NORMAL FAMILIES AND SHARED VALUES OF HOLOMORPHIC FUNCTIONS

Let F be a family of holomorphic functions in a domain D, k be a positive integer, a, b（≠0）, c（≠0） and d be finite complex numbers. If, for each f∈F, all zeros of f-d have multiplicity at least k, f^（k） = a whenever f=0, and f=c whenever f^（k） = b, then F is normal in D. This result extends the well-known normality criterion of Miranda and improves some results due to Chen-Fang, Pang and Xu. Some examples are provided to show that our result is sharp.

NORMAL FAMILIES OF MEROMORPHIC FUNCTIONS WITH SHARED VALUES

We obtain some normality criteria of families of meromorphic functions sharing values related to Hayman conjecture, which improves some earlier related results.

BOREL'S DIRECTIONS AND SHARED VALUES

In this article, we study the problems of Borel＇s directions of meromorphic func- tions concerning shared values and prove that if two meromorphie functions of infinite order share three distinct values, their Borel＇s directions are same.

Normal families and shared values of meromorphic functions

We studied the normality conditions in families of meromorphic functions, improved the results of Fang and Zalcman [Fang ML, Zalcman L, Normal families and shared values of meromorphic functions, Computational Methods and Function Theory, 2001, 1 （1）： 289-299], and generalized two new normality criterions. Let F be a family of meromorphic functions in a domain D, a a non-zero finite complex number, B a positive real number, and k and m two positive integers satisfying m〉2k＋4. If every function denoted by f belonging to F has only zeros with multiplicity at least k and satisfies f^m（z）f^（k）（Z）=α→ ｜^f（k）（z）｜ ≤B or f^m（z）f^（k）（z）=α→｜f（z）｜ ≥, then F is normal in D.

Normality and Shared Value Concerning Differential Polynomials

Let F be a family of functions meromorphic in a domain D, let m, n k , k be three positive integers and b be a finite nonzero complex number. Suppose that, (1) for eachf∈F, all zeros of f have multiplicities at least k ; (2) for each pair of functions f, g ∈F,P(f)H(f) and P(g)H(g) share b, where P(f) and H(f) were defined as (1.1) and (1.2) and nk ≥ max 1≤i≤k-1 {n i }; (3) m ≥ 2 or nk ≥ 2, k ≥ 2, then F is normal in D.

Shared Value in Practice： TOMS Business Model

The article deals with the idea of building a company around the corporate social responsibility （CSR） principles and designing a business model that is simultaneously economically viable, lucrative, and of social benefits. TOMS is analyzed as an example of a company that took this approach and succeeded. The main research questions try to examine the basic assumptions of TOMS business model in light of shared value concept and uncover the reasons of company＇s success. The authors claim that two factors played there a crucial role. Firstly the leader whose passion was so contagious that he/she managed to build the team and set up a company based on his/her ideas. Secondly TOMS business model concept （One for One） is convincing for customers who eagerly join the movement. This business case illustrates how a company that is based on social values can grow.

Shared Values of Certain Nonlinear Differential Polynomials of Meromorphic Functions

In this paper, by using the idea of truncated counting functions of meromorphic functions, we deal with the problem of uniqueness of the meromorphic functions whose certain nonlinear differential polynomials share one finite nonzero value.

Curating Spaces of Hope: Coproducing Shared Values in Uncertain Times

This paper sets out a new paradigm of faith based organisation(FBO)called Curating Spaces of Hope.The paper sets out the paradigm and the interdisciplinary literatures into which the paradigm is applied namely,the diversifying belief landscape in the UK,the postsecular,the redefinition of FBOs,and liminality as the new norm in policy.The paper then turns to ethnographic research to evidence the ability of the paradigm to map and coproduce shared values,before considering applications of Curating Spaces of Hope in post-pandemic contexts in the north west of England through case studies with ecumenical Christian,non-religious,and Turkish Muslim and interfaith contexts.

On meromorphic functions sharing one value

The uniqueness of meromorphic functions with one sharing value and an equality on deficiency is studied. We show that if two nonconstant meromorphic functions f(z) and g(z) satisfy δ(0,f)+δ(0,g)+δ(∞,f)+δ(∞,g)=3 or δ 2(0,f)+δ 2(0,g)+δ 2(∞,f)+δ 2(∞,g)=3, and E(1,f)=E(1,g) then f(z),g(z) must be one of five cases.

UNIQUENESS OF ENTIRE FUNCTIONS SHARING ONE VALUE

We deal with the problem of entire functions sharing one value weakly. Moreover, we improve and generalize some former results obtained by J.-F.Chen, et al. [6], Y.Xu and H.L.Qiu [4], M.L. Fang [5], C.C. Yang, and X.H. Hua [3].

ON THE SHARING VALUES OF ALGEBROID FUNCTIONS AND THEIR DERIVATIVES

In this paper, we define the shared value of an algebroid function and its derivative on its Riemann surface. By considering the relationship between the shared values and the branch points of algebroid functions and their derivatives, we obtain some uniqueness theorems of algebroid functions sharing values with their derivatives, which extend 3 IM shared values theorem of nonconstant meromorphic functions and their derivatives obtained by Mues-Steinmetz and Gundersen.

UNIQUENESS OF MEROMORPHIC FUNCTIONS WHOSE NONLINEAR DIFFERENTIAL POLYNOMIALS SHARE ONE VALUE OR HAVE THE SAME FIXED POINTS IN AN ANGULAR DOMAIN

In this article, we study the uniqueness question of nonconstant meromorphic functions whose nonlinear differential polynomials share 1 or have the same fixed points in an angular domain. The results in this article improve Theorem 1 of Yang and Hua [26], and improve Theorem 1 of Fang and Qiu [6].

Uniqueness Problems for Meromorphic Functions that Share Three Values

This paper investigate the uniqueness problems for meromorphic functions that share three values CM and proves a uniqueness theorem on this topic which can be used to improve some previous related results.

Normal Families of Holomorphic Functions Concerning Shared a Polynomial

In this paper,we study normal families of holomorphic function concerning shared a polynomial.Let F be a family of holomorphic functions in a domain D,k(2)be a positive integer,K be a positive number andα(z)be a polynomial of degree p(p 1).For each f∈F and z∈D,if f and f sharedα(z)CM and|f(k)(z)|K whenever f(z)-α(z)=0 in D, then F is normal in D.

Normality Criteria of Meromorphic Functions Concerning Shared Fixed-points

In this paper, we study the normality criteria of meromorphic functions concerning shared fixed-points, we obtain: Let F be a family of meromorphic functions defined in a domain D. Let n, k ≥ 2 be two positive integers. For every f ∈ F, all of whose zeros have multiplicity at least (nk+2)/(n-1). If f(f(k))nand g(g(k))nshare z in D for each pair of functions f and g, then F is normal.

Entire Functions That Share Pair of Values with Their Derivatives

This paper deals with problems of the uniqueness of entire functions that share one pair of values with their derivatives. The results in this paper generalize and improve a result of Jank, Mues and Volkmann, a result of YANG L Z and a result of R Brück.

Normal families related to shared sets

We studied the normality criterion for families of meromorphic functions related to shared sets. Let F be a family of meromorphic functions on the unit disc △, a and b be distinct non-zero values, S={a,b}, and k be a positive integer. If for every f∈ F, i） the zeros of f（z） have a multiplicity of at least k＋ 1, and ii） E^-f（k）（S） lohtain in E^-f（S）, then F is normal on .4. At the same time, the corresponding results of normal function are also proved.

Partial or Truncated Sharing Values of Meromorphic Functions with Their Shifts

We mainly study the periodicity theorems of meromorphic functions having truncated or partial sharing values with their shifts, where meromorphic functions are of hyper order less than 1 and N（r, f） aT（r; f） for some positive number a.

The Expression of Slow-growing Meromorphic Functions Sharing One Value CM with Their Derivatives

In this paper, we study the relations between meromorphic functions and their derivatives with one shared values, and obtain a concrete expression of meromorphic functions of zero order that share one value CM with their derivatives by a new method. Our main result is the supplementary of a related result due to Li and Yi(Li X M, Yi H X. Uniqueness of meromorphic functions sharing a meromorphic function of a small order with their derivatives. Ann. Polon. Math., 2010, 98(3):201–219).

Entire functions sharing a value with its weight

A uniqueness theorem for entire functions sharing one finite complex value with weight two is proved by using Nevanlinna theory , and this improves the result of Fang and Hua.