亚洲免费av电影一区二区三区,日韩爱爱视频,51精品视频一区二区三区,91视频爱爱,日韩欧美在线播放视频,中文字幕少妇AV,亚洲电影中文字幕,久久久久亚洲av成人网址,久久综合视频网站,国产在线不卡免费播放

        ?

        Normal functions,normal families and uniformly normal families

        2023-01-03 07:48:02ZhuTingYangLiu

        Zhu Ting,Yang Liu

        (School of Mathematics and Physics,Anhui University of Technology,Maanshan 243032,China)

        Abstract:In this paper,the definition of the spherical derivative of meromorphic functions is extended to more general form concerning higher-order derivatives.By the wellknown Zalcman Lemma and the extended spherical derivative,we study the normality criteria for families of meromorphic functions.The theorems obtained extend Lappan′s five-point theorems and Marty′s criterion,and improve some previous results on normal functions and normal families.

        Keywords:meromorphic functions,normal functions,normal family,spherical derivative,value distribution theory

        1 Introduction

        LetFbe a family of meromorphic functions on a domainD?C.ThenFis said to be normal onDin the sense of Montel,if each sequence ofFcontains a subsequence which converges spherically uniformly on each compact subset ofDto a meromorphic function which may be∞identically[1].By the famous normality criterion of Reference[2],a familyFof meromorphic functions onDis normal if and only if the family{f#:f∈F}of the corresponding spherical derivativesis locally uniformly bounded.

        As to generalizations of Marty′s theorem to higher derivatives,Reference[3]obtained the following result.

        Theorem 1.1Letkbe a natural number andFa family of functions meromorphic onDall of whose zeros have multiplicity at leastk.ThenFis normal if and only if the familyis locally uniformly bounded.

        To simplify its statement,we use the following notation.For a meromorphic functionfinDand a positive integerk,the expression

        represents an extension of the spherical derivative off.

        Let?rdenote the disc of radiusrin the complex plane.We will denote the unit disc by?.Iffis a meromorphic function in?,thenfis said to be normal if the set of functions{f?T}forms a normal family in?whereTranges over the conformal mappings of? onto?.A necessary and sufficient condition for a functionfmeromorphic in?to be normal in?is that the real function

        is bounded in?[4].

        Reference[5]posed the following interesting question:ifM>0 is given,does there exist a finite setEsuch that iffis meromorphic in?then the condition that(1?|z|2)f#(z)≤Mfor eachz∈f?1(E)implies thatfis a normal function?In 1974,Reference[6]gave an affirmative answer to the question.

        Theorem 1.2LetEbe any set consisting of five complex numbers, finite or in finite.Iffis a meromorphic function on?such that

        thenfis a normal function.

        In the same paper,Lappan also commented on the sharpness of the number five in Theorem 1.2,showing that there are at least some cases in which “ five” can not be replaced by “four”.

        Recently,for a meromorphic functionfin?,Reference[7]considered not only the bound of the real function Φf(z)but also the bound of the spherical derivative off′and reduce the number five.

        Theorem 1.3LetEbe any set consisting of four complex numbers, finite or in finite.Iffis a meromorphic function on?such that

        thenfis a normal function.

        Note that iffis a normal function,then there exists a constantM(f)such that

        for eachz∈?.In general,the constantM(f)depends on the functionf.By the idea duce to Reference[8],Reference[9]gave the following.

        Definition 1.1LetFbe a family of meromorphic functions in the unit disc?.We call the familyFis uniformly normal on?if

        Theorem 1.4LetFbe a family of meromorphic functions in the unit disc?such that all the zeros offinFare of multiplicity at leastk,and letEbe any set consisting ofk+4 complex numbers,finite or infinite.If

        thenFis uniformly normal on?.

        Motivated by the accomplishment of Marty′s criterion and Lappan′s five-point theorem,many authors studied the problems relate to spherical derivative(see References[10-12]).Inspire by the results above,in this paper we prove the following.

        Theorem 1.5LetFbe a family of meromorphic functions in the unit disc?such that all the zeros offinFare of multiplicity at leastk,and letEbe any set consisting ofcomplex numbers,finite or infinite.If

        thenFis uniformly normal on?.Here and in the following,[x]denotes the greatest integer less than or equal tox.

        Obviously,ifFis an uniformly normal family on?,then each functionfinFmust be a normal function.Therefore,we obtain

        Corollary 1.1Letfbe a meromorphic functions in the unit disc?such that all of whose zeros have multiplicity at leastk,and letEbe any set consisting ofcomplex numbers,finite or infinite.If

        thenfis a normal function.

        In addtion,takingk=1,we get the following corollary of Theorem 1.5 which is an improvement of Theorem 1.3.

        Corollary 1.2LetFbe a family of meromorphic functions in the unit disc?,and letEbe any set consisting of four complex numbers,finite or infinite.If

        thenFis uniformly normal on?.

        An analogous for normal families of meromorphic functions is also obtained as follows.

        Theorem 1.6LetFbe a family of meromorphic functions in the domainDsuch that all the zeros offinFare of multiplicity at leastk.Assume that for each compact subsetK?D,there exists a setE=E(K)be any set consisting ofcomplex numbers,finite or infinite.If

        thenFis normal onD.

        The plan of this paper is as follows.In Section 2,we state a number of auxiliary results.In Section 3,we give the proofs of theorems.

        2 Lemmas

        To prove our results,we require some lemmas.We assume the standard notation of value distribution theory.For details,see References[1,13-14].

        Lemma 2.1[15](Zalcman′s Lemma)LetFbe a family of meromorphic functions of a domainDin C.IfFis not normal at a pointz0∈D,then there exist sequenceswithzn→z0,andwith?n→0+,such that

        converges uniformly on compact subsets of C to a nonconstant meromorphic functionF(ξ)of C.

        Lemma 2.2[13-14](First Main Theorem) Suppose thatfis meromorphic in C andais any complex number.Then forr>0 we have

        Lemma 2.3[13-14](Second Main Theorem) Suppose thatfis a non-constant meromorphic in C andaj(1≤j≤q)areq(≥3)distinct values inbC.Then

        Lemma 2.4[13-14]Suppose thatfis a non-constant meromorphic in C andkis a positive integer.Then

        3 Proofs

        人妻在线中文字幕| 草草影院发布页| 日本大片免费观看视频| 国产看黄网站又黄又爽又色| 日韩免费高清视频网站| 青草草视频在线观看华人免费| 亚洲av无码专区国产不卡顿| 末发育娇小性色xxxxx视频| 精品无吗国产一区二区三区av| 日本国产精品高清在线| 亚洲人成网站在线播放2019 | 成人做爰视频www| 国产视频在线一区二区三区四区| 亚洲一区域二区域三区域四| 伊人久久精品无码av一区| 久热综合在线亚洲精品| 久久与欧美视频| 亚洲一区二区三区熟妇| 国产成人精品999视频| 久久精品国产9久久综合| 日本一区二区三区在线播放 | 99久久精品国产成人综合| 欧美zozo另类人禽交| 国产免费操美女逼视频| 少妇熟女天堂网av| 亚欧AV无码乱码在线观看性色 | 韩国女主播一区二区三区在线观看 | 女人一级特黄大片国产精品 | 亚洲国产人成自精在线尤物| 激情综合色综合啪啪开心| 人妻无码一区二区三区四区| 无码AⅤ最新av无码专区| 日本最新视频一区二区| 亚洲av不卡一区二区三区| 伊人一道本| 国产一区二区美女主播| 337p粉嫩日本欧洲亚洲大胆| 中文字幕无码精品亚洲资源网久久| 亚洲va精品va国产va| 国产禁区一区二区三区| 三男一女吃奶添下面|