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Criteria of normality concerning the sequence of omitted functions
Qiaoyu Chen,Jianming Qi 대한수학회 2016 대한수학회보 Vol.53 No.5
In this paper, we research the normality of sequences of meromorphic functions concerning the sequence of omitted functions. The main result is listed below. Let $\{f_{n}(z)\}$ be a sequence of functions meromorphic in $D$, the multiplicities of whose poles and zeros are no less than $k+2,~k\in \mathbb N$. Let $\{b_{n}(z)\}$ be a sequence of functions meromorphic in $D$, the multiplicities of whose poles are no less than $ k+1$, such that $b_{n}(z)\overset\chi\Rightarrow b(z)$, where $b(z)(\neq 0)$ is meromorphic in $D$. If $f^{(k)}_{n}(z)\ne b_{n}(z)$, then $\{f_{n}(z)\}$ is normal in $D$. And we give some examples to indicate that there are essential differences between the normal family concerning the sequence of omitted functions and the normal family concerning the omitted function. Moreover, the conditions in our paper are best possible.
CRITERIA OF NORMALITY CONCERNING THE SEQUENCE OF OMITTED FUNCTIONS
Chen, Qiaoyu,Qi, Jianming Korean Mathematical Society 2016 대한수학회보 Vol.53 No.5
In this paper, we research the normality of sequences of meromorphic functions concerning the sequence of omitted functions. The main result is listed below. Let {$f_n(z)$} be a sequence of functions meromorphic in D, the multiplicities of whose poles and zeros are no less than k + 2, $k{\in}\mathbb{N}$. Let {$b_n(z)$} be a sequence of functions meromorphic in D, the multiplicities of whose poles are no less than k + 1, such that $b_n(z)\overset{\chi}{\Rightarrow}b(z)$, where $b(z({\neq}0)$ is meromorphic in D. If $f^{(k)}_n(z){\neq}b_n(z)$, then {$f_n(z)$} is normal in D. And we give some examples to indicate that there are essential differences between the normal family concerning the sequence of omitted functions and the normal family concerning the omitted function. Moreover, the conditions in our paper are best possible.
A linear operator and associated families of meromorphically multivalent functions of order a
M. K. Aouf,H. M. Srivastava 장전수학회 2006 Advanced Studies in Contemporary Mathematics Vol.13 No.1
Making use of a linear operator, which is defined here by means of the Hadamard product (or convolution), we introduce two novel subclasses $Q_{a,c}(p,\alpha ;A,B)$ and $Q_{a,c}^{+}(p,\alpha ;A,B)$ of meromorphically multivalent functions of order $\alpha$ $(0\leqq \alpha <p)$ in the punctured unit disk $\mathbb{U}^{\ast}$. The main object of the present paper is to investigate the various important properties and characteristics of these subclasses of meromorphically multivalent functions. We extend the familiar concept of neighborhoods of analytic functions to these subclasses of meromorphically multivalent functions. We also derive many interesting results for the Hadamard products of functions belonging to the function class $Q_{a,c}^{+}(p,\alpha;A,B)$.
SOME RESULTS ON MEROMORPHIC SOLUTIONS OF CERTAIN NONLINEAR DIFFERENTIAL EQUATIONS
Li, Nan,Yang, Lianzhong Korean Mathematical Society 2020 대한수학회보 Vol.57 No.5
In this paper, we investigate the transcendental meromorphic solutions for the nonlinear differential equations $f^nf^{(k)}+Q_{d_*}(z,f)=R(z)e^{{\alpha}(z)}$ and f<sup>n</sup>f<sup>(k)</sup> + Q<sub>d</sub>(z, f) = p<sub>1</sub>(z)e<sup>α<sub>1</sub>(z)</sup> + p<sub>2</sub>(z)e<sup>α<sub>2</sub>(z)</sup>, where $Q_{d_*}(z,f)$ and Q<sub>d</sub>(z, f) are differential polynomials in f with small functions as coefficients, of degree d<sub>*</sub> (≤ n - 1) and d (≤ n - 2) respectively, R, p<sub>1</sub>, p<sub>2</sub> are non-vanishing small functions of f, and α, α<sub>1</sub>, α<sub>2</sub> are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of these kinds of meromorphic solutions and their possible forms of the above equations.
SUFFICIENT CONDITIONS FOR UNIVALENCE AND STUDY OF A CLASS OF MEROMORPHIC UNIVALENT FUNCTIONS
Bhowmik, Bappaditya,Parveen, Firdoshi Korean Mathematical Society 2018 대한수학회보 Vol.55 No.3
In this article we consider the class ${\mathcal{A}}(p)$ which consists of functions that are meromorphic in the unit disc $\mathbb{D}$ having a simple pole at $z=p{\in}(0,1)$ with the normalization $f(0)=0=f^{\prime}(0)-1$. First we prove some sufficient conditions for univalence of such functions in $\mathbb{D}$. One of these conditions enable us to consider the class ${\mathcal{A}}_p({\lambda})$ that consists of functions satisfying certain differential inequality which forces univalence of such functions. Next we establish that ${\mathcal{U}}_p({\lambda}){\subsetneq}{\mathcal{A}}_p({\lambda})$, where ${\mathcal{U}}_p({\lambda})$ was introduced and studied in [2]. Finally, we discuss some coefficient problems for ${\mathcal{A}}_p({\lambda})$ and end the article with a coefficient conjecture.
Two meromorphic functions sharing sets concerning small functions
Ting-Bin Cao 대한수학회 2009 대한수학회보 Vol.46 No.6
The main purpose of this paper is to deal with the uniqueness of meromorphic functions sharing sets concerning small functions. We obtain two main theorems which improve and extend strongly some results due to R. Nevanlinna, Li-Qiao, Yao, Yi, Thai-Tan, and Cao-Yi. The main purpose of this paper is to deal with the uniqueness of meromorphic functions sharing sets concerning small functions. We obtain two main theorems which improve and extend strongly some results due to R. Nevanlinna, Li-Qiao, Yao, Yi, Thai-Tan, and Cao-Yi.
Sufficient conditions for univalence and study of a class of meromorphic univalent functions
Bappaditya Bhowmik,Firdoshi Parveen 대한수학회 2018 대한수학회보 Vol.55 No.3
In this article we consider the class $\mathcal{A}(p)$ which consists of functions that are meromorphic in the unit disc $\ID$ having a simple pole at $z=p\in (0,1)$ with the normalization $f(0)=0=f'(0)-1 $. First we prove some sufficient conditions for univalence of such functions in $\ID$. One of these conditions enable us to consider the class $\mathcal{V}_{p}(\lambda)$ that consists of functions satisfying certain differential inequality which forces univalence of such functions. Next we establish that $\mathcal{U}_{p}(\lambda)\subsetneq \mathcal{V}_{p}(\lambda)$, where $\mathcal{U}_{p}(\lambda)$ was introduced and studied in \cite{BF-1}. Finally, we discuss some coefficient problems for $\mathcal{V}_{p}(\lambda)$ and end the article with a coefficient conjecture.
Tanmay Biswas 한국수학교육학회 2019 純粹 및 應用數學 Vol.26 No.4
Orders and types of entire and meromorphic functions have been actively investigated by many authors. In the present paper, we aim at investigating some basic properties in connection with sum and product of relative (p,q)-ϕ order, relative (p,q)-ϕ type, and relative (p,q)-ϕ weak type of meromorphic functions with respect to entire functions where p,q are any two positive integers and ϕ : [0,+∞)→(0,+∞) be a non-decreasing unbounded function.
Certain subclass of strongly meromorphic close-to-convex functions
Gagandeep Singh,Gurcharanjit Singh,Navyodh Singh 강원경기수학회 2024 한국수학논문집 Vol.32 No.1
The purpose of this paper is to introduce a new subclass of strongly meromorphic close-to-convex functions by subordinating to generalized Janowski function. We investigate several properties for this class such as coefficient estimates, inclusion relationship, distortion property, argument property and radius of meromorphic convexity. Various earlier known results follow as particular cases.
Value Distribution of L-functions and a Question of Chung-Chun Yang
Li, Xiao-Min,Yuan, Qian-Qian,Yi, Hong-Xun Department of Mathematics 2021 Kyungpook mathematical journal Vol.61 No.3
We study the value distribution theory of L-functions and completely resolve a question from Yang [10]. This question is related to L-functions sharing three finite values with meromorphic functions. The main result in this paper extends corresponding results from Li [10].