In this paper, we give a sufficient and necessary condition for an analytic function $f(z)$ on the unit disc $\mathbb{D}$ with Hadamard gaps, that is, $f(z)=\sum\limits_{k=1}^{\infty}a_kz^{n_k}$, where $\frac{n_{k+1}}{n_k}\geq\lambda>1$ for all $k\in \mathbb{N}$, belongs to the Bloch--Orlicz space $ \mathcal{B}^{\varphi}$. As an application of our results, the compactness of composition operator are discussed.
References
R. Aulaskari and P. Lappan, Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal Complex analysis and its applications (Hong Kong, 1993), 136–146, Pitman Res. Notes Math. Ser., 305, Longman Sci. Tech., Harlow, 1994. MR1278928 0826.30027 R. Aulaskari and P. Lappan, Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal Complex analysis and its applications (Hong Kong, 1993), 136–146, Pitman Res. Notes Math. Ser., 305, Longman Sci. Tech., Harlow, 1994. MR1278928 0826.30027
A. II. Baernstein, Analytic functions of bounded mean oscillation, Aspects of contemporary complex analysis (Proc. NATO Adv. Study Inst., Univ. Durham, Durham, 1979), pp. 3–36, Academic Press, London-New York, 1980. MR623463 0492.30026 A. II. Baernstein, Analytic functions of bounded mean oscillation, Aspects of contemporary complex analysis (Proc. NATO Adv. Study Inst., Univ. Durham, Durham, 1979), pp. 3–36, Academic Press, London-New York, 1980. MR623463 0492.30026
L. Brown and A. L. Shields, Multipliers and cyclic vectors in the Bloch space, Michigan Math. J. 38 (1991), 141–146. MR1091517 10.1307/mmj/1029004269 euclid.mmj/1029004269
L. Brown and A. L. Shields, Multipliers and cyclic vectors in the Bloch space, Michigan Math. J. 38 (1991), 141–146. MR1091517 10.1307/mmj/1029004269 euclid.mmj/1029004269
H. Chen and P. Gauthier, Composition operators on $\mu$-Bloch spaces, Canad. J. Math. 61 (2009), 50–75. MR2488449 10.4153/CJM-2009-003-1 1171.47020 H. Chen and P. Gauthier, Composition operators on $\mu$-Bloch spaces, Canad. J. Math. 61 (2009), 50–75. MR2488449 10.4153/CJM-2009-003-1 1171.47020
J. Choa, Some properties of analytic functions on the unit ball with hadamard gaps, Complex Variables Theory Appl. 29 (1996), no. 3, 277–285. MR1388363 10.1080/17476939608814895 J. Choa, Some properties of analytic functions on the unit ball with hadamard gaps, Complex Variables Theory Appl. 29 (1996), no. 3, 277–285. MR1388363 10.1080/17476939608814895
H. Essén and M. Wulan, On analytic and meromorphic functions and spaces of $\mathcal{Q}_K$-type, Illinois J. Math. 46 (2002), no. 4, 1233–1258. MR1988261 1048.30017 euclid.ijm/1258138477
H. Essén and M. Wulan, On analytic and meromorphic functions and spaces of $\mathcal{Q}_K$-type, Illinois J. Math. 46 (2002), no. 4, 1233–1258. MR1988261 1048.30017 euclid.ijm/1258138477
P. Galanopoulos, On $\mathcal{B}_{\log}$ to $\mathcal{Q}_{\log}$ pullbacks, J. Math. Anal. Appl. 337 (2008), no. 1, 712–725. MR2356105 1137.30015 10.1016/j.jmaa.2007.02.049 P. Galanopoulos, On $\mathcal{B}_{\log}$ to $\mathcal{Q}_{\log}$ pullbacks, J. Math. Anal. Appl. 337 (2008), no. 1, 712–725. MR2356105 1137.30015 10.1016/j.jmaa.2007.02.049
K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (1995), 2679–2687. MR1273508 0826.47023 10.1090/S0002-9947-1995-1273508-X K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (1995), 2679–2687. MR1273508 0826.47023 10.1090/S0002-9947-1995-1273508-X
J. Miao, A property of analytic functions with hadamard gaps, Bull. Aust. Math. Soc. 45 (1992), no. 01, 105–112. MR1147249 0739.30040 10.1017/S0004972700037059 J. Miao, A property of analytic functions with hadamard gaps, Bull. Aust. Math. Soc. 45 (1992), no. 01, 105–112. MR1147249 0739.30040 10.1017/S0004972700037059
J. Ramos Fernández, Composition operators on Bloch–Orlicz type spaces, Appl. Math. Comput. 217 (2010), no. 7, 3392–3402. MR2733781 1204.30047 10.1016/j.amc.2010.09.004 J. Ramos Fernández, Composition operators on Bloch–Orlicz type spaces, Appl. Math. Comput. 217 (2010), no. 7, 3392–3402. MR2733781 1204.30047 10.1016/j.amc.2010.09.004
M. Rao and Z. Ren, Theory of orlicz spaces, M. Dekker New York, 1991. MR1113700 1196.46001 M. Rao and Z. Ren, Theory of orlicz spaces, M. Dekker New York, 1991. MR1113700 1196.46001
W. Schachermayer, Some remarks on integral operators and equimeasurable sets, Springer, 1986. MR864715 0611.47024 10.1007/BFb0076303 W. Schachermayer, Some remarks on integral operators and equimeasurable sets, Springer, 1986. MR864715 0611.47024 10.1007/BFb0076303
S. Stević, On new Bloch-type spaces, Appl. Math. Comput. 215 (2009), 841–849. MR2561542 10.1016/j.amc.2009.06.009 1176.30064 S. Stević, On new Bloch-type spaces, Appl. Math. Comput. 215 (2009), 841–849. MR2561542 10.1016/j.amc.2009.06.009 1176.30064
H. Wulan and P. Wu, Characterizations of $\mathcal{Q}_T$ spaces , J. Math. Anal. Appl. 254 (2001), 484–497. MR1805519 10.1006/jmaa.2000.7204 H. Wulan and P. Wu, Characterizations of $\mathcal{Q}_T$ spaces , J. Math. Anal. Appl. 254 (2001), 484–497. MR1805519 10.1006/jmaa.2000.7204
H. Wulan and J. Zhou, $\mathcal{Q}_K$ type spaces of analytic functions, J. Funct. Spaces Appl. 4 (2006), no. 1, 73–84. MR2194636 1098.30027 10.1155/2006/910813 H. Wulan and J. Zhou, $\mathcal{Q}_K$ type spaces of analytic functions, J. Funct. Spaces Appl. 4 (2006), no. 1, 73–84. MR2194636 1098.30027 10.1155/2006/910813
J. Xiao, Holomorphic $\mathcal{Q}$ classes, no. 1767, Springer, 2001. MR1869752 0983.30001 J. Xiao, Holomorphic $\mathcal{Q}$ classes, no. 1767, Springer, 2001. MR1869752 0983.30001
S. Yamashita, Gap series and $\alpha$-Bloch functions, (1980). MR623746 0467.30001 S. Yamashita, Gap series and $\alpha$-Bloch functions, (1980). MR623746 0467.30001
K. Zhu, Bloch type spaces of analytic functions, Rocky Mountain J. Math. 23 (1993), 1143–1177. MR1245472 10.1216/rmjm/1181072549 euclid.rmjm/1181072549
0787.30019 K. Zhu, Bloch type spaces of analytic functions, Rocky Mountain J. Math. 23 (1993), 1143–1177. MR1245472 10.1216/rmjm/1181072549 euclid.rmjm/1181072549
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