In this paper, we prove that if the positive part of a harmonic function in a half-plane satisfies a slowly growing condition, then its negative part can also be dominated by a similarly growing condition. Further, a solution of the Dirichlet problem in a half-plane for a fast growing continuous boundary function is constructed by the generalized Dirichlet integral with this boundary function.
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
1 Introduction and main theorem
Let R be the set of all real numbers and let C denote the complex plane with points , where . The boundary and closure of an open set Ω are denoted by ∂ Ω and , respectively. The upper half-plane is the set , whose boundary is .
We use the standard notations , , and is the integer part of the positive real number d. For positive functions and , we say that if for some positive constant M.
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Given a continuous function f in , we say that h is a solution of the (classical) Dirichlet problem in with f, if in and for every .
The classical Poisson kernel in is defined by
where and .
It is well known (see [1]) that the Poisson kernel is harmonic for and has the expansion
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which converges for . We define a modified Cauchy kernel of by
where m is a nonnegative integer.
To solve the Dirichlet problem in , as in [2], we use the modified Poisson kernel defined by
We remark that the modified Poisson kernel is harmonic in . About modified Poisson kernel in a cone, we refer readers to papers by I Miyamoto, H Yoshida, L Qiao and GT Deng (e.g. see [3‐11]).
Put
where is a continuous function in .
For any positive real number α, We denote by the space of all measurable functions in satisfying
(1.1)
and by the set of all measurable functions in such that
(1.2)
We also denote by the set of all continuous functions in , harmonic in with and .
About the solution of the Dirichlet problem with continuous data in , we refer readers to the following result (see [12, 13]).
Theorem ALetube a real-valued function harmonic inand continuous in . If , then there exists a constantsuch thatfor all .
Inspired by Theorem A, we first prove the following.
Theorem 1Ifand , then .
Then we are concerned with the growth property of at infinity in .
Theorem 2Ifand , then
(1.3)
We say that u is of order λ if
If , then u is said to be of finite order. See Hayman-Kennedy [[14], Definition 4.1].
Our next aim is to give solutions of the Dirichlet problem for harmonic functions of infinite order in . For this purpose, we define a nondecreasing and continuously differentiable function on the interval . We assume further that
(1.4)
Remark For any ϵ (), there exists a sufficiently large positive number R such that , by (1.4) we have
Let be the set of continuous functions f in such that
(1.5)
where β is a positive real number.
Theorem 3If , then the integralis a solution of the Dirichlet problem inwithf.
The following result immediately follows from Theorem 2 (the case ) and Theorem 3 (the case ).
Corollary 1Iffis a continuous function insatisfying
thenis a solution of the Dirichlet problem inwithfsatisfying
For harmonic functions of finite order in , we have the following integral representations.
Corollary 2Let () and letmbe an integer such that .
(I)
If , thenis a harmonic function inand can be continuously extended tosuch thatfor . There exists a constantsuch thatfor all .
(II)
If , thenis a harmonic function inand can be continuously extended tosuch thatfor . There exists a harmonic polynomialof degree at mostwhich vanishes insuch thatfor all .
Finally, we prove the following.
Theorem 4Letube a real-valued function harmonic inand continuous in . If , then we have
for all , whereis an entire function inand vanishes continuously in .
2 Main lemmas
The Carleman formula refers to holomorphic functions in (see [15, 16]).
Lemma 1Ifand () is a harmonic function inwith continuous boundary in , then we have
Letbe a harmonic function insuch thatvanishes continuously in . If
thenin , whereis a polynomial ofof degree less thanmand even with respect to the variabley.
3 Proof of Theorem 1
We distinguish the following two cases.
Case 1. .
If , Lemma 1 gives
(3.1)
Since , we obtain
from (1.1) and hence
(3.2)
Then from (1.2), (3.1), and (3.2) we have
which gives
Thus from .
Case 2. .
Since , we see from (1.1) that
(3.3)
and we see from (1.2) that
(3.4)
We have from (3.3), (3.4), and Lemma 1
Set
We have
from the L’Hospital’s rule and hence we have
So
Then from . We complete the proof of Theorem 1.
4 Proof of Theorem 2
For any , there exists such that
(4.1)
from Theorem 1. For any fixed and , we write
where
By (2.1), (2.2), (2.3), and (4.1), we have the following estimates:
(4.2)
(4.3)
(4.4)
(4.5)
Combining (4.2)-(4.5), (1.3) holds. Thus we complete the proof of Theorem 2.
5 Proof of Theorem 3
Take a number r satisfying , where is a sufficiently large positive number. For any ϵ (), we have
from the remark, which shows that there exists a positive constant dependent only on r such that
(5.1)
for any .
For any and , we have and
from (1.5), (2.2), and (5.1). Thus is finite for any . is a harmonic function of for any fixed . is also a harmonic function of .
Now we shall prove the boundary behavior of . For any fixed , we can choose a number such that . We write
where
Since is the Poisson integral of , it tends to as . Clearly, vanishes in . Further, , which tends to zero as . Thus the function can be continuously extended to such that for any . Then Theorem 3 is proved.
6 Proof of Corollary 2
We prove (II). Consider the function . Then it follows from Corollary 1 that this is harmonic in and vanishes continuously in . Since
(6.1)
for any and
(6.2)
from (1.1), for every we have
from (6.1), (6.2), Corollary 1, and Lemma 3, where is a polynomial in of degree at most and even with respect to the variable y. From this we evidently obtain (II).
If , then for . (II) shows that there exists a constant such that
Put
It immediately follows that for every , which is the conclusion of (I). Thus we complete the proof of Corollary 2.
7 Proof of Theorem 4
Consider the function , which is harmonic in , can be continuously extended to and vanishes in .
The Schwarz reflection principle [[12], p.68] applied to shows that there exists a harmonic function in satisfying such that for . Thus for all , where is an entire function in and vanishes continuously in . Thus we complete the proof of Theorem 4.
Acknowledgements
The authors are very thankful to the anonymous referees for their valuable comments and constructive suggestions, which helped to improve the quality of the paper. This work is supported by the Academy of Finland Grant No. 176512.
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.