Planar Lebesgue Measure of Exceptional Set in Approximation of Subharmonic Functions
We consider the pointwise approximation of a subharmonic function having ¯nite order by the logarithm of the modulus of an function up to a bounded quantity. We prove an estimate from below of the planar Lebesgue measure of the exceptional Set in such approximation.
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irk-123456789-1065472016-10-01T03:01:58Z Planar Lebesgue Measure of Exceptional Set in Approximation of Subharmonic Functions Girnyk, M. We consider the pointwise approximation of a subharmonic function having ¯nite order by the logarithm of the modulus of an function up to a bounded quantity. We prove an estimate from below of the planar Lebesgue measure of the exceptional Set in such approximation. 2009 Article Planar Lebesgue Measure of Exceptional Set in Approximation of Subharmonic Functions / M. Girnyk // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 4. — С. 347-358. — Бібліогр.: 13 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106547 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We consider the pointwise approximation of a subharmonic function having ¯nite order by the logarithm of the modulus of an function up to a bounded quantity. We prove an estimate from below of the planar Lebesgue measure of the exceptional Set in such approximation. |
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Girnyk, M. Planar Lebesgue Measure of Exceptional Set in Approximation of Subharmonic Functions Журнал математической физики, анализа, геометрии |
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Girnyk, M. |
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Girnyk, M. |
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Planar Lebesgue Measure of Exceptional Set in Approximation of Subharmonic Functions |
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Planar Lebesgue Measure of Exceptional Set in Approximation of Subharmonic Functions |
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Planar Lebesgue Measure of Exceptional Set in Approximation of Subharmonic Functions |
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Planar Lebesgue Measure of Exceptional Set in Approximation of Subharmonic Functions |
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Planar Lebesgue Measure of Exceptional Set in Approximation of Subharmonic Functions |
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planar lebesgue measure of exceptional set in approximation of subharmonic functions |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Planar Lebesgue Measure of Exceptional Set in Approximation of Subharmonic Functions / M. Girnyk // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 4. — С. 347-358. — Бібліогр.: 13 назв. — англ. |
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Журнал математической физики, анализа, геометрии |
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AT girnykm planarlebesguemeasureofexceptionalsetinapproximationofsubharmonicfunctions |
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2025-07-07T18:38:00Z |
| last_indexed |
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| _version_ |
1837014430305484800 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2009, vol. 5, No. 4, pp. 347–358
Planar Lebesgue Measure of Exceptional Set
in Approximation of Subharmonic Functions
Markiyan Girnyk
Lviv Academy of Commerce
8 Tuhan-Baranovskyi Str., Lviv, 79005, Ukraine
E-mail:hirnyk@lac.lviv.ua
Received November 17, 2008
We consider the pointwise approximation of a subharmonic function
having finite order by the logarithm of the modulus of an function up to a
bounded quantity. We prove an estimate from below of the planar Lebesgue
measure of the exceptional Set in such approximation.
Key words: subharmonic functions, entire functions, approximation.
Mathematics Subject Classification 2000: 31C05, 30E10.
Results on approximation of a subharmonic function by the logarithm of the
modulus of an entire function have numerous applications in complex analysis
and potential theory (see, for example, [1–6]). The pointwise approximation
is possible only outside an exceptional set, and for this reason, the principal
question concerning its minimal size arises. In this article we prove that the planar
Lebesgue measure of an exceptional set in approximation of the subharmonic
function |z|ρ by the logarithm of the modulus of an entire function of at most
order ρ and normal type cannot be arbitrary small in a certain sense.
We use the main results and standard notations of potential theory [7] and
theory of distribution of values [8]. Let us recall some of them. We denote
by D(a, r) := {z : |z − a| < r}, C(a, r) := {z : |z − a| ≤ r}, S(a, r) := {z :
|z − a| = r}, A(t, T ] := {z : t < |z| ≤ T}, md the Lebesgue measure on Rd,
letters C with indices stand for positive constants, in parentheses we indicate
dependence on parameters. As usually, a+ = max(a, 0), a− = max(−a, 0). Let u
be a subharmonic function, then µu is its Riesz measure, B(r, u) := max{u(z) :
z ∈ C(0, r)} is the maximum, n(a, r, u) := µ(C(a, r)), n(r) := n(0, r, u) are the
counting functions of the Riesz measure, h(z, u,D) is the minimal harmonic ma-
jorant of the function u in domain D (which is sometimes omitted in notations),
T (r, u) := 1
2π
∫ 2π
0 u+(reiϕ) dϕ is the Nevanlinna characteristic of u. We notice
c© Markiyan Girnyk, 2009
Markiyan Girnyk
that the Nevanlinna characteristic of a meromorphic function f is also denoted
by T (r, f). It will not make any difficulties for the readers as it is clear from the
context which characteristic is used. We denote by Λ the class of nondecreas-
ing slowly changing functions λ : [1,∞) 7→ [1,∞) (in particular, λ(2r) ∼ λ(r) if
r →∞).
The notation a ³ b means that |a| ≤ const · |b| and |b| ≤ const · |a|. Let Θ ⊃ Λ
be the class of nondecreasing functions λ : [1,∞) 7→ [1,∞) having the property:
λ(2R) ³ λ(R). This implies the finiteness of order
τ = lim sup
R→∞
log λ(R)
log R
of the function λ. The content of this paper is closely associated with the theorem
by I. Chyzhykov [9], which strengthens and specifies the result by Yu. Lubarskii
and Eu. Malinnikova [10], as well as with the theorem by R. Yulmukhametov [11].
For the reader’s convenience we quote these theorems in somewhat modificated
but equivalent formulations.
Theorem A. Let u be a subharmonic function with the Riesz measure µu.
If for a function λ ∈ Λ there exists a number R0, such that for every number
R > R0 the condition
µu(A(R, Rλ(R)]) > 1 (1)
holds, then there exists an entire function f and a constant C1 satisfying
∫
A(R,2R]
|u(z)− log |f(z)|| dm2(z) < C1R
2 log λ(R). (2)
Moreover, for every real number ε > 0 there exists a constant C2(ε) and a set
E(ε) such that
|u(z)− log |f(z)|| < C2(ε) log λ(|z|), z /∈ E(ε), (3)
and
m2(E(ε) ∩A(R, 2R])/R2 < ε. (4)
Theorem B. Let u be a subharmonic function of finite order ρ and a number
α > ρ. Then there exists an entire function f , a constant C3 = a0 + a1α, a1 > 1,
depending only on α , and an exceptional set E depending on the functions u, f
and the number α such that
|u(z)− log |f(z)|| ≤ C3(α) log |z|, z /∈ E, (5)
where E ⊂ ∪jD(zj , rj) and
∑
R<|zj |≤2R
rj = o(Rρ−α), R →∞. (6)
348 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Planar Measure of Exceptional Set
Let us formulate the result of our work.
Theorem 1. For all real numbers ε > 0, ρ > 0, and for every entire function
f satisfying the condition
B(r, log |f |) < C4r
ρ, (7)
for every function λ ∈ Θ of order τ , and for any measurable set E, the condition
||z|ρ − log |f(z)|| < C5 log λ(|z|), z /∈ E, (8)
implies the existence of a constant C6(ε) such that
m2(E ∩A(R, 2R]) > C6(ε)Rχ+ρ, R > 1, (9)
where χ = min(2− 2ρ− ε, 2− ρ− 4C5τ − ε).
As the formulation of Theorem 1 is cumbersome, we write its statement in
the important case of a bounded function λ (its order τ = 0, then χ = 2−2ρ−ε):
m2(E ∩A(R, 2R]) > C6(ε)R2−ρ−ε, R > 1.
Let us comment the content of Theorem 1. It states that the number ε on
the right-hand side of (4) cannot be replaced by an arbitrary function ε(R) →
0, R →∞. Then condition (7) seems to be natural because in Theorem 1 it holds
for subharmonic functions of finite order ρ and normal type 1, following from (2).
Indeed, let us consider the inequalities
2πT (r, f) ≤
2π∫
0
| log |f(reiϕ)|| dϕ + O(1)
≤
2π∫
0
(| log |f(reiϕ)| − u(reiϕ)|+ u(reiϕ)) dϕ + O(1),
which imply
2πR2T (R, f) < 2π
2R∫
R
T (r, f)r dr ≤
∫
A(R,2R]
| log |f(z)| − u(z)| dm2(z)
+
∫
A(R,2R]
u(z) dm2(z) + O(R2) ≤ C1R
2 log λ(R) + O(Rρ+2) + O(R2),
R →∞,
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 349
Markiyan Girnyk
and growths of the functions B(r, log |f |) and T (r, f) are equal. We also note the
possibility of good approximation of a subharmonic function with finite order by
the logarithm of the modulus of some entire function with infinite order. We give
a simple example. Let
||u(z)− log |f(z)|| < C5 log λ(|z|), z /∈ E,
where u is a subharmonic function of finite order, f is an entire function, then
||u(z)− log |f(z) · V (z)|| < C5 log λ(|z|) + 2,
z /∈ E
⋃
S := {z = x + iy : x ≥ 1, |y| ≤ π},
where (see [8, p. 256–258]) the function
V (z) =
{
exp(exp z) + Ψ1(z)/z, z ∈ S;
Ψ2(z)/z, z 6∈ S;
and |Ψj(z)| < 2, |z| > r0 > 1, j = 1, 2. We also notice that in [11] and [12] the
estimates from below of the sum of radii for any disk covering of the exceptional
set are obtained, but no estimate from below of the planar measure follows from
those estimates.
We cite two above mentioned results.
Theorem C. Let a number ε > 0 and an entire function f satisfy the in-
equality
||z| − log |f(z)|| = o(log |z|), E 63 z →∞,
where E ⊂ ⋃
j{z : |z − zj | < rj}, and radii rj are uniformly bounded. Then the
estimate ∑
R≤|zj |<2R
rj > R1−ε, R > R(ε),
holds.
Theorem D. Let numbers ρ > 0, ε > 0, and an entire function f satisfy the
inequality
||z|ρ − log |f(z)|| < C5 log |z|, z /∈ E,
where E ⊂ ⋃
j{z : |z − zj | < rj}, rj ≤ |zj |1−ρ/2+ε. Then the estimate
∑
R≤|zj |<2R
rj > R1+ρ/2−2C5−ε, R > R(ε),
holds.
350 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Planar Measure of Exceptional Set
Let us consider the accuracy of Theorem 1. From Theorem B it follows
that there exists an entire function f and an exceptional set E satisfying
(8) with λ(R) = R, τ = 1, and m2(E
⋂
A(R, 2R]) = o(R2ρ−2C5/a1−2a0/a1),
R → ∞. We draw a conclusion that the planar Lebesgue measure of
the exceptional set in the annulus A(R, 2R] is a power function of R with the
exponent ³ C5. The dependence of the exponent on ρ is not clear.
P r o o f o f T h e o r e m 1. We begin with the idea of the proof. At first, we
prove that any disk of the form D(a, |a|1−ρ/2), where f(a) = 0, contains a rather
big portion of the exceptional set, namely, the estimate
m2(E ∩D(a, |a|1−ρ/2)) > C7|a|χ (10)
holds. The proof of estimate (10) is the key point of the proof of Theorem 1.
Here we follow the arguments from [12]. A new approach is that we use the
theorem by Edrei and Fuchs on the integral over a small set [13, 8]. Next, it is
proved that every disk with a somewhat greater radius has the same property
without the demand that the center of the disk is zero of the function f . More
exactly, we prove that for every b, |b| > R1, and every ε > 0 the inequality
m2(E ∩D(b, |b|1−ρ/2+ε/2)) > C8|b|χ (11)
holds. To finish the proof, we put sufficiently many nonoverlapping disks with
enlarged radii into the annulus A(R, 2R). By comparing the areas of the annulus
and the disks we obtain estimate (9).
We denote r(a) := |a|1−ρ/2, v(z) := |z|ρ. Let h(z, v) and h(z, log |f |) be the
minimal harmonic majorants, respectively v and log |f |, in the disk D(a, t), t ∈
[(1 − ε/4)r(a), r(a)]. Under the conditions of Theorem 1 we prove the estimate
(0 < δ < 2π)
|h(z, v)− h(z, log |f |)| ≤ C9 log λ(|a|) + C10δ|a|ρ log |a| log
(
2πe
δ
)
,
z ∈ D(a, ε./4r(a)). (12)
By the Poisson–Jensen formula for the disk D(a, t) we have
|h(z, v)− h(z, log |f |)|
=
1
2π
∣∣∣∣∣∣
2π∫
0
(| a + teiϕ|ρ − log |f(a + teiϕ)|)< teiϕ + z − a
teiϕ − z + a
dϕ
∣∣∣∣∣∣
≤ 1
2π
2π∫
0
∣∣| a + teiϕ|ρ − log |f(a + teiϕ)|∣∣ r(a) + r(a)ε/4
(1− ε/4)r(a)− r(a)ε/4
dϕ
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 351
Markiyan Girnyk
≤ 1 + ε
2π
2π∫
0
∣∣| a + teiϕ|ρ − log |f(a + teiϕ)|∣∣ dϕ. (13)
By E(t, a) we denote the set
{
ϕ ∈ [0, 2π] : || a + teiϕ|ρ − log |f(a + teiϕ)|| ≥ C5 log λ(| a + teiϕ|)
> C5(1− ε) log λ(|a|)} (14)
(the last inequality in definitions (14) and the ones below follows from the pro-
perties of the function λ ∈ Θ and restrictions on t for all sufficiently large values
|a|). Its complement is
[0, 2π] \ E(t, a) :=
{
ϕ ∈ [0, 2π] : || a + teiϕ|ρ − log |f(a + teiϕ)||
< C5 log λ(| a + teiϕ|) < C5(1 + ε)λ(|a|)} .
Now we continue estimate (13):
|h(z, v)− h(z, log |f |)| ≤ 1 + ε
2π
∫
E(t,a)
∣∣| a + teiϕ|ρ − log |f(a + teiϕ)|∣∣ dϕ
+
1 + ε
2π
∫
[0,2π]\E(t,a)
∣∣| a + teiϕ|ρ − log |f(a + teiϕ)|∣∣ dϕ
≤ 1 + ε
2π
∫
E(t,a)
(| a + teiϕ|ρ + log+ |f(a + teiϕ)|+ log− |f(a + teiϕ)|) dϕ
+(1 + ε)2C5 log λ(|a|). (15)
We use the theorem by Edrei and Fuchs [13], [8, p. 58] quoted below for the
reader’s convenience.
Theorem E. Let f be a meromorphic function , k and δ be real numbers,
k > 1, 0 < δ < 2π, r > 1. For any measurable set Er ⊂ [0, 2π], such that
m1(Er) ≤ δ, the relation
∫
Er
log+ |f(reiϕ)| dϕ ≤ 6k
k − 1
δ log
2πe
δ
T (kr, f) (16)
holds.
352 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Planar Measure of Exceptional Set
We notice that the analysis of the proof of Theorem E shows that its
δ-subharmonic version is valid. The assumption r > 1 is of technical character;
without it the term
δ log
2
√
kπe
δ
n(0, f)| log r|/ log
√
k
should be added to the right-hand side of (16) (see the proof of Lemma 7.1 in
[8, p. 55]). For more completeness we give a proof of the above mentioned
modification of Theorem E. We begin with the inequalities (R′ > R > 0)
N(R′) ≥
R′∫
R
n(t, f)− n(0, f)
t
dt + n(0, f) log R′
≥ (n(R, f)− n(0, f)) log
R′
R
+ n(0, f) log R,
from which the estimate
n(R, f) ≤ N(R′, f)
log R′
R
− n(0, f) log R
log R′
R
follows. In the proof of Lemma 7.1 it is supposed that R > 1, and because of
this the negative term −n(0, f) log R is omitted, but here it is taken into account.
Then, in the proof of Theorem E in [8] the term
n(
√
kr, f)δ log
2
√
keπ
δ
is obtained. To estimate n(
√
kr, f) we apply the previous inequality with R′ = kr,
R =
√
kr and obtain
n(
√
kr, f)δ log
2
√
keπ
δ
≤ δ log
2
√
keπ
δ
(
N(kr)
log
√
k
− n(0, f) log(
√
kr)
log
√
k
)
≤ δ log
2
√
keπ
δ
(
N(kr)
log
√
k
− n(0, f) log r
log
√
k
)
≤ δ log
2
√
keπ
δ
(
N(kr)
log
√
k
+
n(0, f)| log r|
log
√
k
)
,
then we follow the proof in [8].
Now continue to estimate (15). The integral
∫
E(t,a)
|a + teiϕ|ρ dϕ ≤ (1 + ε)|a|ρδ (17)
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 353
Markiyan Girnyk
if |a| is sufficiently large. To estimate the integral
∫
E(t,a)
(
log+ |f(a + teiϕ)|+ log− |f(a + teiϕ)|) dϕ
we apply more precise Theorem E, putting k = 2 and taking into account the
relation T (r, f) = T (r, 1/f) + O(1) and (7). We obtain the estimate
∫
E(t,a)
(
log+ |f(a + teiϕ)|+ log− |f(a + teiϕ)|) dϕ
=
∫
E(t,a)
(
log+ |f(a + teiϕ)|+ log+
∣∣∣∣
1
f(a + teiϕ)
∣∣∣∣
)
dϕ
≤ 12δ log
2πe
δ
(2T (2t, f(z + a))+O(1))+ δ log
2
√
2πe
δ
n(0, f(z + a))| log t|/ log
√
2
< 24δ log
2πe
δ
3ρC4|a|ρ + 6δ log
2
√
2πe
δ
C42ρ|a|ρ|1− ρ/2| log |a|. (18)
Combining (15), (17), and (18), we have
|h(z, v)− h(z, log |f |)| ≤ C11δ log
2πe
δ
|a|ρ log |a|+ C5(τ + ε) log |a| (19)
if |a| is sufficiently large and z ∈ D(a, εr(a)/4).
The next step is to find the upper bound of the difference log |f(z)|−h(z, log |f |)
for z ∈ D(a, εr(a)/4)\E. This estimate is obtained only in indirect
way. Using the standard tools of calculus, we can prove (see [12]) that for
z ∈ A := A(R− r(R), R + r(R)] the inequality
|v(z)− h(z, v, A)| ≤ C12 (20)
holds, where
h(z, v, A) := (R + r(R))ρ log |z| − log(R− r(R))
log(R + r(R))− log(R− r(R))
+(R− r(R))ρ − log |z|+ log(R + r(R))
log(R + r(R))− log(R− r(R))
is the minimal harmonic majorant of the function v(z) := |z|ρ in the annulus A.
From (20) and the definition of the minimal harmonic majorant it follows that
|v(z)− h(z, v,D(a, t))| ≤ C12. (21)
354 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Planar Measure of Exceptional Set
Applying (19), (21), and (8), we obtain the estimate
| log |f(z)| − h(z, log |f |)| ≤ |h(z, v)− h(z, log |f |)|+ |v(z)− h(z, v)|
+|v(z)− log |f(z)|| ≤ C11δ log
2πe
δ
|a|ρ log |a|+ C5(τ + ε) log |a|+ C12
+C5(τ + ε) log |a| (22)
if z ∈ D(a, εr(a)/4) \ E.
Now we prove the estimate from below for the difference | log |f(z)|−h(z, log |f |)|.
By the Poisson–Jensen formula
log |f(z)| = h(z, log |f |, D(a, t))−
∑
an∈D(a, t)
g(z, an), (23)
where g(z, an) is the Green function of the disk D(a, t) with the pole in zero an
of the function f . Using the known properties of the Green function, from (23)
we obtain
| log |f(z)| − h(z, log |f |, D(a, t))| ≥ g(z, a) = log
t
|z − a| . (24)
We face the alternative: either for every t ∈ [(1 − ε)r(a), r(a)] the measure
m1(E(t, a)) ≥ δ, or there exists t ∈ [(1 − ε)r(a), r(a)] for which the measure
m1(E(t, a)) < δ, where δ = ε(|a|ρ log |a|)−1. In the first case the planar Lebesgue
measure
m2(E ∩D(a, r(a))) ≥ εr(a) · εr(a)(|a|ρ log |a|)−1 = ε2 |a|2−2ρ
log |a| . (25)
In the second case, as it follows from (24), for z ∈ D(a, (1− ε)r(a)/|a|κ),
κ = 2C5(τ + ε) + 2C11ε, the estimate
| log |f(z)| − h(z, log |f |)| ≥ κ log |a| (26)
takes place, and from (22) we obtain that
| log |f(z)| − h(z, log |f |)| ≤ C11ε log |a|+ 2C5(τ + ε) log |a|, (27)
if z ∈ D(a, εr(a)/4) \ E. Comparing (26) and (27), we conclude that the disk
D(a, (1 − ε)r(a)/|a|κ) ⊂ E. Since its area equals π(1 − ε)2|a|2−ρ−2κ, and the
planar Lebesgue measure of the portion of the exceptional set E ∩ D(a, r(a))
does not exceed ε2|a|2−2ρ/ log2 |a| in the first case, then in any case (it is clear
that const · ε can be replaced by ε)
m2(D(a, r(a)) ∩ E) ≥ C13|a|χ, χ = min(2− 2ρ− ε, 2− ρ− 4C5τ − ε). (28)
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 355
Markiyan Girnyk
We put r1(b) := r(b)|b|ε/2. For arbitrary disk of the form D(b, r1(b)) with
sufficiently large |b| we prove the estimate
m2(D(b, r1(b)) ∩ E) ≥ C14|b|χ. (29)
Without lost of generality, we may suppose b /∈ E. In the opposite case
either D(b, r(b)) ⊂ E and then (29) holds, or there exists c ∈ D(b, r(b)) \ E.
It is important that the disk D(c, 3
4r1(c)) ⊂ D(b, r1(b)), this is used under.
Suppose the disk D(c, 3
4r1(c)) does not contain zeros of the entire function f ,
then n(c, t, log |f |) = 0 for t ∈ [0, 3
4r1(c)].
The Poisson–Jensen formula for the difference |z|ρ − log |f(z)| in the disk
D(c, t/2) has the form
−|c|ρ + log |f(c)|+ 1
2π
2π∫
0
(∣∣∣∣c +
1
2
teiϕ
∣∣∣∣
ρ
− log
∣∣∣∣f
(
c +
1
2
teiϕ
)∣∣∣∣
)
dϕ
=
t/2∫
0
n(c, s, |z|ρ)
s
ds. (30)
The estimating of the integral on the left-hand side of (30) similarly to the
one in (13) results
∣∣∣∣∣∣
1
2π
2π∫
0
(∣∣∣∣c +
1
2
teiϕ
∣∣∣∣
ρ
− log
∣∣∣∣f(c +
1
2
teiϕ)
∣∣∣∣
)
dϕ
∣∣∣∣∣∣
≤ C11δ log
2πe
δ
T (t, f(w + c)) + C5(τ + ε) log |c|+ C15δ|c|ρ, (31)
where m1(E(t/2, c)) ≤ δ. Next, for t ∈ [0, 3
4r1(c)] the estimate
1
2π
2π∫
0
log+
∣∣∣∣f
(
c +
1
2
teiϕ
)∣∣∣∣ dϕ ≤ C4(|c|+ t/2)ρ ≤ C42ρ|c|ρ (32)
takes place if |c| is sufficiently large. Here we make use of (7). From (32) and the
definition of the Nevanlinna characteristic of a meromorphic function we obtain
the estimate
T (t, f(w + c)) ≤ 2ρC4|c|ρ. (33)
We put δ := |c|−ρ. Again we face the alternative: either for every t ∈
[12r1(c), 3
4r1(c)] the measure m1(E(t/2, c)) ≥ δ, or there exists t ∈ [12r1(c), 3
4r1(c)]
for which m1(E(t/2, c)) < δ. In the first case
m2(E ∩D(b, r1(b))) ≥ m2
(
E ∩D
(
c,
3
4
r1(c)
))
≥ |c|−ρ 1
4
r1(c)
1
4
r1(c)
356 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Planar Measure of Exceptional Set
³ |c|2−2ρ+ε ³ |b|2−2ρ+ε.
In the second case (31) and (33) imply the inequality
∣∣∣∣∣∣
1
2π
2π∫
0
∣∣∣∣c +
1
2
teiϕ
∣∣∣∣
ρ
− log+
∣∣∣∣f
(
c +
1
2
teiϕ
)∣∣∣∣ dϕ
∣∣∣∣∣∣
≤ C16 log |c|+ C5(τ + ε) log |c|.
(34)
Since c /∈ E, therefore
||c|ρ − log |f(c)|| ≤ C5(τ + ε) log |c|. (35)
Combining (34) and (35), we conclude that the right-hand side of (30) does
not exceed C17 log |c|.
On the other hand, for the function v(z) = |z|ρ its Riesz measure dµv(z) =
1
2π∆v ³ |z|ρ−2 dm2(z), and because of this n(c, s, v) ³ |c|ρ−2s2 if s ≤ 3
4r1(c), and
the right-hand side of (30), i.e.,
∫ t/2
0
n(c,s,v)
s ds ³ |c|ρ−2r1(c)2 ³ |c|ε(t ≥ 1
2r1(c)),
what contradicts the previous estimate. We draw a conclusion that there exists
a zero a of the entire function f such that a ∈ D(c, 3
4r1(c)). If |b| is a sufficiently
large number, then D(a, r(a)) ⊂ D(b, r1(b)). In any case, the measure
m2(E ∩D(b, r1(b))) ≥ C8|b|χ.
To finish the proof of Theorem 1, into the annulus A[R, 2R) we put nonoverlap-
ping disks D(b, r1(b))) at a rate of ³ R2
R2−ρ+ε = Rρ−ε. The union of these disks
contains such a portion of the exceptional set E that
m2(E ∩A[R, 2R)) ≥ C6(ε)Rχ+ρ−ε.
Theorem 1 is proved.
Acknowledgement. I would like to thank all the referees of the paper.
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