlet series n=i

given by (1.1) of lower order (R)X (0 < X < °°) such that (1.4) is satisfied. Then

x.l og \ l , ,

X == max Urn inf—p————— (1.7)

{S> p^0 0 log I ^p I

(x^ - ^ n - l ^ ^ n - l

X == max lim inf —^p———^p———————— (1.8)

{\j p-^

⁹⁰

^'S-i/S'

2. Preliminary Discussions.

Let jLi(o) denote the maximum term of f(s) for Re(^) = o and

\(a\ = max{X^ : jLi(a) = \a^\eJ n}. Let { p ^ } be the sequence of jump points of \(Q) (points of discontinuity of \,^), every jump

point is listed with multiplicity equal to size of the jump, such that

Pi < Pi ^ ?3 ^ • • • Pn ^ " ' • since \(a) ~" °° as ° ~' 00' Pn ^ 00

as n -> oo. We denote b y { X ^ } the range of \/^, so that \(y\ = ^ for a = ?„ . Then' " f c

f) ° < P ^ < P ^ + i =^ ^ + 2 = • • • = Pn^i9 k= 1'2'3- - ' ii) X^^ = X^, when p^^ < a < p^^ , /; = 1 , 2 , 3 , . . . .

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These arguments are analogue to those for entire power series, given and used by Gray and Shah in their works [1,2,3].

<j\^ a^n

Since \a^ \e k-l and | a^ \ e k are the two consecutive maximum terms, we have

p-, = l o g | ^ A^ |/(X, - X , ) . (2.1)

^fc^{& 1} ^k-lⁿkⁿkⁿk-l

Also, we need the following : LEMMA

logx^ lim inf. ———— = X . fe-^oo p

• " f c + i Proof. - Since ([5], Theorem B)

log \^

lim inf ———— = X, (0 < X < oo) ,

a —>•<» OT

there exists a sequence { x ^ } , x^ -^{> 00} with / such that

^^(x,)

lim ————— = X . i -> °° Xf

It is always possible to find a subsequence {?„ } o f { p ^ } which satisfies the inequalities :

P ^ . < ^ , < P ^ . ^ > ' - 1 , 2 . 3 . . . . In either of the cases, for / > /o = ;o(e), e > 0, we have

!

^og

^<!

^ss ^a A ~r e ,^ogx

^<,^

p., x;

'nk^\ l

which implies

lim inf ———- < X .^^i.

k->°° P-,

" f c + l

The reverse inequality is obvious. Hence the lemma is proved.

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3. Proof of the Theorem.

Using (2.1) in the lemma, we. get

, ,. . , ( ^ - ^ ,)log^ ,

A = hm inf —-———AL-J————fc—1 (3.1) fc->- log | a la^

^ k - l ' ^ k1

We have proved (3.1) for a particular subsequence {\^ } which is the range of the rank function \,^. Thus the theorem will be proved completely if we establish, for any arbitrary subsequence (say){\, } o f { X ^ } , the following inequalities :

\ log X^

X > lim inf. —p———p—1 > lim inf. x (3 2) p-^-oo log \a^ | ' p-^oo

p

( \ - \ _ . ) l o g ^ l o g l a ^ ^ / a ^ l 7'roo/ o/ the first inequality in (3.2) : Let

\,log\,_, Um inf —-———•— = a P ->•" log | a^ | 1

Assume that a. > 0, for otherwise the result is trivially true. Therefore, for any e > 0, 3 a N = N(e) such that

IS^^P'JI ' (P^N) .

Let. e0" = 2X^-_^ , p = 1 , 2 , 3 ,. . . . So if Op ^ a < °p-n > we have log M(o) > log | a^ | + Op \,

> log | a^ | + Op X,

>\. log 2

»p ^""p

= .( a-£ )^+' l o g 2 / 2a-£,

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i.e.

log log M (or) > (a - e) a^+i + log log 2 - (a - e) log 2 . which gives

. log log M (a) A = hm inf. —————— > a .

a -> <» 0

Proof of the Second Inequality in ( 3 . 2 ) : Let

(\, - V i)¹⁰^,, , Urn inf —————p-1 . p-1 = P . p->- l o g | ^ _ ^ / ^ )

Again, without any loss of generality, assume j3 > 0, so that S"S-i

'S-i/S^S-^

^?

for p ^ PQ = PQ (e), e > 0. This implies

'PO PU ^o-^1 'P-l

' P O - ^1 I I PQ^2

\^ -\^

nm nm-\

< n \ ^

m=pQ+l nm-\

1 P

———— V (\, "^ )logX.,

^ log|^ |-1 < 0(1) + _{^ _ p ^}

v "w " w - l 0 nm-\

P^e w = p o + l

log I a, 1-1p < o ( l ) + 1

' s. ¹⁰⁸ ^

^ - e ^ - e X ^ l o g X ^ _ ^

\i ^g^' " p ^ p - l

P - I

^ -e) Z ^ ( ^ ^ - l o g X , )_{w -1} W = P O + I

s^s-i

log I ^ ^1-1 lim sup ————^e-

P-"" Xn lOgX

•"p-⁶-^-!

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Hence

\ ^g ^ On - ^n ) ^g ^ np "p-1 "p "p-1 np-l

lim inf. ——————:T"'~ > lim inf. -——————————————— ? p-^- log |^| p-^- l o g | a ^ _ ^ / f l ^

Remark. — If, in addition to the hypothesis of our theorem, (1.5) is satisfied, then our result (1.7) reduces to (1.6). Further, if logX^.^ ~ log X ^ , as n -> °°, the result of Rahman [6] is also obtained.

Justification. — Since (1.5) is satisfied, each term of f(s) is a maximum term and so X^ = X ^ , for A: = 1 , 2 , . . . . Therefore, the result (3.1) reduces to

(X^ - \-i) Icg^-i

X = lim inf. —————-——— • (3.3) fc-^oo l o g | ^ _ i / ^ |

Further, as the result (3.2) is true for every subsequence {\ } of {X^}, it is also true for the sequence {\J, since {\} may be regarded as a subsequence of{\,}. Hence

X ^ l o g X ^ _ i (\, - X ^ _ i ) l o g X ^ _ i

"\ "^ 1 • . / • » ' » • • i •^^ i . • f ' f t A / •^ A \

\ > lim mf —————rr ^ I1111 ln^ ————————;———(3.4) M->⁰⁰ log |^| "-»⁰⁰ l o g | ^ _ i / ^ |

Thus, the result (1.6) follows from (3.3) and (3.4).

Furthermore, if log \+i ~ log X^ , as ^2 -^ o°, then (1.6) implies that

, . . \^\-i , . . ^ z1 0^ ^ A = lim inf. ——————r = llm ml —————"i i->00 log |^| n->- log |^|

which is a result of Rahman [6].

BIBLIOGRAPHY

[1] Alfred GRAY and S.M. SHAH, Asymptotic values of Holomorphic Functions of irregular growth, Bull. Amer. Math. Soc., 5, 71 (1965), 747-749.

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[2] Alfred GRAY and S.M. SHAH, Holomorphic Functions with Gap Power Series, Math. Zeit, 86 (1965), 375-394.

[3] Alfred GRAY and S.M. SHAH, Holomorphic Functions with Gap Power Series, (II), Math. Anal. and Appl. 2, 13, (1966).

[4] O.P. JUNEJA and Prem SINGH, On the Lower order of an entire function defined by Dirichlet series, Math. Ann. 184 (1969), 25-29,

[5] P.K. KAMTHAN, On entire functions represented by Dirichlet series (IV), Ann. Inst. Fourier, Grenoble, 16, 2 (1966), 209-223.

[6] Q.I. RAHMAN, On the lower order of entire functions defined by Dirichlet series. Quart. J. Math. Oxford (2), 7 (1956), 96-99.

Manuscrit re^u Ie 25 janvier 1973 accepte par M. Brelot.

D.R. JAIN. P.K. JAIN,

Department of Mathematics, Department of Mathematics, A.R.S.D. College, Hans Raj College,

Delhi University, NEW-DELHl-23, India Delhi University, DELHI-7, India.