Bajpai, S. K.

Numérisation de Documents Anciens Mathématiques

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

Annales de l’institut Fourier (AIF)

ANNALES DE L’INSTITUT FOURIERS. K. BAJPAIOn the mean values of an entire functionrepresented by Dirichlet seriesAnnales de l’institut Fourier, tome 21, no 2 (1971), p. 31-34<http://www.numdam.org/item?id=AIF_1971__21_2_31_0>© Annales de l’institut Fourier, 1971, tous droits réservés.L’accès aux archives de la revue « Annales de l’institut Fourier »(http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé-nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa-tion commerciale ou impression systématique est constitutive d’une in-fraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/Ann. Inst. Fourier, Grenoble21, 2 (1971), 31-34.ON THE MEAN VALUES OF AN ENTIRE FUNCTIONREPRESENTED BY DIRICHLET SERIESby S. K. BAJPAI1. Let f (s) be an entire function of the complex variables == <T + it defined everywhere in the complex plane byabsolutely convergent Dirichlet series(1.1) f{s}= ia^»n==owhere 0 ^ ̂  < \ < - • • < \ —>• oo as n —> oo.As usual, the symbols M(o, /*), ^(o-, f) and v((y, /*) denotethe maximum modulus, the maximum term and the rankof the maximum term respectively for f{s)^ and can be foundin [8]. We defineA^)=l im— F [f{a+itrdtT-^oo 2i 1 J _^andW,.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 byHadamard [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' XJune] a [6] also proved (1.3) by using a different techniquefor k == 2 and y == 0. In [7] another extension of (1.3)32 S. K. BAJPAIhas 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 presentpaper, 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 haveTHEOREM 1. — If f{s), defined by Dirichlet series (1.1),be an entire function of finite Rift-order p, lower order Xtype T and lower type t, then, for 0 < k < oop - lim snp Iog log Afc(cr) - Hm sup log log w^^X <T->oo inf a 0^.00 inf aandkT = 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 offinite Ritt-order p, then, for 0 < k < oolog A,{a) - log W^,(a) ~ k log M(a, /•).Proof. — It is well known(2.1) |aJ^=' /1 "Tlim — f^ /•(o + ^)e-1^ ̂T->oo Zl J_TFirst, let k ^ 1, then by Holder's inequality, we have(2.2)wherer 1 /'IT ^l/*a^» ^ CJ lim — ( (/-(o + K)|* ̂ |LT^-« ^l j_j, j/ 1 if k == 1C.= 4/ 1 A- Nr (-L + "_V2__2(^-1) ,L^O^;fc—1'' i f /c > 1.ON THE MEAN VALUES OF AN ENTIRE FUNCTION 33Further, from (2.2) we have{H^ ^ CUim— F \f(x+ it^dtT^oo 21 J_THence, for y ^ 0/o 3^ {Kl̂  |̂aJ^^ ^ ̂  . ̂  Q ,(2-0) ^+8 - to^+s < ^^.s.^0) ^ yM (^ /)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„|*+le«fc+l^+8^v"07 ( /c+1^+8 ( /c+1)^+8< C^lim^ r r ^5|A^+ it)\^dtdxT*o» ^1 Jo ^-T< C^M(o, /•)W,,6,,((r) < -^C%Mk+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 thelemma. Proof of theorem 1, follows immediately from theabove lemma.THEOREM 2. — If f(s) is an entire function of finite Ritt-order p and is defined by (1.1) thenlim ^P log{W..5.,(<r,f)/W,,5.,(a,/-)} ̂  kp(T^oo inf a /cXprovided k is an even integer.For k = 2, this result is due to Juneja [6] and for k evenit follows with the use of Minkowski's inequality and the234 S. K. BAJPAIfact that W^§ ̂ ((T) is an increasing convex function of (Tfor k even.In the end author wishes to express his sincere thanks toProfessor R. S. L. Srivastava for his kind guidance. Theauthor is also thankful to the referee for his suggestions.REFERENCES[1] A. G. AZPEITIA, On the maximum modulus and the maximum term ofan 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 byDirichlet series; Acta. Math. Acad. Sci. Hung., 15 (1964), 133-136.[4] P. K. KAMTHAN, On the mean values of entire function represented byDirichlet 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 entirefunction 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 ganzenFunctionen 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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On the mean values of an entire function represented by Dirichlet series

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