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On the growth of solutions of second order linear differential equations with extremal coefficients
Abstract
In this paper, we consider the differential equation f″ +A(z)f′ +B(z)f = 0, where A and B ≢ 0 are entire functions. Assume that A is extremal for Yang’s inequality, then we will give some conditions on B which can guarantee that every non-trivial solution f of the equation is of infinite order.
... for any nonzero constant C and for any non-constant polynomial B(z). There are many conditions on coefficients A(z) and B(z) so that all non-trivial solutions of Eq. (1) are of infinite order [19,22,23]. In this context, the authors [11] have proved that if B(z) is a transcendental entire function in Question 1 such that coefficients of Eq. (1) are of different order then conclusion of Question 1 holds true. ...
... Lemma 11 [19] Suppose that B(z) is an entire function extremal to Yang's inequality and there exists arg ...
We show that all non-trivial solutions of complex differential equation are of infinite order if coefficients A(z) and B(z) are of special type and establish a relation between the hyper-order of these solutions and the orders of coefficients A(z) and B(z). We have also extended these results to higher order complex differential equations.
... Therefore, it is very interesting topic to find condition on A(z) and B(z) guaranteeing every nontrivial solution of (1.1) is of infinite order. Many parallel results written thereafter Theorem 1.1 focus on the case ρ(A) ≥ ρ(B) and ρ(A) > 1 2 ; see, for example, [5,17,19,20,21,22,30,34]. In 2011, Wu and Zhu proved the following result, in which the concept of deficient value is concerned. ...
... , n. It is not hard to see that the function e P (z) has the property that there are n disjoint sectors satisfying δ(P, θ) > 0, and n other disjoint sectors satisfying δ(P, θ) < 0, see [21]. Without loss of generality, set S + j (θ j , φ j ) = {z : δ(P, θ) > 0, θ j < arg z = θ < φ j }, j = 1, 2, . . . ...
... Then every nontrivial solution f of the equation (1) has infinite order. This is a hot research object and a lot of works have sprung up, such as [7,11,15,21,[23][24][25][31][32][33]. Our main purpose is continue to study the above question, try to find conditions which A(z), B(z) should satisfy to ensure that nontrivial solution of (1) has infinite order. ...
In this paper we discuss the classical problem of finding conditions on the entire coefficients A(z) and B(z) guaranteeing that all non-trivial solutions of f + A(z)f + B(z)f = 0 are of infinite order. We assume A(z) is an entire function of completely regular growth and B(z) satisfies three different conditions, then we obtain three results respectively. The three conditions are (1) B(z) has a dynamical property with a multiply connected Fatou component, (2) B(z) satisfies T (r, B) ∼ log M (r, B) outside a set of finite logarithmic measure, (3) B(z) is extremal for Den-joy's conjecture.
... , k − 1, k ≥ 2. The growth of solutions of (1) is very interesting topic after Wittich's work [16], the main tool is Nevanlinna theory of meromorphic functions which can be found in [6,10,18]. Many results have been obtained by many different researchers, for the case of complex plane C, see, for example, [10][11][12][13]17] and therein references, for the case of unit disc D, see, for example [1,2,4,7,14] and therein references. Recently, Fettouch and Hamouda investigated the growth of solutions of equation (1) by using a new idea, in which the coefficients are analytic function except a finite singular point, more details can be found in [3,5]. ...
We investigate the [p,q]-order of growth of solutions of the following complex linear differential equation f(k)+Ak-1(z) f(k-1) + ...+ A1(z) f? + A0(z) f = 0, where Aj(z) are analytic in C? - {z0}, z0 ? C. Some estimations of [p,q]-order of growth of solutions of the equation are obtained, which is generalization of previous results from Fettouch-Hamouda.
... Therefore, finding certain conditions on A(z) and B(z) which can guarantee every non-trivial solution of (1.1) is of infinite order is an interesting topic remained to be discussed. Meanwhile, many parallel results can be found in [2,14,15,17,19]. ...
The growth of solutions of complex linear differential equation is studied, where A(z) and B(z) are entire functions. With some conditions on A(z) and B(z), we prove that every non-trivial solution f of the equation is of infinite order. Moreover, we obtain the lower bound of measure of angular domain, in which the radial order of f is infinite. Some examples are given to illustrate the results.
... In [14], J.R .Long proved that if A(z) is an entire function extremal for Yang's inequality and B(z) a transcendental entire function with ρ(B) = ρ(A), then all solution of the equation (1) are of infinite order. ...
In this paper, we establish transcendental entire function A(z) and polynomial B(z) such that the differential equation f ′′ + A(z)f ′ + B(z)f = 0, has all non-trivial solution of infinite order. We use the notion of critical rays of the function e P (z) , where A(z) = d(z)e P (z) with some restrictions.
In this paper we discuss the classical problem of finding conditions that the entire coefficients A(z) and B(z) should satisfy to ensure all nontrivial solutions of f′′+A(z)f′+B(z)f=0 are of infinite order. Two different approaches are used. In the first approach the entire coefficient B(z) has dynamical property with a multiply connected Fatou component, and A(z) is extremal for Yang’s inequality or a nontrivial solution of w′′+P(z)w=0, where P(z) is a polynomial. In the second approach B(z) satisfies T(r,B)∼logM(r,B) outside a set of finite logarithmic measure, and A(z) has the same properties as in the first approach.
We show that the higher order linear differential equation possesses all solutions of infinite order under certain conditions by extending the work of authors about second order differential equation \cite{dsm2}.
In this paper, we establish transcendental entire function A(z) and polynomial B(z) such that the differential equation f +A(z)f +B(z)f = 0, has all non-trivial solution of infinite order. We use the notion of critical rays of the function e P (z) , where A(z) = d(z)e P (z) with some restrictions.
Let f(z) be an entire function of positive and finite order /z. If /(2) has a finite number of Borel directions of order > /z, then the sum of numbers of finite nonzero deficient values of f(z) and all its primitives does not exceed 2/z. The proof is based on several lemmas and application of harmonic measure.
Suppose g and h are entire functions with the order of h less than the order of g. If the order of g does not exceed ½, it is shown that every (necessarily entire) nonconstant solution f of the differential equation f″ + gf′ + hf = 0 has infinite order. This result extends previous work of Ozawa and Gundersen.
We consider the differential equation /" + A(z)f' B(z)f = 0 where A(z) and B(z) are entire functions. We will find conditions on A(z) and B(z) (Formula present) which will guarantee that every solution 0 of the equation will have infinite order. We will also find conditions on A(z) and B[z) which will guarantee that any finite order solution 0 of the equation will not have zero as a Borel exceptional value. W e will also show that if A(z) and B(z) satisfy certain growth conditions, then any finite order solution of the equation will satisfy certain other growth conditions. Related results are also proven. Several examples are given to complement the theory.
Suppose g and h are entire functions with the order of h less than the order of g. If the order of g does not exceed , it is shown that every (necessarily entire) nonconstant solution f of the differential equation has infinite order. This result extends previous work of Ozawa and Gundersen.
This paper investigates the growth of solutions of the equation f '' +e-zf ' +Q(z)f=0 where the order (Q)=1. When Q(z)=h(z)e bz h(z) is a nonzero polynomial, b≠-1 is a complex constant, every solution of the above equation has infinite order and the hyper-order 1. We improve results due to M. Frei, M. Ozawa, G. Gundersen and J. K. Langley.