On Analytic and Subharmonic Functions in Unit Disc Growing Near a Part of the Boundary
In the paper there was found an analog of the Blaschke condition for analytic and subharmonic functions in the unit disc, which grow at most as a given function j near some subset of the boundary.
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2013
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| Цитувати: | On Analytic and Subharmonic Functions in Unit Disc Growing Near a Part of the Boundary / S.Ju. Favorov, L.D. Radchenko // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 3. — С. 304-315. — Бібліогр.: 10 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1067562025-02-09T15:18:58Z On Analytic and Subharmonic Functions in Unit Disc Growing Near a Part of the Boundary Favorov, S.Yu. Radchenko, L.D. In the paper there was found an analog of the Blaschke condition for analytic and subharmonic functions in the unit disc, which grow at most as a given function j near some subset of the boundary. Найден аналог условия Бляшке для аналитических и субгармонических функций в единичном круге, которые растут не быстрее заданной функции j вблизи некоторого подмножества границы. 2013 Article On Analytic and Subharmonic Functions in Unit Disc Growing Near a Part of the Boundary / S.Ju. Favorov, L.D. Radchenko // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 3. — С. 304-315. — Бібліогр.: 10 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106756 en Журнал математической физики, анализа, геометрии application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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In the paper there was found an analog of the Blaschke condition for analytic and subharmonic functions in the unit disc, which grow at most as a given function j near some subset of the boundary. |
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Favorov, S.Yu. Radchenko, L.D. |
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Favorov, S.Yu. Radchenko, L.D. On Analytic and Subharmonic Functions in Unit Disc Growing Near a Part of the Boundary Журнал математической физики, анализа, геометрии |
| author_facet |
Favorov, S.Yu. Radchenko, L.D. |
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Favorov, S.Yu. |
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On Analytic and Subharmonic Functions in Unit Disc Growing Near a Part of the Boundary |
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On Analytic and Subharmonic Functions in Unit Disc Growing Near a Part of the Boundary |
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On Analytic and Subharmonic Functions in Unit Disc Growing Near a Part of the Boundary |
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On Analytic and Subharmonic Functions in Unit Disc Growing Near a Part of the Boundary |
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On Analytic and Subharmonic Functions in Unit Disc Growing Near a Part of the Boundary |
| title_sort |
on analytic and subharmonic functions in unit disc growing near a part of the boundary |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2013 |
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https://nasplib.isofts.kiev.ua/handle/123456789/106756 |
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On Analytic and Subharmonic Functions in Unit Disc Growing Near a Part of the Boundary / S.Ju. Favorov, L.D. Radchenko // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 3. — С. 304-315. — Бібліогр.: 10 назв. — англ. |
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Журнал математической физики, анализа, геометрии |
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AT favorovsyu onanalyticandsubharmonicfunctionsinunitdiscgrowingnearapartoftheboundary AT radchenkold onanalyticandsubharmonicfunctionsinunitdiscgrowingnearapartoftheboundary |
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2025-11-27T06:55:15Z |
| last_indexed |
2025-11-27T06:55:15Z |
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1849925587124092928 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2013, vol. 9, No. 3, pp. 304–315
On Analytic and Subharmonic Functions in Unit Disc
Growing Near a Part of the Boundary
S.Ju. Favorov and L.D. Radchenko
V.N. Karazin Kharkiv National University
4 Svobody Sq., Kharkiv 61077, Ukraine
E-mail: sfavorov@gmail.com
liudmyla.radchenko@gmail.com
Received January 17, 2012, revised December 20, 2012
In the paper there was found an analog of the Blaschke condition for
analytic and subharmonic functions in the unit disc, which grow at most as
a given function ϕ near some subset of the boundary.
Key words: analytic function, subharmonic function, Riesz measure,
Blaschke condition.
Mathematics Subject Classification 2010: 30D50, 31A05.
1. Introduction
It is well known that the zeroes {zn} of any bounded analytic function in the
unit disk D satisfy the Blaschke condition
∑
(1− |zn|) < ∞. (1)
There have been a number of papers published where there can be found the
analogous conditions for various classes of unbounded analytic and subharmonic
functions (see, for example, [6–9] ). In [5], there was studied the case of analytic
functions in D of an exponential growth near a finite subset E ⊂ ∂D. In [1], the
case of an arbitrary compact subset E ⊂ ∂D was considered. Namely, there were
considered subharmonic functions v in D such that
v(z) ≤ dist(z,E)−q, z ∈ D, (2)
for some 0 < q < ∞. The Riesz measures (generalized Laplacians) µ =
1
2π
4v
are proven to satisfy the condition
∫
D
(1− |λ|)(dist(λ,E))(q−α)+dµ(λ) < ∞ (3)
c© S.Ju. Favorov and L.D. Radchenko, 2013
On Analytic and Subharmonic Functions in Unit Disc
for α such that
2∫
0
m{s ∈ ∂D : dist(s,E) < t}dt
tα+1
< ∞. (4)
Here x+ = max{0, x}, and m is the normalized Lebesgue measure on ∂D, that is,
m(∂D) = 1. If m(E) > 0, then (3) and (4) are valid for any α < 0. If (4) is valid
with some α > 0 and q ≤ α, then µ satisfies the Blaschke condition for bounded
from above in D subharmonic functions
∫
(1− |λ|)dµ(λ) < ∞. (5)
It was also proved in [1] that (3) is not valid for the subharmonic function
v0(z) = dist(z, E)−q in the case of divergent integral in (4).
If f(z) is an analytic function, then log |f(z)| is a subharmonic function with
the Riesz measure
∑
n knδzn , where δzn are the unit masses in the zeros {zn} of
f(z), and kn are the multiplicities of the zeros. Hence, if this is the case, (2) has
the form
|f(z)| ≤ exp dist(z, E)−q, (6)
while (3) has the form
∑
zn
(1− |zn|)dist(zn, E)(q−α)+ < ∞. (7)
Note that condition (6) seems to be too restrictive to be applied to the op-
erator theory [1]. It is natural to consider subharmonic (and analytic) functions
such that
v(z) 6 ϕ(dist(z,E)), (8)
where ϕ(t) is the monotonically decreasing on R+ function, ϕ(t) → +∞ as
t → +0. It is clear that for any subharmonic function v in D, which grows
as dist(z,E) → 0, condition (8) is valid for ϕ(t) = sup{v(z) : dist(z, E) ≥ t}.
The Main Results
To formulate our results we need some notations. Let
ρ(z) = dist(z, E), F (t) = FE(t) = m{ζ ∈ ∂D : ρ(ζ) < t}.
Consider
I(ϕ, E) :=
2∫
0
ϕ(s)dF (s).
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 305
S.Ju. Favorov and L.D. Radchenko
Note that F (t) is continuous on [0, 2] and has a discontinuity at t = 0 if and only
if m(E) > 0.
Theorem 1. Let E be a closed subset of ∂D, v be a subharmonic function in
D, v 6≡ −∞, and v(z) satisfy (8). If I(ϕ,E) < ∞, then the Riesz measure µ of
the function v(z) satisfies (5).
E x a m p l e 1. Let E = {ζ1, . . . , ζk}, ϕ(t) = t−1 log−α(1/t), α > 1.
It is readily seen that F (t) = 2kt/(2π) + o(1) as t → 1. The assumption of
Theorem 1 is fulfilled and the Riesz measure of any subharmonic function v(z) ≤
ρ−1(z) log−α(1/ρ(z)) satisfies the Blaschke condition (5).
Theorem 2. Let E, v and µ satisfy the assumption of Theorem 1, ϕ : R+ →
R+ be an absolutely continuous nonnegative monotonically decreasing function,
ϕ(t) → +∞ as t → +0, and ψ(t) be an absolutely continuous monotonically
increasing function on [0, 2] such that ψ(0) = 0. If
1/25∫
0
ψ(50y)ϕ′(y)F (y)dy > −∞, (9)
then ∫
D
ψ(ρ(λ))(1− |λ|)dµ(λ) < ∞. (10)
R e m a r k 1. In the case m(E) > 0, we always get I(ϕ, E) = ∞ and thus
the assumption of Theorem 1 is not satisfied. Further, (9) takes the form
1/25∫
0
ψ(50y)ϕ′(y)dy > −∞.
In the case E = ∂D, the result coincides with that obtained in [8].
E x a m p l e 2. Let E = {ζ1, . . . , ζk}, ϕ(t) = e1/t. The assumption of
Theorem 2 is satisfied with ψ(t) = e−c/t for c > 50. Hence, if v is a subharmonic
function v(z) such that v(z) ≤ exp(1/ρ(z)), then for c > 50 the integral
∫
exp(−c/ρ(z))(1− |λ|)dµ(λ)
converges.
The following theorem shows the accuracy of Theorem 2.
306 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3
On Analytic and Subharmonic Functions in Unit Disc
Theorem 3. Let E, ϕ, ψ be the same as above and, in addition, ϕ(1/t) be
convex with respect to log t. If
2∫
0
ψ(y)ϕ′(y)F (y)dy = −∞, (11)
then for the Riesz measure µ0 of the function v0(z) = ϕ(ρ(z)) we get
∫
ψ(ρ(λ))(1− |λ|)dµ0(λ) = +∞.
Now consider the case v(z) = log |f(z)| with the analytic function f(z).
Theorem 1′. Let E be a closed subset of ∂D, ϕ(t) be an absolutely continuous
nonnegative monotonically decreasing function on R+, ϕ(t) → +∞ as t → +0,
I(ϕ,E) < ∞, f be an analytic function in D with zeros zn. If
|f(z)| 6 exp(ϕ(ρ(z)),
then its zeros satisfy the Blaschke condition (1).
Theorem 2′. Let E, f , ϕ be the same as in Theorem 1′, but the condition
I(ϕ,E) < ∞ for some absolutely continuous monotonically increasing function ψ
on [0, 2] such that ψ(0) = 0 be replaced by (9). Then
∑
zn
ψ(ρ(zn, E))(1− |zn|) < ∞.
R e m a r k 2. For ϕ(t) = t−q, ψ(t) = ts, Theorems 1–3, 1′, 2′ were proved
earlier in [1].
2. Demonstrations
We begin with the lemmas.
Lemma 1. Let ν be a nonnegative finite measure on X, g(x) be a Borel
function on X, ϕ(t) be a Borel function on R. Then
∫
X
(ϕ ◦ g)(x)dν(x) =
∞∫
−∞
ϕ(y)dH(y),
where H(y) = ν{x : g(x) < y}.
The proof of this lemma can be easily reduced to the case ν(X) = 1 which is
well known (see, for example, [2, formula (15.3.1)]).
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 307
S.Ju. Favorov and L.D. Radchenko
Lemma 2. For each z ∈ D and τ ∈ [0, 1] we have ρ(z) 6 2ρ(τz).
P r o o f. Consider ζ = z/|z| and z′ ∈ [0, ζ] such that ρ(z′) = dist(E, [0, ζ]).
The cases z′ = 0 and z ∈ [0, z′] are trivial. Suppose z ∈ (z′, ζ). We have
ρ(z) 6 ρ(z′) + |z′ − z| 6 ρ(z′) + |z′ − ζ|.
Consider ζ ′ ∈ E such that ρ(z′) = |z′ − ζ ′| and the triangle with vertexes on z′,
ζ, ζ ′. The angle in z′ is right, and the angle in ζ is greater than π/4. Thus,
|z′ − ζ| 6 |z′ − ζ ′| and therefore ρ(z) 6 2ρ(z′) ≤ 2ρ(τz).
Note that the result of the lemma was used in [1, p.41] without proof.
P r o o f of Theorem 1. First, suppose v(0) = 0. Let I(ϕ,E) < ∞. Using
Lemma 1 with the measure m(ζ) on ∂D and taking into account the equalities
F (y) ≡ 0 for y < 0 and F (y) ≡ 1 for y > 2, we get
∫
∂D
(ϕ ◦ ρ)(ζ)dm(ζ) =
∞∫
0
ϕ(y)dF (y) =
2∫
0
ϕ(y)dF (y) = I(ϕ,E) < ∞. (12)
First prove that there exists a harmonic majorant for v in D. Using the
properties of the Poisson integral
U(z) =
∫
∂D
1− |z|2
|ζ − z|2 ϕ(ρ(ζ)) dm(ζ),
we obtain
lim
z→ζ
U(z) = ϕ(ρ(ζ)), ζ ∈ ∂D\E.
Hence limz→ζ(v(z)− U(z)) 6 0 for ζ ∈ ∂D\E and limz→ζ∈E U(z) = +∞.
Let Ωt, t ∈ (0, 1), be the connected component of the set {z ∈ D : ρ(z) > t}
such that 0 ∈ Ωt. Put
Γt = {z ∈ D : ρ(z) = t}, Et = {ζ ∈ ∂D : ρ(z) < t}, Ec
t = ∂D\Et.
Since Et is a finite union of disjoint open sets, it is seen that Ec
t is a finite union
of disjoint closed sets. Moreover, ∂Ωt ⊂ Ec
t ∪ Γt. Note that for the connected
{z ∈ D : ρ(z) > t} we have ∂Ωt = Ec
t ∪ Γt.
For z ∈ Γt, take ζ ′ ∈ E such that |z−ζ ′| = t. Put γz = {ζ ∈ ∂D : |ζ−ζ ′| 6 t}.
Clearly, γz ⊂ Et. Since the function ϕ(t) is positive and ϕ(ρ(ζ)) ≥ ϕ(t) on Et ,
we get
U(z) ≥
∫
γz
1− |z|2
|ζ − z|2 ϕ(ρ(ζ))dm(ζ) ≥ ϕ(t)ω(z, γz,D),
308 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3
On Analytic and Subharmonic Functions in Unit Disc
where
ω(λ, γz,D) =
∫
γz
1− |λ|2
|ζ − λ|2 dm(ζ)
is the harmonic measure at the point λ = z with respect to the arc γz (see, for
example, [3, §4.3]). On the other hand, the harmonic measure at any point λ ∈ D
with respect to the arc γz equals β/(2π), where β = β(λ) is the length of the arc
of the unit circle formed by the strait lines passing through λ and the ends of γz
(see [10, chapter I, §5]). By direct calculations, β(z) = π − 2 arcsin(t/2). Since
0 < t ≤ 1, we obtain ω(z, γz,D) ≥ 1/3 and
U(z) > ϕ(t) ω(z, γz,D) > 1
3
ϕ(t) > v(z)
3
, z ∈ Γt.
Therefore,
limz→ζ [v(z)− 3U(z)] ≤ 0, ζ ∈ Γt, limz→ζ [v(z)− U(z)] ≤ 0, ζ ∈ Ec
t .
Using the maximum principle, we get v(z) 6 3U(z) for all z ∈ Ωt.
Furthermore, by the Green formula [3, Theorem 4.5.4], we have
v(z) = ut(z)−
∫
Ωt
GΩt(z, λ)dµ(λ), (13)
where ut is the least harmonic majorant of v in Ωt. We have ut(z) 6 3U(z) for
z ∈ Ωt. The Green function GΩt(z, λ) in Ωt is equal to
GΩt(z, λ) = log 1/(|z − λ|)− ht(z, λ),
where ht(z, λ) is the harmonic function in Ωt with the boundary values log 1/|z−
λ|. By [1, proof of Theorem 1], GΩt(0, λ) ≥ (1−|λ|)/6 for λ ∈ Ωkt with k = 25 >
6π + 3. Consequently,
∫
Ωkt
(1− |λ|)dµ(λ) 6 6
∫
Ωt
GΩt(0, λ)dµ(λ) = 6ut(0)
6 18U(0) = 18
∫
∂D
ϕ(ρ(ζ))dm(ζ) = 18I(ϕ,E).
The later inequality is valid for all t > 0, hence we obtain the statement of the
theorem for the case v(0) = 0.
If v(0) > 0, consider the function v(z)− v(0) instead of v(z).
If −∞ < v(0) < 0, consider the function v1(z) = ϕ(2)
v(z)− v(0)
ϕ(2)− v(0)
. We have
v1(z) 6 ϕ(ρ(z))
1− v(0)/ϕ(ρ(z))
1− v(0)/ϕ(2)
6 ϕ(ρ(z)).
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 309
S.Ju. Favorov and L.D. Radchenko
Notice that the Riesz measure of the function v1(z) coincides with the Riesz
measure of the function v(z) up to a constant depending only on v(0). Hence the
Blaschke condition (5) for v1 implies the same condition for v.
If v(0) = −∞, consider the harmonic function h(z) in the disk {|z| < 1/2}
such that h(z) = v(z) for |z| = 1/2 and put
v1(z) =
{
max(v(z), h(z)), |z| < 1
2
v(z), |z| > 1
2 .
Clearly,
v1(z) 6 max
|z|=1/2
v(z) 6 ϕ(1/2) for |z| ≤ 1/2 and v1(0) 6= −∞.
In addition, v1(z) is subharmonic in D (see [3, Theorem 2.4.5]) and the restriction
of its Reisz measure µ1 to the set {z ∈ D : |z| > 1
2} is equal to µ. Applying the
proved statement to the function ϕ1(z) = max{ϕ(z), ϕ(1/2)}, we obtain
∫
D
(1− |λ|)dµ1(λ) < ∞.
Therefore the integral in (5) is also finite. Theorem 1 is proved.
P r o o f of Theorem 2. Arguing as above, we can consider only the case
v(0) = 0.
Let Ωt, Γt, Et, Ec
t , γz, ω(λ, γz,D) be the same as in the proof of Theorem 1.
For z ∈ D, put
Vt(z) =
∫
Ec
t
1− |z|2
|ζ − z|2 ϕ(ρ(ζ))dm(ζ) + ϕ(t)
∫
Et
1− |z|2
|ζ − z|2 dm(ζ). (14)
Since γz ⊂ Et, we get
Vt(z) > ϕ(t) ω(z, γz,D) > 1
3
ϕ(t).
Therefore,
lim
z→ζ
sup v(z) 6 lim
z→ζ
3Vt(z) = 3Vt(ζ), ζ ∈ ∂Ωt.
Using the maximum principle, we get v(z) 6 3Vt(z) for all z ∈ Ωt, in particular,
v(0) 6 3Vt(0). Clearly, we have
Vt(0) =
∫
∂D
Vt(ζ)dm(ζ) =
∫
Et
ϕ(t)dm(ζ) +
∫
Ec
t
ϕ(ρ(ζ))dm(ζ)
= ϕ(t)F (t) +
∫
Ec
t
ϕ(ρ(ζ))dm(ζ).
310 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3
On Analytic and Subharmonic Functions in Unit Disc
Using Lemma 1 with g(ζ) = ρ(ζ) and taking into account the equality
H(y) = m{ζ : ρ(ζ) < y} −m{ζ : ρ(ζ) 6 t} = F (y)− F (t),
we get
ϕ(t)F (t) +
∫
Ec
t
ϕ(ρ(ζ))dm(ζ) = ϕ(t)F (t) +
2∫
t
ϕ(y)dF (y).
Integrating by parts, we get
Vt(0) = ϕ(t)F (t) + ϕ(2)F (2)− ϕ(t)F (t)−
2∫
t
ϕ′(y)F (y)dy
= ϕ(2)−
2∫
t
ϕ′(y)F (y)dy. (15)
Therefore, using the Green formula (13) and the estimate ut(z) ≤ 3Vt(z) of the
least harmonic majorant ut in Ωt, we get
∫
Ωt
GΩt(0, λ)dµ(λ) = ut(0) 6 3Vt(0) = 3
ϕ(2)−
2∫
t
ϕ′(y)F (y)dy
. (16)
By [1, proof of Theorem 1], GΩt(0, λ) ≥ (1−|λ|)/6 for λ ∈ Ωkt with k = 25 >
6π + 3. Therefore,
∫
Ωkt
(1− |λ|)dµ(λ) 6 18
ϕ(2)−
2∫
t
ϕ′(y)F (y)dy
(17)
if kt ∈ (0, 1). In particular, the measure (1− |λ|)dµ(λ) of the set {λ ∈ D : ρ(λ) >
ε} is finite for each ε > 0. Applying Lemma 1 with the function g = ρ and taking
into account that ρ(λ) ≤ 2, we get
∫
{λ∈D :ρ(λ)>ε}
ψ(ρ(λ))(1− |λ|)dµ(λ) =
2∫
ε
ψ(t)dG(t)
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 311
S.Ju. Favorov and L.D. Radchenko
with G(t) =
∫
{λ∈D :ε6ρ(λ)<t}
(1− |λ|)dµ(λ). We have
2∫
ε
ψ(t)dG(t) = −
2∫
ε
ψ(t)d
∫
{λ:ρ(λ)>t}
(1− |λ|)dµ(λ)
= ψ(ε)
∫
{λ:ρ(λ)≥ε}
(1− |λ|)dµ(λ) +
2∫
ε
ψ′(t)
∫
{λ:ρ(λ)>t}
(1− |λ|)dµ(λ)
dt. (18)
We claim that under the condition
2∫
0
ψ′(t)
∫
{λ:ρ(λ)>t}
(1− |λ|)dµ(λ)
dt < ∞
we have
ψ(ε)
∫
{λ:ρ(λ)≥ε}
(1− |λ|)dµ(λ) → 0 as ε → 0. (19)
Indeed, for any η > 0 and sufficiently small positive δ < ε < 2,
ε∫
δ
ψ′(t)
∫
{λ:ρ(λ)>t}
(1− |λ|)dµ(λ)
dt 6 η.
Therefore,
(ψ(ε)− ψ(δ))
∫
{λ:ρ(λ)≥ε}
(1− |λ|)dµ(λ) =
ε∫
δ
ψ′(t)dt
∫
{λ:ρ(λ)≥ε}
(1− |λ|)dµ(λ)
6
ε∫
δ
ψ′(t)
∫
{λ:ρ(λ)>t}
(1− |λ|)dµ(λ)
dt 6 η.
Passing to the limit δ → 0, we obtain
ψ(ε)
∫
{λ:ρ(λ)≥ε}
(1− |λ|)dµ(λ) 6 η,
which proves (19).
312 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3
On Analytic and Subharmonic Functions in Unit Disc
By Lemma 2, if ρ(z) > t, then ρ(τz) > t/2 for all z ∈ D and 0 < τ < 1. Hence
the interval [0, z] belongs to the set {ζ : ρ(ζ) > t/2}, and {z : ρ(z) > t} ⊂ Ωt/2.
Therefore,
2∫
ε
ψ′(t)
∫
{ρ(λ)>t}
(1− |λ|)dµ(λ)
dt 6
2∫
ε
ψ′(t)
∫
Ωt/2
(1− |λ|)dµ(λ)
dt.
By (18), to prove the convergence of the integral
∫
D
ψ(ρ(λ))(1− |λ|)dµ(λ),
it is sufficient to show the convergence of the integral
2∫
0
ψ′(t)
∫
Ωt/2
(1− |λ|)dµ(λ)
dt = 2k
1/k∫
0
ψ′(2kt)
∫
Ωkt
(1− |λ|)dµ(λ)
dt.
We have
1/k∫
0
ψ′(2kt)
2∫
t
ϕ′(y)F (y)dy
dt =
1
2k
ψ(2)
2∫
1/k
ϕ′(y)F (y)dy
− lim
t→0
ψ(2kt)
2k
2∫
t
ϕ′(y)F (y)dy +
1
2k
1/k∫
0
ψ(2ky)ϕ′(y)F (y)dy
> const +
1
2k
1/k∫
0
ψ(2ky)ϕ′(y)F (y)dy.
By the condition of the theorem, the last integral is finite. The proof is complete.
P r o o f of Theorem 3. Note that the function − log ρ(z) is subharmonic,
hence the function ϕ(ρ(z)) = ϕ
(
1
e− log ρ(z)
)
is subharmonic as well.
Using the Green formula (13) for the function v0(z) in the domain Ωt, we get
ϕ(1) = v0(0) = u0
t (0)−
∫
Ωt
(log 1/|λ| − ht(0, λ)) dµ0(λ), (20)
where u0
t (z) is the least harmonic majorant for v0(z) in Ωt, and ht(0, λ) is the
solution of Dirichlet problem in Ωt with the boundary values log 1/|λ|. Clearly,
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 313
S.Ju. Favorov and L.D. Radchenko
ht(0, λ) > 0. On the other hand, if Vt(z) is the defined in (14) harmonic function
in D, then limz→ζ Vt(z) = v0(ζ) for ζ ∈ Ec
t and Vt(z) 6 ϕ(t) in D. Hence,
Vt(z) ≤ v0(z) on Γt and Vt(z) 6 u0
t (z) in Ωt. Combining equality (15) for Vt(0)
with (20), we get
ϕ(2)−
2∫
t
ϕ′(y)F (y)dy 6 ϕ(1) +
∫
Ωt
log
1
|λ|dµ0(λ).
Note that Ωt ⊂ {λ : ρ(λ) > t} and log
1
|λ| 6 2(1− |λ|) for |λ| ≥ 1/2. Hence,
−
2∫
t
ϕ′(y)F (y)dy 6 2
∫
{λ:ρ(λ)>t}
(1− |λ|)dµ0(λ) + const. (21)
On the other hand, arguing as in the proof of Theorem 2, we get
∫
{λ:ρ(λ)≥ε}
ψ(ρ(λ))(1− |λ|)dµ0(λ)
= ψ(ε)
∫
{λ:ρ(λ)≥ε}
(1− |λ|)dµ0(λ) +
2∫
ε
ψ′(t)
∫
{λ:ρ(λ)>t}
(1− |λ|)dµ0(λ)
dt.
Using (21), we get
−ψ(ε)
2∫
ε
ϕ′(y)F (y)dy −
2∫
ε
ψ′(t)
2∫
t
ϕ′(y)F (y)dy
dt
6 2
∫
{λ:ρ(λ)≥ε}
ψ(ρ(λ))(1− |λ|)dµ0(λ) + const.
Integrating by parts in t, we obtain
−
2∫
ε
ψ(t)ϕ′(t)F (t)dt 6 2
∫
{λ:ρ(λ)≥ε}
ψ(ρ(λ))(1− |λ|)dµ0(λ) + const.
Proceeding here to the limit as ε → 0 and using the assumption of the theorem,
we complete the proof.
314 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3
On Analytic and Subharmonic Functions in Unit Disc
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