Математичнi Студiї. Т.14, №1 Matematychni Studii. V.14, No.1
УДК 517.576
O. M. Muliava
A CONVERGENCE CLASS FOR ENTIRE DIRICHLET SERIES
OF SLOW GROWTH
Dedicated to the 70th anniversary of Prof. A. A. Gol’dberg
O. M. Muliava. A convergence class for entire Dirichlet series of slow growth, Matematychni
Studii, 14 (2000) 49–53.
For an entire Dirichlet series with nonnegative increasing to +∞ exponents a connection
between the growth of maximum modulus and the behaviour of coefficients is established in
the terms of certain convergence class.
О. М. Мулява. Класс сходимости для целых рядов Дирихле медленного роста // Матема-
тичнi Студiї. – 2000. – Т.14, №1. – C.49–53.
Для целого ряда Дирихле с неотрицательными возрастающими к +∞ показателями в
терминах определенного класса сходимости установлена связь между ростом максимума
модуля и поведением коэффициентов.
Let 0 = λ0 < λn ↑ +∞ and
F (s) =
∞∑
n=0
an exp(sλn), s = σ + it, (1)
be an entire (absolutely convergent in C) Dirichlet series. We say that this series of is of slow
growth if lnM(σ) = (1+o(1))σl(σ) (σ → +∞), where M(σ) = sup{|F (σ+it)| : t ∈ R} and l
is a slowly increasing function, i.e. l is a positive increasing to +∞ function on [x0,+∞) such
that xl′(x)/l(x)→ 0, x→∞. Further, for simplicity, we consider that x0 = 1. Generalizing
a result of Valiron [1] on the belonging of an entire function of finite order to classical
convergence class, Kamthan [2] indicated conditions on the exponents and the coefficients of
series (1) in order that
∫∞
0 exp{−%σ} ln M(σ)dσ < +∞. This results is generalized in [3–4],
where generalized convergence classes are introduced and studied. Here we supplement the
results from [3–4] for entire Dirichlet series of slow growth.
Let α be a slowly increasing function. We say that Dirichlet series (1) belongs to a con-
vergence α-class provided
∫ ∞
1
lnM(σ)dσ
σ2α(σ)
< +∞, if
∫ ∞
1
dt
tα(t)
< +∞, (2)
2000 Mathematics Subject Classification: 30B50.
c©O. M. Muliava, 2000
50 O. M. MULIAVA
and ∫ ∞
1
dσ
α(σ) lnM(σ)
< +∞, if
∫ ∞
1
dt
tα(t)
= +∞. (3)
Suppose that ln n = O(λn), n→∞. Then [5, p. 184] there exists K > 0 and τ > 0 such
that M(σ) ≤ Kµ(σ + τ) for all σ ≥ 1, where µ(σ) = max{|an| exp{σλn} : n ≥ 0} is the
maximal term of series (1). Hence in view of the Cauchy inequality µ(σ) ≤M(σ) [5, c. 125]
it follows that in (2) and (3) we can put lnµ(σ) instead of ln M(σ).
Let ν(σ) = max{n : |an| exp{σλn} = µ(σ)} be the central index of series (1). Then [5,
c. 182] lnµ(σ) − lnµ(σ0) =
∫ σ
σ0
λν(t)dt, whence in view of nondecreasing of λν(σ) we easily
obtain the following estimates (σ/2)λν(σ/2) ≤ lnµ(σ) − lnµ(σ0) ≤ σλν(σ). Thus, Dirichlet
series (1) belongs to the convergence α-class iff
∫ ∞
1
λν(σ)dσ
σα(σ)
< +∞, 1 ≤ σ0 < +∞, if
∫ ∞
1
dt
tα(t)
< +∞, (4)
and ∫ ∞
1
dσ
σα(σ)λν(σ)
< +∞, if
∫ ∞
1
dt
tα(t)
= +∞. (5)
Finally, let aon be the coefficients of Newton majorant of Dirichlet series (1) and κ
o
n =
ln aon−ln a
o
n+1
λn+1−λn
. Then [5, p. 180–183] |an| ≤ aon, κ
0
n ↗ +∞, and if κ
o
n−1 ≤ σ < κ
o
n then
λν(σ) = λn. For simplicity, we suppose that κo0 ≥ 1. Therefore,
∫ ∞
1
λν(σ)dσ
σα(σ)
+ const =
∞∑
n=1
∫ κon
κon−1
λν(σ)dσ
σα(σ)
=
=
∞∑
n=1
λn
∫ κon
κon−1
dσ
σα(σ)
=
∞∑
n=1
λn(β1(κon−1)− β1(κ
o
n))) =
=
∞∑
n=1
(λn − λn−1)β1(κon−1) + const, β1(x) =
∫ ∞
x
dσ
σα(σ)
. (6)
By analogy,
∫ ∞
1
dσ
σα(σ)λν(σ)
+ const =
∞∑
n=1
1
λn
∫ κon
κon−1
dσ
σα(σ)
=
∞∑
n=1
1
λn
(β2(κon)− β2(κ
o
n−1)) =
=
∞∑
n=1
(
1
λn−1
−
1
λn
)
β2(κon−1) + const, β2(x) =
∫ x
1
dσ
σα(σ)
. (7)
Thus, we need to investigate the convergence of the last series in (6) and (7).
From the definition of κon−1 it follows that
1
λn
ln
1
a0n
=
κo0λ
∗
1 + · · ·+ κ
o
n−1λ
∗
n
λ∗1 + · · ·+ λ∗n
, λ∗n = λn − λn−1, (8)
whence we obtain the inequality 1λn ln
1
a0n
≤ κon−1. Therefore, if
∞∑
n=1
(λn − λn−1)β1
(
1
λn
ln
1
a0n
)
< +∞, (9)
A CONVERGENCE CLASS FOR ENTIRE DIRICHLET SERIES OF SLOW GROWTH 51
then, in view of decrease of the function β1, relation (4) holds, and if (5) holds then in view
of increase of the function β2 we have
∞∑
n=1
(
1
λn−1
−
1
λn
)
β2
(
1
λn
ln
1
a0n
)
< +∞. (10)
In order to obtain a converse to the first assertion, we use the following
Lemma [4]. Let p > 1, q = pp−1 and f be a positive function on (A,B), −∞ ≤ A < B ≤
+∞, such that the function f 1/p is convex on (A,B). Let (λ∗n) be a sequence of positive
numbers, (an) be a sequence of numbers from (A,B) and An =
λ∗1a1+···+λ
∗
nan
λ∗1+···+λ
∗
n
. Finally, let
(µn) be a positive nonincreasing sequence. Then
∞∑
n=1
µnλ
∗
nf(An) ≤ q
p
∞∑
n=1
µnλ
∗
nf(an). (11)
Since the function α is increasing to +∞, then it is easy to show that the function β1/21 (x)
is convex on an interval where α(x) ≥ 1. Therefore, if we put in Lemma p = 2, µn ≡ 1 and
λ∗n = λn − λn−1, then in view of (8) we have
∞∑
n=1
(λn − λn−1)β1
(
1
λn
ln
1
a0n
)
≤ 4
∞∑
n=1
(λn − λn−1)β1(κon−1).
Thus, in view of (6) we have proved that (4) holds iff (9) holds.
In order to obtain an converse to the second assertion, we remark that in view of (8)
1
λn
ln 1a0n ≥
λn−λn−1
λn
κon−1 and
β2(e
2x) =
∫ 2x
0
dt
α(et)
=
∫ x
0
dt
α(et)
+
∫ 2x
x
dt
α(et)
=
=
∫ x
0
dt
α(et)
+
∫ x
0
dt
α(et+x)
≤ 2
∫ x
0
dt
α(et)
= 2β2(e
x).
Therefore,
β2(κon−1) ≤ β2
(
exp
{
ln
(
1
λn
ln
1
a0n
)
+ ln
λn
λn − λn−1
})
≤
≤ β2
(
exp
{
2 max
{
ln
(
1
λn
ln
1
a0n
)
, ln
λn
λn − λn−1
}})
≤
≤ 2β2
(
max
{
ln
(
1
λn
ln
1
a0n
)
, ln
λn
λn − λn−1
})
=
= 2 max
{
β2
(
1
λn
ln
1
a0n
)
, β2
(
ln
λn
λn − λn−1
)}
≤
≤ 2
(
β2
(
1
λn
ln
1
a0n
)
+ β2
(
ln
λn
λn − λn−1
))
.
Hence if
∞∑
n=1
(
1
λn−1
−
1
λn
)
β2
(
ln
λn
λn − λn−1
)
< +∞, (12)
52 O. M. MULIAVA
then from (10) we obtain the convergence of the last series in (7).
Thus, we have proved that if the sequence (λn) satisfies condition (12), then (5) holds iff
(10) holds.
Remarking that |an| ≤ aon for all n ≥ 0 and |an| = a
o
n for all n ≥ 0 provided κn =
ln |an|−ln |an+1|
λn+1−λn
↗ +∞, we therefor come to the following
Theorem. Let the exponents of entire Dirichlet series (1) satisfy the condition ln n = O(λn)
(n→∞), and α be a slowly increasing function. Then:
i) if
∫∞
1
dt
tα(t) < +∞ then in order that Dirichlet series belong to the convergence α-class,
it is necessary and in the case when κn ↗ +∞ it is sufficient that condition (9) hold
with |an| instead a0n;
ii) if
∫∞
1
dt
tα(t) = +∞ then in order that Dirichlet series belong to the convergence α-class,
in the case κn ↗ +∞ it is necessary and in case when the exponents satisfy condition
(12) it is sufficient that condition (10) hold with |an| instead a0n.
Remark 1. In the proof of necessity of condition (9) in the first assertion of Theorem the
condition ln n = O(λn) (n→∞) is not used. In the proof of sufficiency we can replace this
condition by the following condition
lim
n→∞
ln n
− ln |an|
= h < 1. (13)
Indeed, if (13) holds then for all σ ≥ 0 the following estimate is true [6, c. 23] M(σ) ≤
A0(ε)µ
(
σ
1−h−ε
)
for every ε ∈ (0, 1− h) and some A0(ε) > 0, and since α is slowly increasing
then, in view of the Cauchy inequality, the integrals
∫∞
1
ln M(σ)
σ2α(σ) dσ and
∫∞
1
lnµ(σ)
σ2α(σ)dσ are either
convergent or divergent simultaneously. The further proof of sufficiency is analogous to that
given above.
Further, if
∫∞
1
lnµ(σ)
σ2α(σ)dσ < +∞ then using L’Hospital rule and the slow increase of α we
obtain for all enough large σ
1
2
≥
∫ ∞
σ
lnµ(x)dx
x2α(x)
≥ lnµ(σ)
∫ ∞
σ
dσ
x2α(x)
dx = (1 + o(1))
lnµ(σ)
σα(σ)
, σ → +∞,
that is lnµ(σ) ≤ σα(σ), σ ≥ σ0, and ln |an| ≤ σα(σ)−σλn for all n ≥ 0 and σ ≥ σ0. Putting
here σ = ϕ(λn), where ϕ(x) is a solution of the equation α(σ) + σα′(σ) = x, we have for
n ≥ n0 the inequality ln |an| ≤ −ϕ(λn)α′(ϕ(λn)), that is (13) holds provided
lim
n→∞
ln n
ϕ(λn)α′(ϕ(λn))
< 1. (14)
Thus, the condition ln n = O(λn) (n→∞) in the first assertion of Theorem can be replaced
by condition (14).
Remark 2. The condition ln n = O(λn) (n → ∞), is used only to prove the necessity of
condition (10). We can replace it by condition (13), but it is impossible to find a condition
similar to (15) because the convergence of integral
∫∞
1
dσ
α(σ) lnµ(σ) for lnµ(σ) it can yield only
an estimate from below.
Condition (12) appeared as a result of the applied method. We could not find out whether
this condition is superfluous.
A CONVERGENCE CLASS FOR ENTIRE DIRICHLET SERIES OF SLOW GROWTH 53
REFERENCES
1. Valiron G. General theory of integral functions, Toulouse, 1923, 382 pp.
2. Kamthan P. K. A theorem of step functions (III), Istambul univ. fen. fac. mecm. A 28 (1963), 65–69.
3. Мулява О. М. Класи збiжностi в теорiї рядiв Дiрiхле, Доп. НАН України, сер. А. (1999), В3, 35–39.
4. Мулява О. М. Про класи збiжностi рядiв Дiрiхле, Укр. мат. журн. 51 (1999), В11, 1485-1494.
5. Леонтьев А. Ф. Ряды экспонент, М.: Наука, 1976, 536 c.
6. Шеремета M.M. Цiлi ряди Дiрiхле, K.: IСДО, 1993, 168 с.
Ivan Franko Drohobych State Pedagogic University,
Physics and Mathematics Faculty
Received 1.07.1999