Bulletin of the. Iranian Mathematical Society

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1 ISSN: 7-6X Print) ISSN: Online) Bulletin of the Iranian Mathematical Society Vol 4 6), No, pp 95 Title: A note on lacunary series in Q K spaces Authors): J Zhou Published by Iranian Mathematical Society

2 Bull Iranian Math Soc Vol 4 6), No, pp 95 Online ISSN: A NOTE ON LACUNARY SERIES IN Q K SPACES J ZHOU Communicated by Ali Abkar) Abstract In this paper, under the condition that K is concave, we characterize lacunary series in Q K spaces We improve a result due to H Wulan and K Zhu Keywords: Q K spaces; lacunary series; concave MSC): Primary: 3H5; Secondary: 3B, 3H5 Introduction Let D be the unit disk in the complex plane C and denote by D the boundary of D As usual, HD) is the class of functions analytic in D The Green function in the unit disk with singularity at a D is given by gz, a) = log σ a z), z D Here σ a z) = a z az, is the Möbius transformation of D Throughout this paper, we assume that K : [, ) [, ) is an increasing function A function f HD) belongs to the space Q K if f Q K = sup f z) K gz, a)) daz) <, a D D where da is the element of Euclidean area on D normalized so that daz) = π dxdy Q K spaces are Möbius invariant in the sense that f σ a QK = f QK for every f Q K and a D See [3 5] for more results of Q K spaces If Kt) = t p, p <, then the space Q K reduces to the space Q p cf [7,8]) In particular, Q is the Dirichlet space; Q = BMOA, the space of bounded mean oscillation; Q p is the Bloch space for all p > Article electronically published on February, 6 Received: 7 September 4, Accepted: 9 November 4 95 c 6 Iranian Mathematical Society

3 if A note on lacunary series in Q K spaces 96 Recall that a function fz) = a kz HD) is called a lacunary series λ = inf k + > Such series are often used to give examples of functions in various analytic function spaces It is well known that a lacunary series belongs to BMOA if and only if it is in the Hardy space H see []) By [], if < p <, then the lacunary series f Q p if and only if n p k a k < H Wulan and K Zhu [6] characterized lacunary series in Q K spaces as follows Theorem A [6]) Let K satisfy φ K s) s ds <, ) where φ K s) = sup Kst)/Kt), < s < t Then a lacunary series fz) = a k z belongs to Q K if and only if ) a k K < By [3], the space Q K only depends on the weight function K in a neighbourhood of the origin If K t) = t log e t, < t <, then Q K is the analytic version of Q D) space see [7] and [9]) An elementary calculation shows that φ K s) = s when s Thus, K does not satisfy the condition ) In other words, Theorem A misses the case of the analytic version of Q D) space It was pointed out in [6] that Theorem A also misses the classical case of BMOA The goal of this article is to characterize lacunary series in Q K spaces under a weaker condition of K Our result covers the cases of BMOA and the analytic version of Q D) space We write A B if there exists a constant C such that A CB, in addition, the symbol A B means that A B A Main result In [6], we can find many nice estimates of the weight function K under the condition ) In recent years, the study of Q K spaces benefits from these estimates In particular, if K satisfies ), then there exists an increasing function K on, ) such that K t) Kt) for all t, ) Moreover,

4 97 Zhou K is twice differentiable on, ) and concave Namely, K t) for t, ) Next we state the main result of this paper Theorem Let K be a concave function Then a lacunary series belongs to Q K if and only if fz) = a k z ) a k K < ) Proof We prove the result by following the proof of Theorem A in [6] First suppose that fz) = a kz Q K Then D f z) K log z Bearing in mind that K is increasing, we get > ) daz) < n k a k r nk K log ) dr r n k a k e nkt Kt)dt ) a k K On the other hand, suppose that the condition ) holds Since K is concave, the estimates in [6, page 6] show that sup f z) Kgz, a))daz) r a k r k ) n K log ) dr r Write I n = {k : k < +, k N} The Hölder inequality gives a k r k) n a k r I n / r n n/ r n a k I n log ) / n/ r n a k r I n

5 A note on lacunary series in Q K spaces 98 Thus, sup f z) Kgz, a))daz) n/ a k I n Since K is increasing, we see that K e r ) / n = n K r n log ) / K log ) dr r r log ) / K log ) dr r r e nt t dt ) e s s ds If K), by [3], we can assume that K is constant Of course, Kt)/t is decreasing on, ) If K) =, we claim that Kt)/t is decreasing on, ) Choose s, t, ) such that s < t By the basic property of a concave function, we have Thus Ks) s Ks) s Kt) t = Ks) K) s = tks) skt) st Kt) Ks) t s = [sks) Kt)) + t s)ks)] st = t s [ ] Ks) Kt) Ks) t s t s Hence, Therefore, e K r n log ) / K log ) dr r r ) = n K sup f z) Kgz, a))daz) / e n t t dt ) e s s ds ) n K n k a k I n

6 99 Zhou If I n, then Using the monotonicity of Kt)/t, one gets ) ) K K This gives sup f z) Kgz, a))daz) n ) a k K I n ) a k K I n Since f is a lacunary series, then there exists a positive constant λ such that + λ > for all k It is well known that the Taylor series of fz) has at most [log λ ] + terms a k z such that I n for all n N By Hölder inequality, we obtain The proof is complete sup f z) Kgz, a))daz) [log λ ] + ) a k K I n ) a k K < ) Note that Kt) = t p, p is concave The following result follows easily from Theorem Corollary [7]) Let p [, ] Then a lacunary series fz) = a k z belongs to Q p if and only if n p k a k < Since K t) = t log e t is also concave, we have the following statement Corollary 3 [9]) A lacunary series fz) = a k z

7 A note on lacunary series in Q K spaces belongs to Q D) if and only if log + ) a k < Acknowledgment The author thanks Dr G Bao for providing a number of helpful suggestions and corrections The work is supported by NFS of Anhui Province of China No 6885MA) References [] R Aulaskari, J Xiao and R Zhao, On subspaces and subsets of BMOA and UBC, Analysis 5 995), no, [] A Baernstein, Analytic functions of bounded mean oscillation Aspects of contemporary complex analysis Proc NATO Adv Study Inst, Univ Durham, Durham, 979), 3 36, Academic Press, London-New York, 98 [3] M Essén and H Wulan, On analytic and meromorphic functions and spaces of Q K - type, Illinois J Math 46 ), no 4, [4] M Essén, H Wulan and J Xiao, Several function-theoretic characterizations of Möbius invariant Q K spaces, J Funct Anal 3 6), no, 78 5 [5] J Pau, Bounded Möbius invariant Q K spaces, J Math Anal Appl 338 8), no, 9 4 [6] H Wulan and K Zhu, Lacunary series in Q K spaces, Stud Math 78 7), no 3, 7 3 [7] J Xiao, Holomorphic Q Classes, Lecture Notes in Mathematics, 767, Springer- Verlag, Berlin, [8] J Xiao, Geometric Q p functions, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 6 [9] J Zhou and G Bao, Analytic version of Q D) space, J Math Annal Appl 4 5), no, 9 Jizhen Zhou) School of Sciences, Anhui University of Science and Technology, Huainan, Anhui 3, China address: hope89@63com