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1
Content available remote Entire and meromorphic solutions for systems of the differential difference equations
Xu Hong Yan, Li Hong, Ding Xin
Demonstratio Mathematica
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2022
|
Vol. 55, nr 1
676--694
EN
With the help of the Nevanlinna theory of meromorphic functions, the purpose of this article is to describe the existence and the forms of transcendental entire and meromorphic solutions for several systems of the quadratic trinomial functional equations: {f(z)2+2αf(z)g(z+c)+g(z+c)2=1,g(z)2+2αg(z)f(z+c)+f(z+c)2=1, {f(z+c)2+2αf(z+c)g'(z)+g'(z)2=1,g(z+c)2+2αg(z+c)f'(z)+f'(z)2=1, and {f(z+c)2+2αf(z+c)g′′(z)+g′′(z)2=1,g(z+c)2+2αg(z+c)f′′(z)+f′′(z)2=1. We obtain a series of results on the forms of the entire solutions with finite order for such systems, which are some improvements and generalizations of the previous theorems given by Gao et al. Moreover, we provide some examples to explain the existence and forms of solutions for such systems in each case.
2
Content available Fast growing solutions to linear differential equations with entire coefficients having the same ρϕ-order
Belaϊdi Benharrat
Journal of Mathematics and Applications
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2019
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Vol. 42
63--77
EN
This paper deals with the growth of solutions of a class of higher order linear differential equations f(k)+Ak-1(z)f(k-1)+ … +A1(z)f’+A0(z)f=0; k≥2 when most coefficients Aj (z) (j = 0, ..., k-1) have the same ρϕ-order with each other. By using the concept of τϕ-type, we obtain some results which indicate growth estimate of every non-trivial entire solution of the above equations by the growth estimate of the coefficient A0 (z). We improve and generalize some recent results due to Chyzhykov-Semochko and the author.
3
Content available remote Normal families and shared function II
Chengxiong Sun
Fasciculi Mathematici
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2019
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nr 62
141--154
EN
Let k, n ∈ N,l ∈ N\ {1} , m ∈ N U {0}, and a(z)(≠ 0) be a holomorphic function, all of whose zeros have multiplicities at most m. Let F be a family of meromorphic functions in D such that multiplicities of zeros of each f ∈ F are at least k + m. If for f, g ∈ F satisfy fl(f(k))n and gl(g(k))n share a(z), then F is normal in D. The examples are provided to show that the result is sharp. The result extends the related theorems [9,10,12]. we also omit the conditions “m is divisible by n +1” and “all poles of f have multiplicities at least m + 1” in the result due to Meng, Liu and Xu [12] [Journal of Computational Analysis and Applications 27(3)(2019), 511-526].
4
Content available On hyper-regularity and unimodularity of Ore polynomial matrices
Fritzsche K., Röbenack K.
International Journal of Applied Mathematics and Computer Science
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2018
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Vol. 28, no. 3
583--594
EN
We investigate Ore polynomial matrices, i. e., matrices with polynomial entries in d/dt whose coefficients are meromorphic functions in t and as such constitute a non-commutative ring. In particular, we study the properties of hyper-regularity and unimodularity of such matrices and derive conditions which make it possible to efficiently check for these characteristics. In addition, this approach enables computation of hyper-regular left and right and unimodular inverses.
5
Content available remote Normal families and shared functions
Sun C. X.
Fasciculi Mathematici
|
2018
|
Nr 60
173--180
EN
Let k ϵ N, m ϵ N U {0}, and let a(z)( ≡ 0) be a holomorphic function, all zeros of a(z) have multiplicities at most m. Let F be a family of meromorphic functions in D. If for each f ϵ F, the zeros of f have multiplicities at least k + m + 1 and all poles of f are of multiplicity at least m +1, and for f, g ϵ F, ff(k) - a(z) and gg(k) - a(z) share 0, then F is normal in D. Some examples are given to show that the conditions are best, and the result removes the condition “m is an even integer” in the result due to Sun [Kragujevac Journal of Math 38(2), 173-282, 2014].
6
Content available remote Non-linear differential polynomials sharing small function with finite weight
Harina W. P., Rajeshwari S.
Fasciculi Mathematici
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2017
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Nr 58
57--75
EN
The purpose of the paper is to study the uniqueness of entire and meromorphic functions sharing a small function with finite weight. The results of the paper improve and extend some recent results due to Abhijit Banerjee and Pulak Sahoo [3].
7
Content available remote Norm estimates for functions of non-selfadjoint operators nonregular on the convex hull of the spectrum
Gil’ M.
Demonstratio Mathematica
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2017
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Vol. 50, nr 1
267--277
EN
We consider a bounded linear operator A in a Hilbert space with a Hilbert-Schmidt Hermitian component (A−A*)/2i. A sharp norm estimate is established for functions of A nonregular on the convex hull of the spectrum. The logarithm, fractional powers and meromorphic functions of operators are examples of such functions. Our results are based on the existence of a sequence An(n = 1, 2,...) of finite dimensional operators strongly converging to A, whose spectra belongs to the spectrum of A. Besides, it is shown that the resolvents and holomorphic functions of An strongly converge to the resolvent and corresponding function of A.
8
Defective functions of meromorphic functions in the unit disc
Ciechanowicz E.
Journal of Applied Analysis
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2016
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Vol. 22, nr 1
15--26
EN
Let f be a meromorphic function in the unit disc and (aν)kν=1 a set of distinct meromorphic functions small with respect to f. An analogue of the second main theorem for f and (aν)kν=1 is given. Upper limits for the sum of defects of an admissible meromorphic function and an admissible holomorphic function follow. For meromorphic and holomorphic functions in the unit disc and their small functions the analogues of Ullrich's theorem are presented.
9
Content available remote Uniqueness of differential polynomials of meromorphic functions sharing a small function without counting multiplicity
Kieu-Oanh B. T., Thuy N. T. T.
Fasciculi Mathematici
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2016
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Nr 57
121--135
EN
The paper concerns interesting problems related to the field of Complex Analysis, in particular Nevanlinna theory of meromorphic functions. The author have studied certain uniqueness problem on differential polynomials of meromorphic functions sharing a small function without counting multiplicity. The results of this paper are extension of some problems studied by K. Boussaf et. al. in [2] and generalization of some results of S.S. Bhoosnurmath et. al. in [4].
10
Content available remote On an open problem of Xiao-Bin Zhang and Jun-Feng Xu
Majumder S.
Demonstratio Mathematica
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2016
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Vol. 49, nr 2
161--182
EN
The purpose of the paper is to study the uniqueness of meromorphic functions sharing a nonzero polynomial. The results of the paper improve and generalize the recent results due to X. B. Zhang and J. F. Xu [19]. We also solve an open problem as posed in the last section of [19].
11
Content available remote Growth of meromorphic solutions of finite logarithmic order of linear difference equations
Belaïdi B.
Fasciculi Mathematici
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2015
|
Nr 54
5--20
EN
In this paper, we deal with the growth and the oscillation of solutions of the linear difference equation an (z) f (z + n) + an-1 (z) f (z + n - 1) + ··· + a1 (z) f (z + 1) + a0 (z) f (z) = 0, where an(z), ···, a0(z) are meromorphic functions of finite logarithmic order such that an(z)a0(z) ≠0.
12
Content available remote Non linear differential polynomials sharing fixed points with finite weights
Banerjee A., Sahoo P.
Demonstratio Mathematica
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2013
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Vol. 46, nr 1
51--62
EN
We employ the notion of weighted sharing to investigate the uniqueness of meromorphic functions when two nonlinear differential polynomials share fixed points. The results of the paper improve and generalize the recent results due to Xu–Lu–Yi [10].
13
Content available remote Non linear differential polynomials sharing one value
Banerjee A.
Demonstratio Mathematica
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2007
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Vol. 40, nr 1
65-75
EN
We prove three uniqueness theorems concerning non linear dierential polynomials which will improve and supplement some earlier results given by Yang and Hua, Lahiri.
14
Notes on uniqueness of meromorphic functions
Lü W., Cao J.
Journal of Applied Analysis
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2006
|
Vol. 12, nr 2
259-265
EN
In this paper, we study the problem of uniqueness on meromorphic functions involving in differential polynomials and obtain some results which extend and improve the theorems of M. Fang and W. Hong et al.
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