Benharrat Belaïdi - Academia.edu (original) (raw)
Papers by Benharrat Belaïdi
Vestnik Udmurtskogo universiteta, Aug 31, 2023
We study the solutions of the differential equation f (n) + An−1(z)f (n−1) + • • • + A1(z)f + A0(... more We study the solutions of the differential equation f (n) + An−1(z)f (n−1) + • • • + A1(z)f + A0(z)f = 0, where the coefficients are entire functions. We find conditions on the coefficients so that every solution that is not identically zero has infinite order.
Rad Hrvatske akademije znanosti i umjetnosti Matematičke znanosti
This paper is devoted to studying the growth and the oscillation of solutions of the second order... more This paper is devoted to studying the growth and the oscillation of solutions of the second order non-homogeneous linear differential equation f ′′ + Ae a 1 z f ′ + B (z) e a 2 z f = F (z) e a 1 z , where A, a 1 , a 2 are complex numbers, B (z) (̸ ≡ 0) and F (z) (̸ ≡ 0) are entire functions with order less than one. Moreover, we investigate the growth and the oscillation of some differential polynomials generated by solutions of the above equation.
DOAJ (DOAJ: Directory of Open Access Journals), Aug 1, 2019
In this paper, we deal with the growth of solutions of homogeneous linear complex differential eq... more In this paper, we deal with the growth of solutions of homogeneous linear complex differential equation by using the concept of lower [p,q]-order and lower [p,q]-type in a sector of the unit disc instead of the whole unit disc, and we obtain similar results as in the case of the unit disc.
DOAJ (DOAJ: Directory of Open Access Journals), 2016
In this article, we study the uniqueness of entire functions that share small functions of finite... more In this article, we study the uniqueness of entire functions that share small functions of finite order with their difference operators. In particular, we give a generalization of results in [3, 4, 13]. c (∆ c f (z)), n ∈ N, n ≥ 2. In particular, ∆ n c f (z) = ∆ n f (z) for the case c = 1. Let f and g be two meromorphic functions and let a be a finite nonzero value. We say that f and g share the value a CM provided that f − a and g − a have the same zeros counting multiplicities. Similarly, we say that f and g share a IM provided that f − a and g − a have the same zeros ignoring multiplicities. It is well-known that if f and g share four distinct values CM, then f is a Möbius transformation of g. Rubel and Yang [15] proved that if an entire function f shares two distinct complex numbers CM with its derivative f , then f ≡ f. In 1986, Jank et al [10] proved that for a nonconstant meromorphic function f , if f , f and f share a finite nonzero value CM, then f ≡ f. This result suggests the following question: Question 1 in [17]. Let f be a nonconstant meromorphic function, let a be a finite nonzero constant, and let n and m (n < m) be 2010 Mathematics Subject Classification. 30D35, 39A32.
Journal of Inequalities in Pure & Applied Mathematics, 2004
In this paper, we study the possible orders of transcendental solutions of the differential equat... more In this paper, we study the possible orders of transcendental solutions of the differential equation f (n) + a n−1 (z) f (n−1) + • • • + a 1 (z) f + a 0 (z) f = 0, where a 0 (z) ,. .. , a n−1 (z) are nonconstant polynomials. We also investigate the possible orders and exponents of convergence of distinct zeros of solutions of non-homogeneous differential equation f (n) + a n−1 (z) f (n−1) + • • • + a 1 (z) f + a 0 (z) f = b (z) , where a 0 (z) ,. .. , a n−1 (z) and b (z) are nonconstant polynomials. Several examples are given.
Annals of the University of Craiova - Mathematics and Computer Science Series, Jun 30, 2021
In this paper, we investigate the growth of solutions of higher order linear differential equatio... more In this paper, we investigate the growth of solutions of higher order linear differential equations with analytic coefficients of ϕ-order in the unit disc. We introduce new definitions of the lower order and the type related to the ϕ-order concepts to generalise and extend previous results due to Chyzhykov-Semochko [6], Semochko [14], Belaïdi [1,2,3], Hu-Zheng [12].
Kyungpook Mathematical Journal, 2005
In this paper we will investigate the growth of solutions of certain class of nonhomogeneous line... more In this paper we will investigate the growth of solutions of certain class of nonhomogeneous linear differential equations with entire coefficients having the same order and type. This work improves and extends some previous results in [1], [7] and [9].
Theory and Applications of Mathematics & Computer Science, Aug 29, 2020
International Journal of Mathematics and Statistics, May 15, 2011
Periodica Mathematica Hungarica, Feb 10, 2013
We consider the complex differential equations f +A 1 (z)f +A 0 (z)f = F and where A 0 ≡ 0, A 1 a... more We consider the complex differential equations f +A 1 (z)f +A 0 (z)f = F and where A 0 ≡ 0, A 1 and F are analytic functions in the unit disc Δ = {z : |z| < 1}. We obtain results on the order and the exponent of convergence of zero-points in Δ of the differential polynomials g f = d 2 f + d 1 f + d 0 f with non-simultaneously vanishing analytic coefficients d 2 , d 1 , d 0. We answer a question posed by J. Tu and C. F. Yi in 2008 for the case of the second order linear differential equations in the unit disc.
WSEAS transactions on mathematics, Jun 14, 2022
In this paper, we deal with the complex oscillation of solutions of linear differential equation.... more In this paper, we deal with the complex oscillation of solutions of linear differential equation. We mainly study the interaction between the growth, zeros of solutions with the coefficients of second order linear differential equations in terms of (α, β, γ)-order and obtain some results in general form which considerably extend some results of [5], [18] and [21].
arXiv (Cornell University), Apr 3, 2019
In this paper, we deal with the growth of solutions of homogeneous linear complex differential eq... more In this paper, we deal with the growth of solutions of homogeneous linear complex differential equation by using the concept of lower [p,q]-order and lower [p,q]-type in a sector of the unit disc instead of the whole unit disc, and we obtain similar results as in the case of the unit disc.
arXiv (Cornell University), Dec 24, 2022
In this article, we study the growth of meromorphic solutions of linear delay-differential equati... more In this article, we study the growth of meromorphic solutions of linear delay-differential equation of the form n i=0 m j=0 A ij (z)f (j) (z + c i) = F (z), where A ij (z) (i = 0, 1,. .. , n, j = 0, 1,. .. , m, n, m ∈ N) and F (z) are meromorphic of finite logarithmic order, c i (i = 0,. .. , n) are distinct non-zero complex constants. We extend those results obtained recently by Chen and Zheng, Bellaama and Belaïdi to the logarithmic lower order.
Electronic Journal of Differential Equations
In this article, we study the order of growth for solutions of the non-homogeneous linear delay-d... more In this article, we study the order of growth for solutions of the non-homogeneous linear delay-differential equation sumi=0nsumj=0mAijf(j)(z+ci)=F(z),\sum_{i=0}^n\sum_{j=0}^{m}A_{ij}f^{(j)} (z+c_i)=F(z),sumi=0nsumj=0mAijf(j)(z+ci)=F(z), where \(A_{ij}(z)\) \((i=0,\dots ,n;j=0,\dots ,m)\), \(F(z)$\)are entire or meromorphic functions and \(c_i\) \((0,1,\dots ,n)\) are non-zero distinct complex numbers. Under the condition that there exists one coefficient having the maximal lower order, or having the maximal lower type, strictly greater than the order, or the type, of the other coefficients, we obtain estimates of the lower bound of the order of meromorphic solutions of the above equation. For more information see https://ejde.math.txstate.edu/Volumes/2021/92/abstr.html
IOCMA 2023
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
International Conference on Differential Equations and Dynamical Systems, 2021
Nonlinear Studies, Feb 22, 2021
Nonlinear Studies, 2018
In this paper, we give new conditions on the fast-growing entire and meromorphic coefficients of ... more In this paper, we give new conditions on the fast-growing entire and meromorphic coefficients of linear complex differential equations to estimate the iterated ppp-order and iterated ppp-type of all solutions, where pinmathbbNbackslash0,1p\in \mathbb{N}\backslash \{0,1\}pinmathbbNbackslash0,1. We also, give an improvement to some previous results, as in \cite{hamouda}.
Scientific Publications of the State University of Novi Pazar Series A: Applied Mathematics, Informatics and mechanics, 2018
In this article, we investigate the growth of solutions of second order complex differential equa... more In this article, we investigate the growth of solutions of second order complex differential equations in which the coefficients are analytic in the unit disc with lower [p, q]-order. We've proved similar results as in the case of complex differential equations in the whole complex plane with usual [p, q]-order. We define also new type of order applied on the coefficients to study the growth of solutions.
Vestnik Udmurtskogo universiteta, Aug 31, 2023
We study the solutions of the differential equation f (n) + An−1(z)f (n−1) + • • • + A1(z)f + A0(... more We study the solutions of the differential equation f (n) + An−1(z)f (n−1) + • • • + A1(z)f + A0(z)f = 0, where the coefficients are entire functions. We find conditions on the coefficients so that every solution that is not identically zero has infinite order.
Rad Hrvatske akademije znanosti i umjetnosti Matematičke znanosti
This paper is devoted to studying the growth and the oscillation of solutions of the second order... more This paper is devoted to studying the growth and the oscillation of solutions of the second order non-homogeneous linear differential equation f ′′ + Ae a 1 z f ′ + B (z) e a 2 z f = F (z) e a 1 z , where A, a 1 , a 2 are complex numbers, B (z) (̸ ≡ 0) and F (z) (̸ ≡ 0) are entire functions with order less than one. Moreover, we investigate the growth and the oscillation of some differential polynomials generated by solutions of the above equation.
DOAJ (DOAJ: Directory of Open Access Journals), Aug 1, 2019
In this paper, we deal with the growth of solutions of homogeneous linear complex differential eq... more In this paper, we deal with the growth of solutions of homogeneous linear complex differential equation by using the concept of lower [p,q]-order and lower [p,q]-type in a sector of the unit disc instead of the whole unit disc, and we obtain similar results as in the case of the unit disc.
DOAJ (DOAJ: Directory of Open Access Journals), 2016
In this article, we study the uniqueness of entire functions that share small functions of finite... more In this article, we study the uniqueness of entire functions that share small functions of finite order with their difference operators. In particular, we give a generalization of results in [3, 4, 13]. c (∆ c f (z)), n ∈ N, n ≥ 2. In particular, ∆ n c f (z) = ∆ n f (z) for the case c = 1. Let f and g be two meromorphic functions and let a be a finite nonzero value. We say that f and g share the value a CM provided that f − a and g − a have the same zeros counting multiplicities. Similarly, we say that f and g share a IM provided that f − a and g − a have the same zeros ignoring multiplicities. It is well-known that if f and g share four distinct values CM, then f is a Möbius transformation of g. Rubel and Yang [15] proved that if an entire function f shares two distinct complex numbers CM with its derivative f , then f ≡ f. In 1986, Jank et al [10] proved that for a nonconstant meromorphic function f , if f , f and f share a finite nonzero value CM, then f ≡ f. This result suggests the following question: Question 1 in [17]. Let f be a nonconstant meromorphic function, let a be a finite nonzero constant, and let n and m (n < m) be 2010 Mathematics Subject Classification. 30D35, 39A32.
Journal of Inequalities in Pure & Applied Mathematics, 2004
In this paper, we study the possible orders of transcendental solutions of the differential equat... more In this paper, we study the possible orders of transcendental solutions of the differential equation f (n) + a n−1 (z) f (n−1) + • • • + a 1 (z) f + a 0 (z) f = 0, where a 0 (z) ,. .. , a n−1 (z) are nonconstant polynomials. We also investigate the possible orders and exponents of convergence of distinct zeros of solutions of non-homogeneous differential equation f (n) + a n−1 (z) f (n−1) + • • • + a 1 (z) f + a 0 (z) f = b (z) , where a 0 (z) ,. .. , a n−1 (z) and b (z) are nonconstant polynomials. Several examples are given.
Annals of the University of Craiova - Mathematics and Computer Science Series, Jun 30, 2021
In this paper, we investigate the growth of solutions of higher order linear differential equatio... more In this paper, we investigate the growth of solutions of higher order linear differential equations with analytic coefficients of ϕ-order in the unit disc. We introduce new definitions of the lower order and the type related to the ϕ-order concepts to generalise and extend previous results due to Chyzhykov-Semochko [6], Semochko [14], Belaïdi [1,2,3], Hu-Zheng [12].
Kyungpook Mathematical Journal, 2005
In this paper we will investigate the growth of solutions of certain class of nonhomogeneous line... more In this paper we will investigate the growth of solutions of certain class of nonhomogeneous linear differential equations with entire coefficients having the same order and type. This work improves and extends some previous results in [1], [7] and [9].
Theory and Applications of Mathematics & Computer Science, Aug 29, 2020
International Journal of Mathematics and Statistics, May 15, 2011
Periodica Mathematica Hungarica, Feb 10, 2013
We consider the complex differential equations f +A 1 (z)f +A 0 (z)f = F and where A 0 ≡ 0, A 1 a... more We consider the complex differential equations f +A 1 (z)f +A 0 (z)f = F and where A 0 ≡ 0, A 1 and F are analytic functions in the unit disc Δ = {z : |z| < 1}. We obtain results on the order and the exponent of convergence of zero-points in Δ of the differential polynomials g f = d 2 f + d 1 f + d 0 f with non-simultaneously vanishing analytic coefficients d 2 , d 1 , d 0. We answer a question posed by J. Tu and C. F. Yi in 2008 for the case of the second order linear differential equations in the unit disc.
WSEAS transactions on mathematics, Jun 14, 2022
In this paper, we deal with the complex oscillation of solutions of linear differential equation.... more In this paper, we deal with the complex oscillation of solutions of linear differential equation. We mainly study the interaction between the growth, zeros of solutions with the coefficients of second order linear differential equations in terms of (α, β, γ)-order and obtain some results in general form which considerably extend some results of [5], [18] and [21].
arXiv (Cornell University), Apr 3, 2019
In this paper, we deal with the growth of solutions of homogeneous linear complex differential eq... more In this paper, we deal with the growth of solutions of homogeneous linear complex differential equation by using the concept of lower [p,q]-order and lower [p,q]-type in a sector of the unit disc instead of the whole unit disc, and we obtain similar results as in the case of the unit disc.
arXiv (Cornell University), Dec 24, 2022
In this article, we study the growth of meromorphic solutions of linear delay-differential equati... more In this article, we study the growth of meromorphic solutions of linear delay-differential equation of the form n i=0 m j=0 A ij (z)f (j) (z + c i) = F (z), where A ij (z) (i = 0, 1,. .. , n, j = 0, 1,. .. , m, n, m ∈ N) and F (z) are meromorphic of finite logarithmic order, c i (i = 0,. .. , n) are distinct non-zero complex constants. We extend those results obtained recently by Chen and Zheng, Bellaama and Belaïdi to the logarithmic lower order.
Electronic Journal of Differential Equations
In this article, we study the order of growth for solutions of the non-homogeneous linear delay-d... more In this article, we study the order of growth for solutions of the non-homogeneous linear delay-differential equation sumi=0nsumj=0mAijf(j)(z+ci)=F(z),\sum_{i=0}^n\sum_{j=0}^{m}A_{ij}f^{(j)} (z+c_i)=F(z),sumi=0nsumj=0mAijf(j)(z+ci)=F(z), where \(A_{ij}(z)\) \((i=0,\dots ,n;j=0,\dots ,m)\), \(F(z)$\)are entire or meromorphic functions and \(c_i\) \((0,1,\dots ,n)\) are non-zero distinct complex numbers. Under the condition that there exists one coefficient having the maximal lower order, or having the maximal lower type, strictly greater than the order, or the type, of the other coefficients, we obtain estimates of the lower bound of the order of meromorphic solutions of the above equation. For more information see https://ejde.math.txstate.edu/Volumes/2021/92/abstr.html
IOCMA 2023
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
International Conference on Differential Equations and Dynamical Systems, 2021
Nonlinear Studies, Feb 22, 2021
Nonlinear Studies, 2018
In this paper, we give new conditions on the fast-growing entire and meromorphic coefficients of ... more In this paper, we give new conditions on the fast-growing entire and meromorphic coefficients of linear complex differential equations to estimate the iterated ppp-order and iterated ppp-type of all solutions, where pinmathbbNbackslash0,1p\in \mathbb{N}\backslash \{0,1\}pinmathbbNbackslash0,1. We also, give an improvement to some previous results, as in \cite{hamouda}.
Scientific Publications of the State University of Novi Pazar Series A: Applied Mathematics, Informatics and mechanics, 2018
In this article, we investigate the growth of solutions of second order complex differential equa... more In this article, we investigate the growth of solutions of second order complex differential equations in which the coefficients are analytic in the unit disc with lower [p, q]-order. We've proved similar results as in the case of complex differential equations in the whole complex plane with usual [p, q]-order. We define also new type of order applied on the coefficients to study the growth of solutions.