М. I. Parolya, М. М. Sheremeta ESTIMATES FROM BELOW FOR CHARACTERISTIC FUNCTIONS OF PROBABILITY LAWS

Transcript

1 Математичнi Студiї. Т.39, Matematychni Studii. V.39, No. УДК М. I. Paolya, М. М. Sheemeta ESTIMATES FROM BELOW FOR CHARACTERISTIC FUNCTIONS OF PROBABILITY LAWS M. I. Paolya, M. M. Sheemeta. Estimates fom below fo chaacteistic functions of pobability laws, Mat. Stud , Let ϕ be the chaacteistic function of a pobability law F that is analytic in D R {z : z < R}, 0 < R +, M, ϕ ma { ϕz : z < R} and W F F +F, 0. A connection between the gowth of M, ϕ and the decease it of W F is investigated in tems of estimates fom below. Fo entie chaacteistic functions it is poved, fo eample, that if ln λ ln ln W F fo some inceasing sequence such that + O,, then ln ln M,ϕ + oλ ln as +. М. И. Пароля, М. М. Шеремета. Оценки снизу характеристических функций вероятностних законов // Мат. Студiї Т.39,. C Пусть ϕ аналитическая в D R {z : z < R}, 0 < R +, характеристическая функция вероятностного закона F, M, ϕ ma { ϕz : z < R} и W F F + F, 0. В терминах оценок снизу изучена связь между ростом M, ϕ и убыванием W F. Например, для целых характеристических функций доказано, что если ln λ ln ln W F для некоторой возрастающей последовательности такой, что + O,, то ln ln M,ϕ + oλ ln при +.. Intoduction. A non-deceasing function F continuous on the left on, is said [, p. 0] to be a pobability law, if lim F 0 and lim F, and the function + + ϕz eiz df defined fo eal z is called [, p. 2] a chaacteistic function of this law. If ϕ has an analytic continuation on the dis D R {z : z < R}, 0 < R +, then we call ϕ an analytic in D R chaacteistic function of the law F. Futhe we always assume that D R is the maimal dis of the analyticity of ϕ. It is nown that ϕ is an analytic in D R chaacteistic function of the pobability law F if and only if fo evey [0, R W F : F + F O e, +. Hence it follows that lim + ln W F R. If we put M, ϕ ma { ϕz : z } fo < R then W F e 2M, ϕ fo all 0 and [0, R. Fo R + this inequality is poved in [, p.54] and fo R < + the poof is analogous. Theefoe, if we define see also [2] µ, ϕ sup {W F e : 0} 200 Mathematics Subject Classification: 30D99, 60E0. Keywods: chaacteistic function, pobability law, lowe estimate. c М. I. Paolya, М. М. Sheemeta, 203

2 ESTIMATES FROM BELOW FOR CHARACTERISTIC FUNCTIONS 55 then µ, ϕ 2M, ϕ. Thus, the estimates fom below fo ln M, ϕ follow fom such estimates fo ln µ, ϕ. Fo entie chaacteistic functions N. I. Jaovleva [3] poved that, ln ln/w if lim F + ln ln M,ϕ then lim + λ. Hence it follows that if + ln λ ln + ln λ ln ln, 0, W F then ln M, ϕ ln + oλ ln, +. 2 The question aises, whethe asymptotical inequality 2 is valid if condition holds not necessaily fo all 0 but only fo some inceasing unbounded sequence. In view of the inequality ln M, ϕ ln µ, ϕ ln 2, the following assetion gives a positive answe to this question. Poposition. If thee eists an inceasing to + sequence such that ln λ ln ln W F fo all and ln µ,ϕ + O as then ln +oλ ln as +. We obtain Poposition fom main esults poved below fo chaacteistic functions that ae entie o analytic in the dis chaacteistic functions. 2. Auiliay esults. If ϕ is an entie chaacteistic function and ϕ const then [, p. 45] lim + ln M, ϕ σ 0, + ]. If σ < + then the estimate ln M, ϕ fom below is tivial. It can be shown that σ < + povided W F 0 fo all 0. Theefoe we assume in what follows that W F 0 fo all 0 and, thus, W F 0 +. ln µ,ϕ Then + as +. In the case, when 0 < R < +, the function µ, ϕ may be bounded and it is easy to show that ln µ, ϕ + as R if and only if lim W F e R Futhe we assume that 3 holds and fo the investigation of the gowth of ln µ, ϕ we use the esults fom [4]. By Ω0, R, 0 < R +, we denote the class of positive unbounded functions Φ on [ 0, R fo some 0 [0, R such that the deivative Φ is positively continuously diffeentiable and inceasing to + on [ 0, R. Fo Φ Ω0, R let Ψ Φ be a function associated with Φ in the sense of Newton and φ be the invese Φ function to Φ. Fo the numbes Φ 0 < a < b < + we put G a, b, Φ ab b a b a Φφt dt, G t 2 2 a, b, Φ Φ b a b a φtdt. Lemma. Let Φ Ω0, R, 0 < R +, and ϕ be an analytic in D R chaacteistic function of a pobability law F satisfying condition 3 and ln W F Ψφ 4 fo some inceasing to + sequence of positive numbes. Then [φ, φ+ ] and 0 the following estimates ae valid ln µ, ϕ Φ G 2, +, Φ G, +, Φ, 5

3 56 М. I. PAROLYA, М. М. SHEREMETA and ln µ, ϕ Φ G, +, Φ G 2, +, Φ 6 Φ ln µ, ϕ Φ G 2, +, Φ Φ G, +, Φ, 7 whee Φ is the invese function to Φ. Φ ln µ, ϕ Φ G 2, +, Φ Φ G, +, Φ, 8 Poof. Let P be an abitay function defined on 0, + and diffeent fom + it can tae on the value but P and let Q sup {P + : 0}, < < R, be the function conjugated to P in the sense of Young. By Ω, R, as in [4], we denote the class of positive unbounded functions on, R such that the deivative Φ is positive continuously diffeentiable and inceasing to + on, R. The functions Ψ, φ and the quantities G a, b, Φ, G 2 a, b, Φ we define as above. Then fom Theoem in [4] it follows that if P Ψφ fo some inceasing to + sequence of positive numbes then fo all [φ, φ+ ] and all the estimates 5 8 hold with Q instead of ln µ, ϕ. It is clea that the functions ln µ, ϕ and ln W F ae conjugated in the sense of Young, and in view of the definition of Ω0, R fo each function Φ Ω0, R thee eists Φ Ω, R such that Φ Φ fo [ 0, R. Since Ψ Ψ fo [ 0, R and φ φ fo 0 0 0, Poposition follows fom the quoted esult in [4]. We note in passing that [4] G a, b, Φ < G 2 a, b, Φ and the following lemma is hold [4] [6]. Lemma 2. Fo > a let G G 2 a,, Φ G a,, Φ, G G 2a,,Φ G a,,φ and fo [a, b let G G 2, b, Φ G, b, Φ, G G 2,b,Φ G,b,Φ. Then the functions G and G ae inceasing on a, + and the functions G and G ae deceasing on a, b. 3. Estimates fom below of ln µ, ϕ fo the functions of finite ode. We begin with a theoem, which is a genealization of Poposition. Theoem. Let ϕ be an entie chaacteistic function of a pobability law F such that ln W F, >, T > 0, 9 T fo some inceasing to + sequence of positive numbes. Then if + h fo, whee the function h is positive continuous and nondeceasing on [0, + and h o as +, then ln µ, ϕ T + o 8T 2 h 2 T, + ; 0 2 if + ω+ fo, whee the function ω is continuous and non-deceasing on [0, + and ω > fo all 0, then fo all lage enough ln µ, ϕ T f ωt, fω ωω ω. ω

4 ESTIMATES FROM BELOW FOR CHARACTERISTIC FUNCTIONS 57 Poof. It is easy to chec that fo the function Φ T with > the following equalities ae tue φ T, Ψφ T, ab G a, b, Φ b T a, b a G 2 a, b, Φ b a. T 2 b a Theefoe, G, + h, Φ T + h + h h T + h h { h + 2 h h 3 h 4 } T + h { + 2 h h h 3 } { h 2 h 2 h 3 } +O 3 T + +, G 2, + h, Φ T { h + h } { h h 2 2 h T 2 h h 4 } { h +O 4 T h h 3 } { h 5 h 2 h 3 } +O 3 T + +, as. Hence in view of the condition + h by Lemma 2 we have G 2, +, Φ G, +, Φ G 2, + h, Φ G, + h, Φ h 2 h 3 + o T T h 2, as, and since 9 implies 4 see inequality 5 ln µ, ϕ T + o h 2 2,, 8 T fo all [ /T, + /T ], 0. Fo such we have T + and since the function h is non-deceasing and + + o as then ln µ, ϕ T +o h 2 T T 2 8 T as +, whence 0 follows. The fist pat of Theoem is poved. Fo the poof of the second pat we ema that + G ω+, + ω + +, Φ, T ω+ ω +

5 58 М. I. PAROLYA, М. М. SHEREMETA + G 2 ω+, +, Φ + ω +, T 2 ω + ω+ that is, by Lemma 2 in view of the condition t + t ωt + we have G, +, Φ G 2, +, Φ G + /ω+, +, Φ G 2 + /ω+, +, Φ fωt +. Theefoe, by Lemma ln µ, ϕ T fω + fo all such as in the poof of the fist pat. Since ω+ ωt we need to pove that the function fω is deceasing on [, +. It is easy to chec that fω ξ ω ξ ω, whee the function ξω ω deceases to 0 on, + and ξω as ω, ξω > 0 and ω ξ ω < 0 on [, +. Hence it follows that f ω ξ 2 ω ξω ξ ω < 0, i. e. f is a deceasing function. The second pat of Theoem is poved. Now we pove Poposition. By its assumption, ln W F λ+ λ. Let > be an abitay numbe such that λ+ >. Then fo evey lage enough inequality 9 holds λ with T. Since + O as we have + K, that is, the assumptions of Poposition hold with ω K >, and by Theoem ln µ, ϕ A fo all lage ln µ,ϕ enough, whee A is a positive constant, whence it follows that ln ln +ln A. Letting hee to λ+ we obtain the desied asymptotical inequality. Poposition is poved. Fom Theoem the following poposition also follows. Poposition 2. If thee eists an inceasing to + sequence such that 9 holds and + / then ln M, ϕ + ot as +. Indeed, since + ω fo an abitay ω > and all 0 ω, by poposition 2 of Theoem we have ln µ, ϕ T fω. Since lim fω we obtain hence ω the desied asymptotical inequality. Fo analytic in D R function the following theoem is an analog of Theoem. Theoem 2. Let ϕ be an analytic in D R, 0 < R < +, chaacteistic function of pobability law F such that ln W F R + + T + + fo some inceasing to + sequence of positive numbes. Then if satisfies the assumption of poposition of Theoem then, > 0, T > 0, T + o ln µ, ϕ R 8T + R +2 h 2 T, R; 2 R + 2 if satisfies the assumption of poposition 2 of Theoem, then ln µ, ϕ T + + R f ω T R +, fω ω + ω + ω + ω +. 3

6 ESTIMATES FROM BELOW FOR CHARACTERISTIC FUNCTIONS 59 Poof. It is easy to chec that fo the function Φ T R we have φ T + R, Ψφ R + T + +, T G a, b, Φ + + ab b a a + b + and G2 a, b, Φ T + b a. + Theefoe, as in the poof of Theoem, it b + a + is possible to show that G t, t + ht, Φ } +O h 3 T +, G 2 t, t +ht, Φ T + { + 2+ { h 2+ h 2+ h h 2 +O h and, thus, G 2, + h, Φ G, + h, Φ +o + T 8+, as, whence in view of the condition + + h and lemmas and 2 we obtain + h 2 +2 T T + + o ln µ, ϕ h 2 R +2 +,, fo all [R T / +, R T /+ + ] and all lage enough. Fo such we have T R + + and since the function h is non-deceasing and + + o as 4 implies 2. The fist pat of Theoem 2 is poved. We pove the second pat. Since + T G ω+, + + ω + + +, Φ + + ω+, + G 2 ω+, +, Φ T ω + + ω+ +, ω + + we have G + /ω+, +, Φ + + fω G 2 + /ω+, +, Φ +, and by Lemmas and 2 ln µ, ϕ T + + fω R + fo all such as in Poposition of this theoem. Since ω+ ω T R we need to pove, as above, that the + function f is deceasing on [, +. So, fω a deceasing function. ω + ω Fom Theoem 2 the following two popositions follow. + ω Poposition 3. If a pobability law F satisfies the condition ln lnw F e R ω + ω + } and evey facto is λ λ + ln, λ > 0, 5 fo some inceasing to + sequence of positive numbes and + O,, then fo its chaacteistic function ϕ we have the following asymptotic inequality ln ln M, ϕ + oλ ln, R. 6 R Indeed, 5 implies ln W F R + λ λ+ R + + fo evey < λ and all lage enough, that is, holds and since + K fo all by item 2 of Theoem 2 we have ln µ, ϕ A, whee A is a positive constant, whence ln ln µ, ϕ R ln R + +, R. In view of the abitainess of we obtain 6.

7 60 М. I. PAROLYA, М. М. SHEREMETA Poposition 4. If fo a pobability law F condition holds and + + o as then ln M, ϕ +ot R as R. Poposition 4 easy follows fom item 2 of Theoem 2, because lim fω. We ω + + ema that if in item of Theoems 2 + h const and in item 2 of these theoems + / ω const then we need not use Lemma 2, that is, we need not estimate of G 2 G and G /G 2. Theefoe, in view of the optimality of estimates 5 and 6, which we used in the poof of theoems 2, in the cases whee + h and + / ω estimates 0, 2 and coesponding, 3 ae unimpovable. 4. Genealized esults. Since we not always can find G and G 2 in an eplicit way, the following theoem is useful. Theoem 3. Let 0 < R +, Φ Ω0, R be such that ΦΦ η non-incease on [ 0, R fo some 0 0, R and η [0, +, let ϕ be an analytic in D R chaacteistic function of a pobability law F, which satisfies condition 3 and let inequality 4 hold fo some inceasing to + sequence of positive numbes. Then if + h,, whee a positive and continuous on 0, + function h is such that h o as, the function + h inceases and the function η h non-deceases on 0, +, then ln µ, ϕ Φ + o + η 2 Φ Φ hφ, R; 7 2 if + ω+,, whee a continuous and non-deceasing on 0, + function ω is such that ω > fo > 0, then ln µ, ϕ fo all < R close enough to R. ω η Φ Φ 8 ηω η Φ ωφ Poof. At fist we assume that η > 0 and pove item. Fom the non-incease of we have G, + h, Φ + h h +h + h Φφ + h h η Φφ + h +η { + h η } + h η ηh Φφ + h +η { ηh + ηη h2 + h η ηh Φφ + h + h η η G 2, + h, Φ Φ h Φφ η d +η + h +η + h η η { + η h 2 +h 2 2 h 2 2 h 3 3 } },, φtdt Φφ + h. Φ Φ +η

8 Theefoe, ESTIMATES FROM BELOW FOR CHARACTERISTIC FUNCTIONS 6 G 2, + h, Φ G, + h, Φ { η Φφ + h + η h h 2 } + h 2 2 { + η h 2 h 2 } Φφ + h 2 Φφ + h + η h h 2 2 h 2 ηh h 2 } { + η h 2 2 { + η Φφ + h 2 Hence in view of the condition + inequality 5 we obtain h h 2 2 } Φφ + h + η + h +η 2 h η + o,. + h using Lemma 2 gowth of G and ln µ, ϕ Φ Φφ + h + h +η + η 2 η h + o,, 9 fo all [φ, φ+ ]. Since Φϕtt η non-inceases, η h non-deceases and the inequalities φ φ+ imply the inequalities Φ +, we obtain ln µ, ϕ Φ Φφ + η +η 2 η h + o Φ Φ + η Φ +η 2 Φ η hφ + o i.e., inequality 7 holds. If η 0 then by analogy we have G, + h, Φ + h Φφ + h h + h Φφ + h h h 2 2 G 2, + h, Φ G, + h, Φ Φφ + h + h 2 2 ln + h,, h + o,, 2 whence we obtain 9 with η 0. Hence, as above estimate 7 follows. The fist pat of Theoem 3 is poved. We pove second pat. Fo η > 0 we have + G ω+, + + Φφ +, Φ η d ω+ +η + ω+ + ω+ Φφ+ +η η + η η + ω η G 2 ω+, +, Φ Φφ+. Φφ+ ηω+, ω η +

9 62 М. I. PAROLYA, М. М. SHEREMETA Theefoe, in view of the condition + ω+ using Lemma 2 and inequality 6 fo all [φ, φ+ ] and all 0 we have ω η + ln µ, φ Φ ηω η + ω+ Φ ω η Φ ηω η Φ ωφ, because the function f η η deceases on [, + and + Φ. The inequality 8 is poved. If η 0 then, by analogy, we have + G ω+, + Φφ+ +, Φ ln ω+ Φφ+ ln ω+ ω+ + ω+ and in view of the estimates G + 2 ω+ +, Φ Φφ+, as above, we obtain ln µ, ϕ Φ ln ω+ ω+ Φ ln ωφ ωφ 20 fo all < R close enough to R. Since estimate 8 with η 0. ω η ln ω ηω η ω ω as η 0 estimate 20 coincides with The condition of the non-incease of ΦΦ η can be emoved if we use estimates 7 and 8 fom Lemma. We get the following theoem. Theoem 4. Let Φ Ω0, R, 0 < R +, and ϕ be an analytic chaacteistic function of a pobability law, which satisfies conditions 3 and 4 fo some inceasing to + sequence of positive numbes. Then if φ+ φ h+, whee h is a positive continuous and non-inceasing function on 0, + such that R > φ h R as +, then fo all < R close enough to R ln µ, ϕ Φ hφ ; 2 2 if φ+ φ ω+, whee ω is a positive continuous and non-inceasing function on 0, + such that R > φ R as +, then fo all < R close enough to R ω ln µ, ϕ Φ. 22 ωφ Poof. Since the function Φφt inceases we have Φ G, +, Φ Φ + Φφ + Φ G 2, +, Φ d φ 2, φtdt φ+. Theefoe, fom 7 and 8 fo all [φ, φ+ ] we obtain espectively Φ ln µ, ϕ φ+ φ h+ hφ, Φ ln µ, ϕ ϕ ϕ+ whence the inequalities 2 and 22 follows. ω+ ωφ,

10 ESTIMATES FROM BELOW FOR CHARACTERISTIC FUNCTIONS Coollaies. Let L be a class of continuous inceasing functions α such that α 0 fo 0, α α 0 fo 0 and on [ 0, + the function α inceases to +. We say that α L 0 if α L and α + o + oα as + ; futhe α L si if αc + oα as + fo any c 0, +. It is easy to see that L si L 0. Coollay. Let eithe α L si and β L 0 o α L 0 and β L si and ϕ be an entie chaacteistic function of a pobability law F such that β ln α 23 W F fo some inceasing to + sequence of positive numbes, which satisfies the condition β cα+ /β cα as fo any c, +. Then ln µ, ϕ α + oβ,. 24 Poof. Let at fist α L si, β L 0 and ε 0, be an abitay numbe. Since β L 0, we have [7] β + δ ε εβ, whee δ ε 0 as ε 0 and, thus, β εβ +δ ε, and the condition α L si implies αε +oα as +, that is, fo any δ 2 > 0 and all lage enough the inequality αε +δ 2 α is tue. Theefoe, β α εβ + δ εα εβ + δ ε + δ 2 αε fo all lage enough. On the othe hand 0 β + δ ε + δ 2 αtdt Hence it follows fom 23 that ε β + δ ε + δ 2 αtdt β + δ ε + δ 2 αε ε. ln W F β α εβ + δ ε + δ 2 αε fo each ε 0,, δ 2 > 0 and all 0 0 ε, δ 2. We put Φ 0 α βt +δ φ β + δα and Ψφ 0 β + δ ε + δ 2 αtdt 25 dt, whee +δ < +δ ε+δ 2. Then Φ α β +δ 0 φtdt + const 0 β + δαtdt + const 0 β + δ ε + δ 2 αtdt. Theefoe, inequality 25 implies 4 fo all lage enough. Futhe, since β +δα+ β +δα, thee eists a deceasing to continuous function ω such that φ+ φ ω + fo all. Theefoe, by item 2 of Theoem 4 inequality 22 is tue, that is, in view of the condition β L 0 we have +o β ln µ, ϕ Φ Φ + o α d ωφ 0 + δ β β ε α d α ε + δ 2 + δ 2 ε,

11 64 М. I. PAROLYA, М. М. SHEREMETA fo all lage enough. Since α L si, β L 0 and the numbes ε, δ 2 and δ ae abitay, fom the latte inequality we easily obtain 24. If α L 0 and β L si then α ε α, whee δ +δ ε ε 0 as ε 0 and βε +δ 2 β fo all lage enough. Theefoe, as above β + δ ε + δ 2 αtdt εβ + δ ε + δ 2 α ε 0 εβ + δ 2 α β + δ 2 ββ + δ 2 α β α. Hence it follows fom 23 that ln W F 0 β + δ ε + δ 2 αtdt fo any ε 0,, δ 2 > 0 and all 0 0 ε, δ 2. Theefoe, choosing Φ, as above, and epeating the aguments, we again aive at inequality 24. Fo analytic functions in D R, 0 < R < +, the following coollay is an analog of Coollay. d ln β α d ln Coollay 2. Let α L si, β L si, q < fo all lage enough and α + oα as +, and ϕ be an analytic in D β α R, 0 < R < +, chaacteistic function of a pobability law F, fo which β α lnw F e R 26 fo some inceasing to + sequence of positive numbes such that β α+ Oβ α as. Then αln µ, ϕ + oβ, R. 27 R Poof. Fom 26 it follows that ln W F R +. Since d ln β α β α q <, d ln we have + β α 0 +, and using L Hospital s ule it is easy to show that Theefoe, β α + o q ln W F R + q 0 fo any q q, and all lage enough. We put Φ whee q 2 q,. Then Φ α β 0 α q 2 R 0 dt β αt,. dt β αt 28 q2 β d, 29 R, φ R q 2 β α and dt Ψφ R q 2 0 β αt + const,

12 ESTIMATES FROM BELOW FOR CHARACTERISTIC FUNCTIONS 65 that is, in view of 28 and q < q 2 we obtain 4. Since β α+ Kβ α, K >, fo all, we have. Theefoe, if we put h K β α β α+ β α+ K q 2 β α then φ h R K q 2 β α R as +, hφ K R and φ+ φ h+ fo. Finally, fo function 29 and > ma{ 0, R/2} we have Φ 2 R α β Theefoe, by item of Theoem 4 ln µ, ϕ R + hφ α β q2 d R α q2 β. R 2R KR α β q2 2KR fo all < R close enough to R. But fom the condition α q 2 2R + hφ β α + oα as + it follows that α α βt + oβt as t and since α L t si, β L si the last inequality implies 27. We ema that unde the othe conditions of Coollay 2 the condition β α+ Oβ α as holds povided + O as. The conditions on α and β in Coollay 2 assume that the function α inceases slowe than the function β. In the case whee α inceases quice than β, the following coollay is tue. Coollay 3. Let α L si, β L si, d ln α β q < fo all lage enough, dα β d ln d 0 and α βf Oα β as +, and ϕ be an analytic in D f R, 0 < R < +, chaacteistic function of a pobability law F, fo which αlnw F e R β 30 fo some inceasing to + sequence of positive numbes such that lim f+ f < 2. Then asymptotical inequality 27 holds. Poof. If we put Ψφ R α β then 30 implies 4 and φ Ψφ R dα β R. Hence it follows that d f Φ f, R Φ Φ 0 R f R 0 f +α β R f R d f R td qα β ft f R But fom the condition α βf Oα β as + it follows that α β Kα β f, R R.

13 66 М. I. PAROLYA, М. М. SHEREMETA whee K is a positive constant. Theefoe, Φ K α β R, whee K is a positive constant, fo all < R close enough to R, and if h ar φ, 0 < a <, then Φ hφ K α β + ar. 3 Unde such a choice of the function h the condition φ+ φ h+ is equivalent to the condition f+ + af, and the latte condition follows fom the condition < 2. Theefoe, by item of Theoem 4 inequality 2 is tue and in view of lim f+ f 3 and the conditions α L si, β L si we obtain 27. We ema that fom Coollaies 3 one can obtain analogues of Popositions 4, but we shall not discuss this hee. REFERENCES. Linni Yu.V., Ostovsii I.V. Decomposition of andom vaiables and vectos. Мoscow: Naua, p. in Russian 2. Sasiv O.B., Sooivs yj V.M. On the maimum of module of chaacteistic functions of pobabilistic law// Kaj. Zadachi Dyfe. Rivnyan V.7. P in Uainian 3. Jaovleva N.I. On the gowth of entie chaacteistic functions of pobability laws// Poblems of mathematical physics and funct. analysis. К.: Nauova duma, 976. P in Russian 4. Sheemeta M.M., Sumy О.М. A connection between the gowth of Young conjugated functons// Mat. Stud V.,. P in Uainian 5. Sheemeta M.M., Sumy О.М. Lowe estimates fo the maimal tem of a Diichlet seies// Izv. Vyssh. Uchebn. Zaved., Mat in Russian 6. Sumy О.М. Lowe estimates fo maimal tem of Diichlet seies// Visn. L viv. Univ., se. meh.-math V.53. P in Uainian 7. Sheemeta M.M. On two classes of positive functions and belonging to them of main chaacteistics of entie functions// Mat. Stud V.9,. P Ivan Fano Natonal Univesity of L viv mata069@amble.u m_m_sheemeta@list.u Received