CUBO A Mathematical Journal
Vol.11, No¯ 04, (127–136). September 2009

On an inequality related to the radial growth of subharmonic
functions

Juhani Riihentaus

Department of Physics and Mathematics,
University of Joensuu,

P.O. Box 111, FI-80101 Joensuu, Finland.
email: juhani.riihentaus@joensuu.fi

ABSTRACT

It is a classical result that every subharmonic function, defined and Lp-integrable for
some p, 0 < p < +∞, on the unit disk D of the complex plane C is for almost all
θ of the form o((1 − |z|)−1/p), uniformly as z → eiθ in any Stolz domain. Recently
Pavlović gave a related integral inequality for absolute values of harmonic functions,

also defined on the unit disk in the complex plane. We generalize Pavlović’s result to

so called quasi-nearly subharmonic functions defined on rather general domains in Rn,

n ≥ 2.

RESUMEN

Es un resultado clásico que toda función subarmónica definida y Lp-integrable para
algún p, 0 < p < +∞, sobre el disco unitario D del plano complejo C es para casi todo
θ de la forma o((1 − |z|)−1/p), uniformemente cuando z → eiθ en cualquier dominio
de Stolz. Recientemente, Pavlović encontró una desigualdad integral relacionada para

valores absolutas de funciones armónicas, también definidas en el disco unitario del

plano complejo. Generalizamos el resultado de Pavlović a las así llamada funciones

subarmónicas casi-cercanas definidas en dominios bastante generales en Rn, n ≥ 2.

Key words and phrases: Subharmonic function, quasi-nearly subharmonic function, accessible
boundary point, approach region, integrability condition, radial order.



128 Juhani Riihentaus CUBO
11, 4 (2009)

Math. Subj. Class.: 31B25, 31B05.

1 Introduction

1.1 Previous results. The following theorem is a special case of the original result of Gehring

[4, Theorem 1, p. 77], and of Hallenbeck [5, Theorems 1 and 2, pp. 117-118], and of the later and

more general results of Stoll [23, Theorems 1 and 2, pp. 301-302, 307]:

Theorem A: If u is a function harmonic in D such that

I(u) :=

∫

D

| u(z) |p (1− | z |)β dm(z) < +∞, (1)

where p > 0, β > −1, then
lim

r→1−
| u(reiθ) |p (1 − r)β+1 = 0 (2)

for almost all θ ∈ [0, 2π).

Observe that Gehring, Hallenbeck and Stoll in fact considered subharmonic functions and

that the limit in (2) was uniform in Stolz approach regions (in Stoll’s result in even more general

regions). For a more general result, see [19, Theorem, p. 31], [15, Theorem, p. 233], [10, Theorem 2,

p. 73] and [18, Theorem 3.4.1, pp. 198-199].

With the aid of [12, Theorem A and Theorem 1, pp. 433-434], Pavlović showed that the

convergence in (2) in Theorem A is dominated. At the same time he pointed out that whole

Theorem A follows from his result:

Theorem B ([12, Theorem 1, pp. 433-434]) If u is a function harmonic in D satisfying (1),

where p > 0, β > −1, then

J(u) :=

2π∫

0

sup
0<r<1

| u(reiθ) |p (1 − r)β+1 dθ < +∞.

Moreover, there is a constant C = Cp,β such that J(u) ≤ C I(u).

The purpose of this note is to point out that, with the aid of [19, Theorem, p. 31], one can

extend Theorem B considerably: Instead of absolute values of harmonic functions on the unit disk

D in the complex plane C we will consider nonnegative quasi-nearly subharmonic functions defined

on rather general domains of Rn, n ≥ 2. See Theorems 1 and 2 below.

First the necessary notation and definitions.

1.2 Notation. Our notation is fairly standard, see e.g. [19, 21, 6]. However, for convenience

of the reader we recall the following. The common convention 0 · ∞ = 0 is used. The complex
space Cn is identified with the real space R2n, n ≥ 1. In the sequel D is an arbitrary domain in Rn,



CUBO
11, 4 (2009)

On an inequality related to the radial growth of subharmonic ... 129

n ≥ 2, D 6= Rn, whereas Ω is a bounded domain in Rn whose boundary ∂Ω is Ahlfors-regular with
dimension d, 0 ≤ d ≤ n (for the definition of this see 1.6 below). The distance from x ∈ D to ∂D
is denoted by δ(x). If ρ > 0 write Dρ = {x ∈ D : δ(x) < ρ}. Bn(x,r) is the Euclidean ball in Rn,
with center x and radius r, and B(x) = Bn(x, 1

3
δ(x)). We write Bn = B(0, 1) and Sn−1 = ∂Bn.

m is the Lebesgue measure in Rn, and νn = m(B
n
). L1

loc
(D) is the space of locally (Lebesgue)

integrable functions on D. The d-dimensional Hausdorff (outer) measure in Rn is denoted by Hd,

0 ≤ d ≤ n. Our constants C and K are always positive, mostly ≥ 1 and they may vary from line
to line. (One exception: In the proof of Theorem 2 we write K for ∂Ω, just in order to follow our

previous notation in [19].) On the other hand, C0 and r0 are fixed constants which are involved

with the used (and thus fixed) admissible function ϕ (see 1.5 (5) below). Similarly, if α > 0 is

given, C1 = C1(C0,α), C2 = C2(C0,α) and C3 = C3(C0,α) are fixed constants, coming directly

from [19, Lemma 2.3, pp. 32-33] or [15, Lemma 2.3, p. 234], and thus defined already there.

1.3 Nearly subharmonic functions. We recall that an upper semicontinuous function

u : D → [−∞, +∞) is subharmonic if for all Bn(x,r) ⊂ D,

u(x) ≤ 1
νn r

n

∫

Bn(x,r)

u(y) dm(y).

The function u ≡ −∞ is considered subharmonic.
We say that a function u : D → [−∞, +∞) is nearly subharmonic, if u is Lebesgue measurable,

u
+ ∈ L1

loc
(D), and for all Bn(x,r) ⊂ D,

u(x) ≤ 1
νn r

n

∫

Bn(x,r)

u(y) dm(y).

Observe that in the standard definition of nearly subharmonic functions one uses the slightly

stronger assumption that u ∈ L1
loc

(D), see e.g. [6, p. 14]. However, our above, slightly more

general definition seems to be more useful, see [21, Proposition 2.1 (iii) and Proposition 2.2 (vi),

(vii), pp. 54-55].

1.4 Quasi-nearly subharmonic functions. A Lebesgue measurable function u : D →
[−∞, +∞) is K-quasi-nearly subharmonic, if u+ ∈ L1

loc
(D) and if there is a constant K =

K(n,u,D) ≥ 1 such that for all Bn(x,r) ⊂ D,

uM (x) ≤
K

νn r
n

∫

Bn(x,r)

uM (y) dm(y) (3)

for all M ≥ 0, where uM := sup{u,−M} + M. A function u : D → [−∞, +∞) is quasi-nearly
subharmonic, if u is K-quasi-nearly subharmonic for some K ≥ 1.

A Lebesgue measurable function u : D → [−∞, +∞) is K-quasi-nearly subharmonic n.s. (in
the narrow sense), if u+ ∈ L1

loc
(D) and if there is a constant K = K(n,u,D) ≥ 1 such that for all



130 Juhani Riihentaus CUBO
11, 4 (2009)

Bn(x,r) ⊂ D,
u(x) ≤ K

νn r
n

∫

Bn(x,r)

u(y) dm(y). (4)

A function u : D → [−∞, +∞) is quasi-nearly subharmonic n.s., if u is K-quasi-nearly subhar-
monic n.s. for some K ≥ 1.

Quasi-nearly subharmonic functions (perhaps with a different terminology), or, essentially,

perhaps just functions satisfying a certain generalized mean value inequality, more or less of the

form (3) or (4) above, have previously been considered or used at least in [3, 25, 8, 14, 24, 5, 11, 9,

23, 15, 10, 16, 17, 18, 13, 19, 20, 21, 7]. We recall here only that this function class includes, among

others, subharmonic functions, and, more generally, quasisubharmonic and nearly subharmonic

functions (for the definitions of these, see above and e.g. [6]), also functions satisfying certain

natural growth conditions, especially certain eigenfunctions, polyharmonic functions, subsolutions

of certain general elliptic equations. Also, the class of Harnack functions is included, thus, among

others, nonnegative harmonic functions as well as nonnegative solutions of some elliptic equations.

In particular, the partial differential equations associated with quasiregular mappings belong to

this family of elliptic equations, see Vuorinen [26]. Observe that already Domar [2] has pointed

out the relevance of the class of (nonnegative) quasi-nearly subharmonic functions.

To motivate the reader still further, we recall here the following, see e.g. [13, Proposition 1,

Theorem A, Theorem B, p. 91] and [21, Proposition 2.1 and Proposition 2.2, pp. 54-55]:

(i) A K-quasi-nearly subharmonic function n.s. is K-quasi-nearly subharmonic, but not neces-

sarily conversely.

(ii) A nonnegative Lebesgue measurable function is K-quasi-nearly subharmonic if and only if it

is K-quasi-nearly subharmonic n.s.

(iii) A Lebesgue measurable function is 1-quasi-nearly subharmonic if and only if it is 1-quasi-

nearly subharmonic n.s. and if and only if it is nearly subharmonic (in the sense defined

above).

(iv) If u : D → [0, +∞) is quasi-nearly subharmonic and p > 0, then up is quasi-nearly sub-
harmonic. Especially, if h : D → R is harmonic and p > 0, then | h |p is quasi-nearly
subharmonic.

(v) If u : D → [−∞, +∞) is quasi-nearly subharmonic n.s., then either u ≡ −∞ or u is finite
almost everywhere in D, and u ∈ L1

loc
(D).

1.5 Admissible functions. A function ϕ : [0, +∞) → [0, +∞) is admissible, if it is strictly
increasing, surjective and there are constants C0 = C0(ϕ) ≥ 1 and r0 > 0 such that

ϕ(2t) ≤ C0 ϕ(t) and ϕ−1(2s) ≤ C0 ϕ−1(s) (5)



CUBO
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On an inequality related to the radial growth of subharmonic ... 131

for all s, t, 0 ≤ s, t ≤ r0.

Functions ϕ1(t) = t
τ , τ > 0, or, more generally, nonnegative, increasing surjective functions

ϕ2(t) which satisfy the ∆2-condition and for which the functions t 7→ ϕ2(t)t are increasing, are
examples of admissible functions. Further examples are ϕ3(t) = ct

α
[log(δ + t

γ
)]

β
, where c > 0,

α > 0, δ ≥ 1, and β,γ ∈ R are such that α + βγ > 0. For more examples, see [15, 18].

Let ϕ : [0, +∞) → [0, +∞) be an admissible function and let α > 0. One says that ζ ∈ ∂D is
(ϕ,α)-accessible, shortly accessible, if

Γϕ(ζ,α) ∩ Bn(ζ,ρ) 6= ∅

for all ρ > 0. Here

Γϕ(ζ,α) = {x ∈ D : ϕ(|x − ζ|) < αδ(x) },
and it is called a (ϕ,α)-approach region, shortly an approach region, in D at ζ. Choosing ϕ(t) = t
(in the case of the unit disk D of the complex plane C) one gets the familiar Stolz approach region.

Choosing ϕ(t) = tτ , τ ≥ 1, say, one gets more general approach regions, see [23].

1.6 Let 0 ≤ d ≤ n. A set E ⊂ Rn is Ahlfors-regular with dimension d if it is closed and there
is a constant C4 > 0 so that

C
−1
4
r

d ≤ Hd(E ∩ Bn(x,r)) ≤ C4rd

for all x ∈ E and r > 0. The smallest constant C4 is called the regularity constant for E. Simple
examples of Ahlfors-regular sets include d-planes and d-dimensional Lipschitz graphs. Also certain

Cantor sets and self-similar sets are Ahlfors-regular. For more details, see [1, pp. 9-10].

2 The results

2.1 First a partial generalization to Pavlović’s result [12, Theorem 1, pp. 433-434] or Theo-

rem B above. Observe that though the constant C below in (6) does depend on K, it is, neverthe-

less, otherwise independent of the (K-)quasi-nearly subharmonic function u.

Theorem 1 Let Ω be a domain in Rn, n ≥ 2, Ω 6= Rn, such that its boundary ∂Ω is Ahlfors-
regular with dimension d, 0 ≤ d ≤ n. Let u : Ω → [0, +∞) be a K-quasi-nearly subharmonic
function. Let ϕ : [0, +∞) → [0, +∞) be an admissible function, with constants r0 and C0. Let
α > 0 be arbitrary. Let ρ0 := min{r0/21+α,r0/23αC0,ϕ(r0)/α}. Let γ ∈ R be such that

∫

Ω

δ(x)
γ
u(x) dm(x) < +∞.

Then there is a constant C = C(n, Ω,d,ϕ,α,γ,K) such that for all ρ ≤ ρ0,
∫

∂Ω

sup
x∈Γϕ,ρ(ζ,α)

{δ(x)n+γ [ϕ−1(δ(x))]−du(x) }dHd(ζ) ≤ C
∫

Ωρ′

δ(x)
γ
u(x) dm(x),



132 Juhani Riihentaus CUBO
11, 4 (2009)

where ρ′ = 4
3
ρ and

Γϕ,ρ(ζ,α) = {x ∈ Γϕ(ζ,α) : δ(x) < ρ}.

Proof. Proceeding as in [19, proof of Theorem (with ψ = id), pp. 31-35] (cf. [15, proof of

Theorem, pp. 235-237]) and choosing K = ∂Ω, one obtains
∫

∂Ω

M
∂Ω
ρ (ζ) dH

d
(ζ) ≤ C

∫

Ωρ′

δ(x)
γ
u(x) dm(x)

where ρ′ = 4
3
ρ and M∂Ωρ : ∂Ω → [0, +∞],

M
∂Ω
ρ (ζ) sup

x∈Γϕ,ρ(ζ,α)

δ(x)
n+γ

u(x)

[ϕ−1(δ(x))]d + Hd(Bn(x,C1C2 ϕ
−1(δ(x))) ∩ ∂Ω).

Here and below the constants C1 = C1(C0,α), C2 = C2(C0,α) and C3 = C3(C0,α) are, as

pointed out above, directly from [19, proof of Lemma 2.3, pp. 32-33] or [15, proof of Lemma 2.3,

pp. 234-235]. By this lemma one has, for each ζ ∈ ∂Ω and for each x ∈ Γϕ,ρ(ζ,α), Bn(x,C1C2ϕ−1(δ(x))) ⊂
B

n
(ζ,C1C2C3ϕ

−1
(δ(x))). Since ∂Ω is Ahlfors-regular with dimension d, we have

H
d
(B

n
(ζ,C1C2C3ϕ

−1
(δ(x))) ∩ ∂Ω) ≤ C4[C1C2C3ϕ−1(δ(x))]d

where also C4 is a fixed constant. Therefore

M
∂Ω
ρ (ζ) sup

x∈Γϕ,ρ(ζ,α)

δ(x)
n+γ

u(x)

[ϕ−1(δ(x))]d + Hd(Bn(x,C1C2 ϕ
−1(δ(x))) ∩ ∂Ω)

≥ sup
x∈Γϕ,ρ(ζ,α)

δ(x)
n+γ

u(x)

[ϕ−1(δ(x))]d + Hd(Bn(ζ,C1C2C3 ϕ
−1(δ(x))) ∩ ∂Ω)

≥ sup
x∈Γϕ,ρ(ζ,α)

δ(x)
n+γ

u(x)

[ϕ−1(δ(x))]d + C4(C1C2C3)
d[ϕ−1(δ(x))]d

≥ 1
1 + (C1C2C3)

dC4
sup

x∈Γϕ,ρ(ζ,α)

{δ(x)n+γ [ϕ−1(δ(x))]−du(x) }.

Hence ∫

∂Ω

sup
x∈Γϕ,ρ(ζ,α)

{δ(x)n+γ [ϕ−1(δ(x))]−du(x) }dHd(ζ) ≤ C
∫

Ωρ′

δ(x)
γ
u(x) dm(x),

concluding the proof.

2.2 Theorem 1 seems to be useful in many situations. For example, with the aid of it one gets

the following improvements to Pavlović’s result [12, Theorem 1, pp. 433-434] or Theorem B above:

Theorem 2 Let Ω, d, u, ϕ, α, γ and ρ0 be as above in Theorem 1. Suppose moreover that

H
d
(∂Ω) < +∞. Then there is a constant C = C(n, Ω,d,ϕ,α,γ,K) such that

∫

∂Ω

sup
x∈Γϕ(ζ,α)

{δ(x)n+γ [ϕ−1(δ(x))]−du(x)}dHd(ζ) ≤ C
∫

Ω

δ(x)
γ
u(x) dm(x).



CUBO
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On an inequality related to the radial growth of subharmonic ... 133

Proof. By Theorem 1 (we may clearly assume that
∫
Ω
δ(x)

γ
u(x) dm(x) < +∞),

∫

∂Ω

sup
x∈Γϕ,ρ0 (ζ,α)

{δ(x)n+γ [ϕ−1(δ(x))]−du(x) }dHd(ζ) ≤ C
∫

Ω
ρ′
0

δ(x)
γ
u(x) dm(x).

Write

Γ
c
ϕ,ρ0

(ζ,α) := {x ∈ Γϕ(ζ,α) : δ(x) ≥ ρ0}.

Since

sup
x∈Γϕ(ζ,α)

{δ(x)n+γ [ϕ−1(δ(x))]−du(x) } ≤ sup
x∈Γcϕ,ρ0

(ζ,α)

{δ(x)n+γϕ−1(δ(x))]−du(x) }

+ sup
x∈Γϕ,ρ0 (ζ,α)

{δ(x)n+γϕ−1(δ(x))]−du(x) },

we obtain:

∫

∂Ω

sup
x∈Γϕ(ζ,α)

{δ(x)n+γ [ϕ−1(δ(x))]−du(x) }dHd(ζ)

≤
∫

∂Ω

sup
x∈Γcϕ,ρ0

(ζ,α)

{δ(x)n+γ [ϕ−1(δ(x))]−du(x) }dHd(ζ)

+

∫

∂Ω

sup
x∈Γϕ,ρ0 (ζ,α)

{δ(x)n+γ [ϕ−1(δ(x))]−du(x) }dHd(ζ)

≤
∫

∂Ω

sup
x∈Γcϕ,ρ0

(ζ,α)

{δ(x)n+γ [ϕ−1(δ(x))]−du(x) }dHd(ζ) + C
∫

Ω
ρ′
0

δ(x)
γ
u(x) dm(x)

≤
∫

∂Ω

sup
x∈Γcϕ,ρ0

(ζ,α)

{δ(x)n+γ [ϕ−1(δ(x))]−du(x) }dHd(ζ) + C
∫

Ω

δ(x)
γ
u(x) dm(x).

It remains to show that

∫

∂Ω

sup
x∈Γcϕ,ρ0

(ζ,α)

{δ(x)n+γ [ϕ−1(δ(x))]−du(x)}dHd(ζ) ≤ C
∫

Ω

δ(x)
γ
u(x) dm(x)

for some C = C(n, Ω,d,ϕ,α,γ,K). For all x ∈ Γcϕ,ρ0 (ζ,α) we have

u(x) ≤ K
νn(

δ(x)
3

)n

∫

B(x)

u(y) dm(y).



134 Juhani Riihentaus CUBO
11, 4 (2009)

Using also the facts that 2
3
δ(x) ≤ δ(y) ≤ 4

3
δ(x) for all y ∈ B(x), one gets easily:

∫

∂Ω

sup
x∈Γcϕ,ρ0

(ζ,α)

{δ(x)n+γ [ϕ−1(δ(x))]−du(x)}dHd(ζ)

≤
∫

∂Ω

sup
x∈Γcϕ,ρ0

(ζ,α)

{δ(x)n+γ [ϕ−1(δ(x))]−d K
νn(

δ(x)
3

)n

∫

B(x)

u(y) dm(y)}dHd(ζ)

≤ 3
n
K

νn

∫

∂Ω

sup
x∈Γcϕ,ρ0

(ζ,α)

{δ(x)γ [ϕ−1(δ(x))]−d
∫

B(x)

u(y) dm(y)}dHd(ζ)

≤
(

3

2

)|γ|
3

n
K

νn

∫

∂Ω

sup
x∈Γcϕ,ρ0

(ζ,α)

{[ϕ−1(δ(x))]−d
∫

B(x)

δ(y)
γ
u(y) dm(y)}dHd(ζ)

≤ 3
|γ|+n

K

2|γ|νn
[ϕ

−1
(ρ0)]

−d
H

d
(∂Ω)

∫

Ω

δ(y)
γ
u(y) dm(y).

Thus ∫

∂Ω

sup
x∈Γϕ(ζ,α)

{δ(x)n+γ [ϕ−1(δ(x))]−du(x) }dHd(ζ) ≤ C
∫

Ω

δ(x)
γ
u(x) dm(x),

concluding the proof.

Corollary Let u : Bn → [0, +∞) be a subharmonic function and let p > 0, α > 1 and
γ > −1 − max{ (n − 1)(1 − p), 0 }. Then there is a constant C = C(n,γ,p,α) such that

∫

Sn−1

sup
x∈Γid(ζ,α)

{(1− | x |)γ+1u(x)p}dσ(ζ) ≤ C
∫

Bn

(1− | x |)γu(x)p dm(x).

Here id is the identity mapping of Rn and σ is the spherical (Lebesgue) measure in Sn−1.

Remark Observe that Suzuki [24, Theorem 2, pp. 272-273] has shown the following: If p > 0

and γ ≤ −1 − max{ (n − 1)(1 − p), 0 }, then the only nonnegative subharmonic function on a
bounded domain D of Rn with C2 boundary satisfying

∫

D

δ(x)
γ
u(x)

p
dm(x) < +∞ (6)

is the zero function. On the other hand, if p > 0 and γ > −1 − max{ (n − 1)(1 − p), 0 }, then there
exist nonnegative non-zero subharmonic functions on D = Bn satisfying (6).

Received: August 2008. Revised: November 2008.

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