John Disk And Nehari Class

In this paper we mainly discussed some equivalent conditions of John disk and Nehari class with the growth of extremal metric and some characters of Nehari class.The notion of John disk was introduced by F. John in 1961 when he investigate the theory of the elastic theory. Because John disk is closely connected with several metrics, it has become important investigate object on complex analytic dymamical systems, approximation theory and flexibility and has attracted widespread interest.Since Krass had proved a necessary condition for univalence of a function which is analytic in the unit disc, in 1949, Nehari use the area method, that is, through to the units on the disk to be analytic functions series expansion of the coefficient estimates, proved the result by himself.The theory has played a pioneering role in discuss the connection of Schwarzian derivative and conformal mapping, it reveals the connection of the growth of Schwarzian derivative and univalence. Nehari's work immediately aroused widespread concern. Henceforth, in the analytic function of the univalence in the study also slowly introduced Nehari class of such a concept, likewise, many scholars at home and abroad have aroused great interest to Nehari class.The research based on the definition of John disk, through the properties of Nehari class and Schwarzian derivative , by use of the method of function to meet the general conditions of Nehari univalence of the function f.Investigated the condition of f(D) to be John disk, at the same time we discussed the growth of extremal metric of Nehari clss and the boundary behavior of the image of unit disc under the mapping of Nehari class. This paper is divided into four chapters:Chapter 1,Preface. In this chapter, we briefly introduce the investigate situation of Nehari and John disk, as well as Schwarzian derivatives and related properties.Chapterâ…¡,The equivalent conditions of John disk. John disk can be regarded as a quasidisks to meet the conditions of Unilateral conditions, to meet the general conditions of Nehari univalence of the function f,discussed f(D) to be John disk's conditions.Chapterâ…¢,On the growth of the extremal metric of a Nehari class. In this chapter, we consider locally injective and analytic function f,which satisfy |S_f(z)|≤2a(l+(1-a)|z|²)(1-|z|²)²,1≤a≤2.and f(0)=0,f'(0)=1,f"(0)=0,By considering the extreme metricu(f(rζ))=[(1-r²)²|f'(rζ)|]^-1 investigated the radial growth of extremal metric u, obtain thecorresponding lower bounds of |(?)log u|.Chapterâ…£,Some characteristic properties of Nehari class. In this chapter we relay a equvalence condition of John disk, investigate the bound of spherical metric of Nehari class and local injective analytic function f which meet certain normalized conditions, the boundarybehavior of Nehari class.