Abstract and Applied Analysis

VolumeΒ 2008Β (2008), Article IDΒ 404636, 9 pages

http://dx.doi.org/10.1155/2008/404636

## Some Sufficient Conditions for Analytic Functions to Belong to Space

Department of Mathematics, Jia Ying University, Meizhou 514015, Guangdong, China

Received 31 May 2008; Accepted 7 June 2008

Academic Editor: StevoΒ Stevic

Copyright Β© 2008 Xiaoge Meng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper gives some sufficient conditions for an analytic function to belong to the space consisting of all analytic functions on the unit disk such

#### 1. Introduction

Let be the open unit disk in the complex plane and the space of all analytic functions in . For , letbe the MΓΆbius transformation of and letbe the Green's function on . Let denote the pseudo-hyperbolic metric disk centered at with radius , that is,

It is said that an analytic function is defined by a lacunary series ifFor some results in the topic, see, for example, [1β6] and the references therein.

Given a function we consider the space of all functions such thatBy we denote the space consisting of all such thatwhere , , and .

For , , the space is reduced to (see, e.g., [7]). If , , then (see, e.g., [8, 9]).

Throughout the paper, we assume that the condition holds (see [7])so that the space we study is nontrivial. We also assume that as a nondecreasing function. An important tool in the study of spaces is the auxiliary function defined by (see [10])The following conditionis crucial in this paper. It has played an important role in the study of spaces during the last few years.

In this paper, we give some sufficient conditions for an analytic function to belong to the space .

The followings are our main results in this paper.

Theorem 1.1. *Let , , and let be a monotone increasing function in on such that ,
for .
If **then .*

Theorem 1.2. *For , ,
and .
If satisfies condition (1.7) and is a function with the property that for ,
then a lacunary series belongs to if *

Throughout this paper, stands for a positive constant, whose value may differ from one occurrence to the other. The expression means that there is a positive constant such that .

#### 2. Main Results and Proofs

In this section, we give the proofs of Theorems 1.1 and 1.2. Before formulating the main results, we give some lemmas which are used in the proofs.

Lemma 2.1 (see [7]). * Let , .
Then, if and only if *

Lemma 2.2 (see [5]). *
Let be a function with the property that for .
If satisfies condition (1.8), then there exists a constant such that for .
*

Lemma 2.3 (see [5]). * If satisfies condition (1.8), then we can find another nonnegative function such that and the new function has the following properties: *(a)* is nondecreasing on ;*(b)* satisfies condition ();*(c)* on ;*(d)* is differentiable (up to any given order) on ;*(e)* is concave on ;*(f)* for ;*(g)* on .*

Lemma 2.4 (see [5]). * If satisfies condition (1.8), then for any and ,
one has **where is a constant depending on alone.*

Lemma 2.5 (see [11]). * For , ,
and , **where is a constant.*

*Proof of Theorem 1.1. * Let .
By Lemma 2.3, we may also assume that is concave, so that the following inequality
true holdsFrom the definition of for , ,
we have that Using these facts and polar coordinates, it
follows thatThe last integral exists for
every in view of (1.9), and .
Further, sincethen the last integral tends to
zero for every as .
By Lebesgue's dominated convergence theorem, we obtain .
By Lemma 2.1, we get

From Theorem 1.1, we have the following corollary. Here, we give a different and technical proof.

Corollary 2.6. * Let , , , ,
and let be a monotone increasing function of in such that ,
for .
If **then .*

*Proof. * Let .
We haveFor , ,
henceFor ,
by Lemma 2.5, we haveThe last integral exists since and .
It is clear thatwhich impliesBy Lebesgue's dominated
convergence theorem, we get Now, we consider the case .
Note thatin the case. By a well-known
inequality (see, e.g., [12, page 3]), we have thatand consequently, for ,
we haveBy the monotonicity of ,
we haveThis impliessince the following integral
existsChoosing ,
for every ,
it follows thatThis andimply thatoras .
Consequently,Combining (2.8), (2.13) with (2.23) we
see thatwhich means The proof is complete.

*Proof of Theorem 1.2. * Consider the monotone
increasing functionFor every ,
we haveBy Theorem 1.1, we only need to
proveBy the inequality , ,
and Lemma 2.2, there exists a constant such thatThen for ,
we haveFor ,
assume .
Sincewhich together with HΓΆlder's
inequality givesHence,For , ,
by Lemma 2.4, choosing , ,
we obtainIf ,
then .
The assumption that is nondecreasing and Lemma 2.2 giveSince is a lacunary series, the Taylor series of has most terms such that .
Combining this with the last inequality and HΓΆlder's inequality, we
obtainThis shows that .

#### References

- J. S. Choa, βSome properties of analytic functions on the unit ball with Hadamard gaps,β
*Complex Variables: Theory & Application*, vol. 29, no. 3, pp. 277β285, 1996. View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - J. Miao, βA property of analytic functions with Hadamard gaps,β
*Bulletin of the Australian Mathematical Society*, vol. 45, no. 1, pp. 105β112, 1992. View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - S. Stević, βA generalization of a result of Choa on analytic functions with Hadamard gaps,β
*Journal of the Korean Mathematical Society*, vol. 43, no. 3, pp. 579β591, 2006. View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - S. Stević, βOn Bloch-type functions with Hadamard gaps,β
*Abstract and Applied Analysis*, vol. 2007, Article ID 39176, 8 pages, 2007. View at Publisher Β· View at Google Scholar Β· View at MathSciNet - H. Wulan and K. Zhu, βLacunary series in ${\mathcal{Q}}_{K}$ spaces,β
*Studia Mathematica*, vol. 173, no. 3, pp. 217β230, 2007. View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - S. Yamashita, βGap series and $\alpha $-Bloch functions,β
*Yokohama Mathematical Journal*, vol. 28, no. 1-2, pp. 31β36, 1980. View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - H. Wulan and J. Zhou, β${\mathcal{Q}}_{K}$ type spaces of analytic functions,β
*Journal of Function Spaces and Applications*, vol. 4, no. 1, pp. 73β84, 2006. View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - S. Li and S. Stević, βCompactness of Riemann-Stieltjes operators between $F(p,q,s)$ spaces and $\alpha $-Bloch spaces,β
*Publicationes Mathematicae Debrecen*, vol. 72, no. 1-2, pp. 111β128, 2008. View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - R. Zhao, βOn a general family of function spaces,β
*Annales Academiæ Scientiarum Fennicæ. Mathematica Dissertationes*, no. 105, p. 56, 1996. View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - H. Wulan and K. Zhu, βDerivative-free characterizations of ${Q}_{K}$ spaces,β
*Journal of the Australian Mathematical Society*, vol. 82, no. 2, pp. 283β295, 2007. View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - R. Aulaskari, J. Xiao, and R. H. Zhao, βOn subspaces and subsets of BMOA and UBC,β
*Analysis*, vol. 15, no. 2, pp. 101β121, 1995. View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - J. B. Garnett,
*Bounded Analytic Functions*, vol. 96 of*Pure and Applied Mathematics*, Academic Press, New York, NY, USA, 1981. View at Zentralblatt MATH Β· View at MathSciNet