Hyperbolic Q_k Space

P.Wu study the criterion of normal and Bloch functions with a nondecreasing function in the paper On increasing functions,Bloch functions and normal functions in 1998,on the basic of this paper,P.Wu and H.Wulan formally introduce the definition of Q_k spaces in their paper Characterizations of Q_k spaces.By now Q_k spaces also attracts a lot of attentions of many people.At the same time there is a lot of unsolved problems in this field.R.Aulaskari,P.Lappan,M.Eseen,P.Wu,H.Wulan,J.Xiao and R.Zhao and so on have obtained a number of results on Q_k spaces.In this paper we introduce a new function space by introduce hyperbolic derivative to Q_k spaces.It play an important role in coping with bounded analytic functions. We first state the history of function space and then give some definitions of hyperbolic functions in the introduction part.In the first chapter we investigate some basic property of hyperbolic Q_k spaces,it is interested that we get a relationship between hyperbolic Q_k spaces and Q_k spaces,we also study the relationship between hyperbolic Q_k spaces and other hyperbolic type spaces.By the Carleson measure we study the hyperbolic Q_k spaces in the second chapter.In the third chapter,we study the composition operators in hyperbolic Q_k spaces,our results extend the results of P.Wu and H.Wulan and obtain a equivalent relationship between composition operators in hyperbolic Q_k spacers and Q_k spaces.