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A NNALES DE L ’ INSTITUT F OURIER

S. K. B AJPAI

On the mean values of an entire function represented by Dirichlet series

Annales de l’institut Fourier, tome 21, no2 (1971), p. 31-34

<http://www.numdam.org/item?id=AIF_1971__21_2_31_0>

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21, 2 (1971), 31-34.

ON THE MEAN VALUES OF AN ENTIRE FUNCTION REPRESENTED BY DIRICHLET SERIES

by S. K. BAJPAI

1. Let f (s) be an entire function of the complex variable s == <T + it defined everywhere in the complex plane by absolutely convergent Dirichlet series

(1.1) f{s}= ia^»

n==o

where 0 ^ ^ < \ < - • • < \ —>• oo as n —> oo.

As usual, the symbols M(o, /*), ^(o-, f) and v((y, /*) denote the maximum modulus, the maximum term and the rank of the maximum term respectively for f{s)^ and can be found in [8]. We define

A ^ ) = l i m — F [f{a+itrdt

T-^oo 2i 1 J _^

and

W,.5.,((r) = lim ^ F r \f(x + ^|^ ^ ^; 8 > 0.

T»00 ^ 1 J y J___^

The mean values for k == 2 and y == 0 were defined by Hadamard [2] and also by Kamthan ([3], [4]) who has obtai- ned the following main results for k == 2 and y == 0

(1.2) lim ^P log log Afc(g) == p

<r-^oo inf (T X (1.3) lim sup ^g^g^.s.^) = P

(T-Xao inf O' X

June] a [6] also proved (1.3) by using a different technique for k == 2 and y == 0. In [7] another extension of (1.3)

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32 S. K. BAJPAI

has been obtained for k = 2 and y == — oo. Kamthan [5]

has further attempted to establish (1.2) for every k > 0, but his arguments of the proof are vague. Thus, in the present paper, we restrict ourselves to functions of finite Ritt-order p, establish (1.2) and extend (1.3) for every k > 0 by a diffe- rent technique.

2. We have

THEOREM 1. — If f{s), defined by Dirichlet series (1.1), be an entire function of finite Rift-order p, lower order X type T and lower type t, then, for 0 < k < oo

p - lim snp Iog log Afc(cr) - Hm sup log log w^^

X <T->oo inf a 0^.00 inf a and

kT = lim snp log Afc((Y) = lim sup logw^^

kt <r»oo inf e^ <r^oo inf e^

First we prove the following:

LEMMA. — Let f{s), given by (1.1), be an entire function of finite Ritt-order p, then, for 0 < k < oo

log A,{a) - log W^,(a) ~ k log M(a, /•).

Proof. — It is well known

(2.1) | a J ^ = ' /1lim — f^ /•(o + ^)e- "T 1^ ^

T->oo Z l J _ T

First, let k ^ 1, then by Holder's inequality, we have

(2.2)

where

r 1 /'IT ^l/*

a^» ^ CJ lim — ( (/-(o + K)|* ^|

LT^-« ^l j_j, j

/ 1 if k == 1

C.= 4

/ 1 A- N

r (-L + "

_ V 2 _ _ 2 ( ^ - 1 ) ,

L^O^;

fc—1

'' if/c > 1.

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Further, from (2.2) we have

{H^ ^ CUim— F \f(x+ it^dt

T^oo 2 1 J _ T

Hence, for y ^ 0

/o 3^ {Kl^ ^|aJ^^ ^ ^ . ^ Q ,

(

2

-

0

) ^ + 8

-

to^+s

<

^^.s.^

0

) ^ y

M

(^ /)

The case when 0 < k < 1 may be treated as following:

( \ /»T ^ fc+l

(2.4) |^|^ < C^ lim^ ( \f(a + ^l^^

(T^oo Z 1 J _ T )

^ C^{M((T,/-)A,(<T)|^

Also, from (2.1), for 0 < A- < 1, we have /g ^ {loj^-'}^1 _ e-8J^a„|*+lfc+l^+8^

v"07 ( / c + 1 ^ + 8 ( / c + 1 ) ^ + 8

< C^lim^ r r ^5|A^+ it)\^dtdx

T*o» ^1 J o ^-T

< C^M(o, /•)W,,6,,((r) < -^C%M

k

+

l

((T,/•)

Since /"(^ is of finite Ritt-order p, it follows that ([I],

p. 719), ([9], p. 265)

(2.6) log(x(<T,/-) ~logM(<r,/-)

and log log (i(<r, /") ~ log ^^

Hence (2.2), (2.3), (2.4), (2.5) and (2.6) together imply the lemma. Proof of theorem 1, follows immediately from the above lemma.

THEOREM 2. — If f(s) is an entire function of finite Ritt- order p and is defined by (1.1) then

lim ^P log{W..5.,(<r,f)/W,,5.,(a,/-)} ^ kp (T^oo inf a /cX provided k is an even integer.

For k = 2, this result is due to Juneja [6] and for k even it follows with the use of Minkowski's inequality and the

2

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34 S. K. BAJPAI

fact that W^§ ^((T) is an increasing convex function of (T for k even.

In the end author wishes to express his sincere thanks to Professor R. S. L. Srivastava for his kind guidance. The author is also thankful to the referee for his suggestions.

REFERENCES

[1] A. G. AZPEITIA, On the maximum modulus and the maximum term of an entire Dirichlet series, Proc. Amer. Math. Soc., 12 (1961), 717-721.

[2] E. HILLE, Analytic function theory volume II; Blaisdell Pub. Comp.

1962.

[3] P. K. KAMTHAN, On the mean values of entire functions represented by Dirichlet series; Acta. Math. Acad. Sci. Hung., 15 (1964), 133-136.

[4] P. K. KAMTHAN, On the mean values of entire function represented by Dirichlet series, ibidem, 16 (1965), 89-92.

[5] P. K. KAMTHAN, On entire functions represented by Dirichlet series (IV);

Ann. Inst. Fourier, Grenoble 16 (1966), 209-223.

[6] 0. P. JUNEJA, On the mean values of an entire function and its deriva- tives represented by Dirichlet series, Ann. Polon. Math., 18 (1966), 307-313.

[7] 0. P. JUNEJA and K. N. AWASTHI, On the mean values of an entire function and its derivatives represented by Dirichlet series II, Ann.

Polon. Math., 22 (1969), 89-96.

[8] S. MANDELBBOJT, Dirichlet series, Rice Inst. Pamph, 31, 4 (1944).

[9] K. SUGIMURA, Ubertragung einiger Satze aus der Theorie der ganzen Functionen auf Dirichletsche Reihen, Math. Z., 29 (1928-1929), 264- 277.

Manuscrit recu Ie 24 avril 1970.

S. K. BAJPAI, Department of Mathematics, Indian Institute of Technology,

Kanpur 16, India.

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