Normality Criteria For Families Research Articles

Normality criteria involving differential polynomials

In this paper, we prove normality criteria for families of meromorphic functions involving sharing of a holomorphic function by a certain class of differential polynomials. Results in this paper extend the works of different authors carried out in recent years, for example Charak and Sharma (Turk J Math 41: 404–411, 2017), Fang and Yuan (J Aust Math Soc Ser A 68(3): 321–333, 2000) Jiang and Gao (Acta Math Sci 32(4): 1503–1512, 2012), Meng and Hu (Bull Malays Math Sci Soc 38:1331–1347, 2015), Sun (Kragujev J Math 38(2):273–282, 2014), Xiao et al. (Abstr Appl Anal Article ID 373910, 1–8, 2011), Yuan (Arch Math 96(5): 435–444 2011).

Normal families and multiple values

In this paper, we obtain some normality criteria for families of meromorphic functions, which improve and generalize related results of Gu, Yang–Chang, Schwick, and Datt–Kumar.

Normality criteria for families of zero-free meromorphic functions

AbstractIn this paper, we prove some normality criteria for families of zero-free meromorphic functions. The results of this paper are generalizations of some problems studied by Chang [Acta Math. Sin. (Engl. Ser.) 28 (2012), 707–716] and Dethloff–Tan–Thin [J. Math. Anal. Appl. 411 (2014), 675–683].

ON LIMIT BRODY CURVES IN C<b><i><sup>n </sup></i></b>AND (C<b><sup>∗</sup></b>)<sup>2 </sup>

In this paper, the conjecture on the Zalcmanness of Cn (n � 2) and (C�)2, which is posed in (8), is proved in the case where the derivatives of limit holomorphic curves are bounded. Moreover, several criteria for normality of families of holomorphic mappings are given.

Some normality criteria for families of meromorphic functions

Abstract In the note, we study the normality of families of meromorphic functions. We consider whether a family of meromorphic functions ℱ is normal in D if, for a normal family G and for each function f ∈ F , there exists g ∈ G such that f n f ′ = a i implies g n g ′ = a i for two distinct nonzero points a i , i = 1 , 2 and an integer n. An example shows that the condition in our results is best possible. MSC:30D45, 30D35.

Normality Criteria of Lahiri's Type and Their Applications

We prove two normality criteria for families of some functions concerning Lahiri's type, the results generalize those given by Charak and Rieppo, Xu and Cao. As applications, we study a problem related to R. Bruck's Conjecture and obtain a result that generalizes those given by Yang and Zhang, Lu, Xu and Chen.

Equicontinuous classes of ring Q-homeomorphisms

We give a description of ring Q-homeomorphisms in ℝn, n ≥ 2, and find a series of conditions for normality of families of ring Q-homeomorphisms. For a family to be normal it is sufficient that the dominant Q(x) have logarithmic-type singularities of order at most n-1. Another sufficient condition for normality is that Q(x) has finite mean oscillation at each point; for example, Q(x) has finite mean value over infinitesimal balls. The definition of ring Q-homeomorphism is motivated by the ring definition of Gehring for quasiconformality. In particular, the mappings with finite length distortion satisfy a capacity inequality that justifies the definition of ring Q-homeomorphism. Therefore, deriving consequences of the theory to be presented, we obtain criteria for normality of families of homeomorphisms f with finite length distortion and homeomorphisms of the Sobolev class W loc 1, in terms of the inner dilation K I (x, f). Moreover, the class of strong ring Q-homeomorphisms for a locally summable Q is closed.

Normal families and exceptional functions

In this paper, we obtain some normality criteria for families of meromorphic functions that concern the exceptional functions of derivatives.

Normality of meromorphic functions with multiple zeros and shared values

In this paper, we study the normality of a family of meromorphic functions and general criteria for normality of families of meromorphic functions with multiple zeros concerning shared values are obtained.

Picard values and normal families of meromorphic functions with multiple zeros

In this paper, general modular theorems are obtained for meromorphic functions and their derivatives. The related criteria for normality of families of meromorphic functions are proved.

Normality criteria for families of meromorphic functions

Normality criteria for families of meromorphic functions

Some criteria for normality of families of continuous functions

Some criteria for normality of families of continuous functions