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Growth of solutions of second order linear differential equations
Abstract
This paper is devoted to studying the growth of solutions of equations of type f″+h(z)eazf′+Q(z)f=H(z) where h(z), Q(z) and H(z) are entire functions of order at most one. We prove four theorems of such type, improving previous results due to Gundersen and Chen.
... The conditions on the coe‰cients of equation (1) have been studied so that all solutions of (1) are of infinite order [2,16,17]. In [7,8,9,10,11], certain conditions have been established on the coe‰cients of associated homogeneous di¤erential equation (2) for the existence of non-trivial solutions of infinite order. ...
... The following lemma proved by Wang and Laine [17] provides an estimate for an entire function f in the neighbourhood of the points in which f takes the maximum modulus. ...
... Case (b). Let dðP; y 0 Þ < 0. Again, as dðP; yÞ is a continuous function, therefore 3 2 dðP; y 0 Þ < dðP; y k Þ < 1 2 dðP; y 0 Þ solutions of non-homogeneous linear differential equations for all su‰ciently large k A N. Using part (ii) of Lemma 2, we have exp 3ð1 þ eÞ 2 dðP; y 0 Þr n k a jAðr k e iy k Þj a exp ð1 À eÞ 2 dðP; y 0 Þr n k ð24Þ for all su‰ciently large k A N. Using equations (1), (17), (19) and (24), we obtain jBðr k e iy k Þj a f 00 ðr k e iy k Þ f ðr k e iy k Þ þ jAðr k e iy k Þj ...
... Á 0/ are meromorphic functions of finite order. Some results on the growth of entire solutions of (1.1) have been obtained by several researchers (see [5,6,12,14]). Li and Wang (see [12]) investigated the nonhomogeneous linear differential equation ...
... < 1 2 ; and b is a real constant. They proved that all nontrivial solutions of (1.2) are of infinite order, provided that .H / < 1: After their, Wang and Laine (see [14]) studied the differential equation ...
... [14]). Suppose that A 0 6 Á 0; A 1 6 Á 0; H are entire functions of order less than one, and the complex constants a; b satisfy ab ¤ 0 and a ¤ b: Then every nontrivial solution f of (1.3) is of infinite order.J. ...
This paper is devoted to studying the growth and oscillation of solutions and their derivatives of equations of the type f 00 C A .´/ f 0 C B .´/ f D F .´/ ; where A .´/ ; B .´/ .6 Á 0/ and F .´/ .6 Á 0/ are meromorphic functions of finite order.
... Recently, some mathematicians investigate the non-homogeneous equations of second order and higher order linear equations such as Li and Wang [18], Cao [5], Wang and Laine [22] and proved that every solution of these equation has infinite order. ...
... In 2008, Wang and Laine [22] investigated the non-homogeneous equation related to (2) and obtained the following result. ...
... Theorem C(see. [22,Theorem 1.1]) Suppose that A j ≡ 0(j = 0, 1), H are entire functions of order less than one, and the complex constants a, b satisfy ab = 0 and a = b. ...
In this paper, we deal with the order of growth and the hyper order of solutions of higher order linear differential equations where Bj (z) (j = 0, 1, ..., k - 1) and F are entire functions or polynomials. Some results are obtained which improve and extend previous results given by Z.-X. Chen, J. Wang, T.-B. Cao and C.-H. Li.
... Ilpo and Steve proved several results concerning the particular cases when A(z) is either a polynomial or a rational function in (3.5); see [2], [4]. In [3] they investigated equation (3.1) when A(z) is an entire periodic function. The next theorem is one example of their many results of this type and equation (3.4) was used in the proof. ...
... Theorem 3.4 ([3]). Consider equation (3.1) ...
... Equation (3.7) has been studied by several authors. Theorem 3.5 ([3] ...
... In [19], Wang and Laine have investigated the growth of solutions of some second order non-homogenous linear differential equations and have obtained the following result. ...
... Theorem 1.5. [19] Let A j (z) (̸ ≡ 0) (j = 0, 1) and H (z) be entire functions with max{ρ (A j ) (j = 0, 1) , ρ (H)} < 1, and let a, b be complex constants that satisfy ab ̸ = 0 and a ̸ = b. Then every non-trivial solution f of the equation Thus, the following question arises naturally: Whether the results similar to Theorem 1.5 can be obtained if ρ(H) = 1? ...
... Kumar and Saini [8] consider λ(A) < ρ(A) and B(z) a transcendental entire function satisfying either ρ(B) = ρ(A) or B(z) having Fabry gap and proved that non-trival solutions of (1) are of infinite order. J. Wang and I. Laine [12] consider A(z) = h(z)e −z and B(z) to satisfy ...
... Theorem A [12] Suppose that A(z) and B(z) are entire functions where A(z) = h(z)e −z and ρ(h) < ρ(B) = 1, and that B(z) satisfies (2) in a set of positive upper logarithmic density. Then every non-trivial solution f of equation (1) is of infinite order. ...
In this paper, we will prove that all non-trivial solutions of are of infinite order, where we have some restrictions on entire functions A(z) and B(z).
... Here, we employ two results to handle this proposition, the first one is due to Wang and Laine [25,Lemma 2.4], the latter one is due to Gundersen [9, Corollary 2]. ...
... It is related to a famous differential equation question posed by Gundersen in [11]. We refer to [6,10,25] for some results of the above differential equation. It is natural to generalize the above differential equation to f + Π 1 (z)e az f + Π 2 e bz f = Π 3 , where the infinite product Π i is defined as in the main theorem. ...
The paper is mainly devoted to deriving the relationship between an entire function and its derivative when they share one small function except possibly a set, which is related to the famous Brück conjecture. In addition, two propositions of infinite products are obtained. The first one is the growth property of a certain infinite product. The second one is the property of entire solutions of the differential equation which concerns infinite products.
... Recently in [14], Wang and Laine have investigated the growth of solutions of some second order linear differential equations and have obtained. ...
... Theorem 3 (see [14]). Let A j (z) ( ≡ 0) (j = 0, 1) and F (z) be entire functions with max{ρ (A j ) (j = 0, 1) , ρ (F )} < 1, and let a, b be complex constants that satisfy ab = 0 and a = b. ...
This paper is devoted to studying the growth and the oscillation of solutions of the second-order nonhomogeneous linear differential equation f '' +A 1 (z)e P(z) f ' +A 0 (z)e Q(z) f=F, where P(z),Q(z) are nonconstant polynomials such that deg P = deg Q = n and A j (z)(¬≡0) for j=0,1 and F(z) are entire functions with max{ρ(A j ):j=0,1}<n. We also investigate the relationship between small functions and differential polynomials g f (z)=d 2 f '' +d 1 f ' +d 0 f, where d 0 (z),d 1 (z),d 2 (z) are entire functions such that at least one of d 0 ,d 1 ,d 2 does not vanish identically with ρ(d j )<n for j=0,1,2 generated by solutions of the above equation.
... In[10], Wang and Laine investigated the growth of solutions of some second order nonhomogeneous linear differential equation and obtained. Theorem A([10]). ...
... In[10], Wang and Laine investigated the growth of solutions of some second order nonhomogeneous linear differential equation and obtained. Theorem A([10]). Let A j (z) ( ≡ 0) (j = 0, 1) and F (z) be entire functions with max{σ (A j ) (j = 0, 1) , σ (F )} < 1, and let a, b be complex constants that satisfy ab = 0 and a = b. ...
In this paper, we study the order of growth of solutions to the non-homogeneous linear differential equation az}f
... Wang and Laine investigated the nonhomogeneous equation (4) and got the following; see [7]. ...
... Lemma 6 (see [7]). ...
We prove that the hyperorder of every nontrivial solution of homogenous linear differential equations of type and nonhomogeneous equation of type is one, where are entire functions of order less than one, improving the previous results of Chen, Wang, and Laine.
... The general problem of Ozawa has been proposed by Heittokangas, and given by Gundersen, see [10,Section 5]. More and more papers focus on the existence of finite order solutions of second linear differential equations (1.3) with entire coefficients, even precise form of finite order solutions, see [14,15,17,24,36,39]. ...
... The general problem of Ozawa has been proposed by Heittokangas, and given by Gundersen, see [10,Section 5]. More and more papers focus on the existence of finite order solutions of second linear differential equations (1.3) with entire coefficients, even precise form of finite order solutions, see [14,15,17,24,36,39]. ...
We show a necessary and sufficient condition on the existence of finite order entire solutions of linear differential equations f^{(n)}+a_{n-1}f^{(n-1)}+\cdots+a_1f'+a_0f=0,\eqno(+) where are exponential sums for with all positive (or all negative) rational frequencies and constant coefficients. Moreover, under the condition that there exists a finite order solution of (+) with exponential sum coefficients having rational frequencies and constant coefficients, we give the precise form of all finite order solutions, which are exponential sums. It is a partial answer to Gol'dberg-Ostrovski\v{i} Problem and Problem 5 in \cite{HITW2022} since exponential sums are of completely regular growth.
... We are motivated by Theorem E and replace the condition of B(z) to satisfy some conditions given in Theorem 3. The following Lemma yield us a lower bound for modulus of an entire function in the neighbourhood of θ, where θ ∈ [0, 2π). Lemma 8. [23] Suppose f (z) is an entire function of finite order ρ and M (r, f ) = |f (re ιθr )| for every r. Given ζ > 0 and 0 < C(ρ, ζ) < 1, there exists 0 < l 0 < 1 2 and a set S ⊂ (1, ∞) with log dens(S) ≥ 1 − ζ such that ...
In this article, we study about the solutions of second order linear differential equations by considering several conditions on the coefficients of homogenous linear differential equation and its associated non-homogenous linear differential equation.
... In several papers like [12], [14], [15], [21] and [23] authors have considered A(z) = h(z)e P (z) , where P (z) is a polynomial of degree n satisfying λ(A) < ρ(A) and B(z) with various conditions. Here we have exchanged the conditions of A(z) with B(z) and prove the following result. ...
Considering differential equation f''+A(z)f'+B(z)f=0, where A(z) and B(z) are entire complex functions, our results revolve around proving all non-trivial solutions are of infinite order taking various restrictions on coefficients A(z) and B(z).
... Wang and Laine [8] have established that when ρ(A) = ρ(B) and ρ(H) < max{ ρ(A), ρ(B)} , then all solutions of equation (1) are of infinite order. In our case we have assumed ρ(A) = ρ(B) and proved the following theorem: ...
In this paper, we have considered second order non-homogeneous linear differential equations having entire coefficients. We have established conditions ensuring non-existence of finite order solution of such type of differential equations.
... Proof. Following the argument in [18,Lemma 2.4] and Lemma 2, we obtain ...
We investigate transcendental entire solutions of complex differential equations f″+A(z)f=H(z), where the entire function A(z) has a growth property similar to the exponential functions, and H(z) is an entire function of order less than that of A(z). We first prove that the lower order of the entire solution to the equation is infinity. By using our result on the lower order, we prove the entire solution does not bear any Baker wandering domains.
... The proof deviates from that of [8, Theorem 6] in the sense that we make use of [2], which is not needed in proving [8, Theorem 6]In the remaining results the functions T (r, A) and log M (r, A) are asymptotically comparable, where A(z) is the coefficient function in (2). Prior to Theorem D, the same conclusion was proved under the stronger assumptions that ρ(B) < ρ(A) < ∞ and T (r, A) ∼ log M (r, A) as r → ∞ outside of a set of finite logarithmic measure, see [16]. Next we state an analogous result involving the lower order and a class of entire functions A(z) satisfying T (r, A) ∼ α log M (r, A), r → ∞, ...
... The problem of finding conditions on A(z) and B (z) for the case ρ(A) ≥ ρ(B ) under which all nontrivial solutions of (1.1) are of infinite order has raised considerable interest, and many parallel results on this question written after Theorem A, see, for example, [3,4,14,15,17,18,22]. It would be interesting to get some relations between the growth of solutions of (1.1) and some deep results in Nevanlinna theory of meromorphic functions. ...
The classical problem of finding conditions on the entire coefficients A(z) and B(z) guaranteeing that all nontrivial solutions of f″ + A(z) f′+ B(z) f = 0 are of infinite order is discussed. Some such conditions which involve deficient value, Borel exceptional value and extremal functions for Denjoy's conjecture are obtained.
... holds for all m ≥ 0 and all r / ∈ F . Ä ÑÑ 2.6º (see [23]) Let f (z) be an entire function of finite order ρ, and M (r, f ) = f (re iθ r ) for every r. Given ζ > 0 and 0 < C(ρ, ζ) < 1, there exists a constant 0 < l 0 < 1 2 and a set E ζ of lower logarithmic density greater than 1 − ζ such that ...
In this paper, we consider a problem of meromorphic functions that share an arbitrary set having two elements with their derivatives. We obtain a uniqueness result which is an improvement of some related theorems given by Lü and Xu’s. We also generalize the famous Brück conjecture with the idea of sharing a set.
... What can we have if there exists one middle coefficient í µí°´íµí°´í µí± (í µí± §) (1 ≤ í µí± ≤ í µí±−1) such that í µí°´íµí°´í µí± (í µí± §) grows faster than other coefficients in (12) or (13)? Many authors have investigated this question when í µí°´íµí°´í µí± (í µí± §) is of finite order and obtained many results (e.g., see131415). Here, our question is that under what conditions can we obtain similar results with Theorems B-C if í µí°´íµí°´í µí± (í µí± §) (1 ≤ í µí± ≤ í µí± − 1) is of finite iterated order and grows faster than other coefficients in (12) or (13). ...
The authors introduce the lacunary series of finite iterated order and use them to investigate the growth of solutions of higher-order linear differential equations with entire coefficients of finite iterated order and obtain some results which improve and extend some previous results of Belaidi, 2006, Cao and Yi, 2007, Kinnunen, 1998, Laine and Wu, 2000, Tu and Chen, 2009, Tu and Deng, 2008, Tu and Deng, 2010, Tu and Liu, 2009, and Tu and Long, 2009.
... Chen [3] got the same conclusion when a cb c > 1 , and Chen and Shon [5] investigated the more general equations with meromorphic coefficients. Under the same assumption of Theorem A, if A 1 z and A 0 z are meromorphic functions with σ A j < 1 j 0, 1 , then there is the same conclusion with Theorem A. In 2008, Wang and Laine [6] extended Theorem A to nonhomogeneous second-order linear differential equations. ...
This paper is devoted to studying the growth of solutions of second-order nonhomogeneous linear differential equation with meromorphic coefficients. We also discuss the relationship between small functions and differential polynomials L(f)=d2f″+d1f′+d0f generated by solutions of the above equation, where d0(z),d1(z), and d2(z) are entire functions that are not all equal to zero.
... Our paper is motivated by the papers [11,12]. In these papers, the authors considered the special differential equation, f + A 1 (z)e P 1 (z) f + A 0 (z)e P 0 (z) f = 0, (4.5) ...
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.
The growth of solutions of second order complex differential equations f + A(z)f + B(z)f = 0 with transcendental entire coefficients is considered. Assuming that A(z) has a finite deficient value and that B(z) has either Fabry gaps or a multiply connected Fatou component, it follows that all solutions are of infinite order of growth.
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}.
For a transcendental entire function f(z) in the complex plane, we study its divided differences Gn(z). We partially prove a conjecture posed by Bergweiler and Langley under the additional condition that the lower order of f(z) is smaller than 12. Furthermore, we prove that if zero is a deficient value of f(z), then δ(0,G)<1, where G(z)=(f(z+c)−f(z))/f(z).
MSC:
30D35.
We investigate the hyper-order of solutions of two class of complex linear differential equations. We investigate the growth of solutions of higher order and certain second order linear differential equations, and we obtain some results which improve and extend some previous results in complex oscillations.
MSC: 30D35, 34M10.
The main purpose of this paper is to consider the differential equation where P is a polynomial with in general complex coefficients. Let be the zeros of a nonzero solution u to that equation. We obtain bounds for the sums which extend some recent results proved by Gil'.
The main purpose of this paper is to consider the differential equation u((m)) = P (z)u (m >= 2) where P is a polynomial with complex, in general, coefficients. Let z(k)(u), k = 1, 2, ... be the zeros of a nonzero solution u to that equation. We obtain bounds for the sums Sigma(j)(k=1) 1/vertical bar z(k)(u)vertical bar (j is an element of N) which extend some recent results proved by Gil'.
In this paper, we investigate the growth of meromorphic solutions of the equations f ( k ) + A k − 1 ( z ) f ( k − 1 ) + ⋯ + A 1 ( z ) f ′ + A 0 ( z ) f = 0 , f ( k ) + A k − 1 ( z ) f ( k − 1 ) + ⋯ + A 1 ( z ) f ′ + A 0 ( z ) f = F ( z ) , where A 0 ( z ) (≢0), A 1 ( z ) , … , A k − 1 ( z ) and F ( z ) (≢0) are entire functions of finite order. We find some conditions on the coefficients to guarantee that every nontrivial meromorphic solution of such equations is of infinite order. MSC:30D35, 34M10.
We consider the equation y '' =F(z)y(z∈ℂ) with an entire function F satisfying the condition |F(z)|≤Aexp|z| p p(p≥1,A=const>0)· Let z k (y),k=1,2,⋯, be the zeros of a solution y(z) to the above equation. Bounds for the sum ∑ k=1 j 1 |z k (y)|(j=1,2,⋯) are established. Some applications of these bounds are also considered.
We investigate the growth properties of solutions and the existence of subnormal solutions for a class of higher linear differential equations, and obtain several results which improve and extend some results of Chen, Hamouda and Belaïdi.
We give some growth properties for solutions of linear complex differential equations which are closely related to the Brück Conjecture. We also prove that the Brück Conjecture holds when certain proximity functions are relatively small.
We consider the value distribution of a class of the functions analytic in the exterior of a disc and their applications to complex oscillation theory of differential equations with periodic coefficients in the complex plane.
We investigate the growth of solutions of higher-order nonhomogeneous linear differential equations with meromorphic coefficients. We also discuss the relationship between small functions and solutions of such equations.
Biconfluent Heun equation (BHE) is a confluent case of the general Heun
equation which has one more regular singular points than the Gauss
hypergeometric equation on the Riemann sphere . Motivated by
a Nevanlinna theory (complex oscillation theory) approach, we have established
a theory of \textit{periodic} BHE (PBHE) in parallel with the Lam\'e equation
verses the Heun equation, and the Mathieu equation verses the confluent Heun
equation. We have established condition that lead to explicit construction of
eigen-solutions of PBHE, and their single and double orthogonality, and a
related first-order Fredholm-type integral equation for which the corresponding
eigen-solutions must satisfy. We have also established a Bessel polynomials
analogue at the BHE level which is based on the observation that both the
Bessel equation and the BHE have a regular singular point at the origin and an
irregular singular point at infinity on the Riemann sphere ,
and that the former equation has orthogonal polynomial solutions with respect
to a complex weight. Finally, we relate our results to an equation considered
by Turbiner, Bender and Dunne, etc concerning a quasi-exact solvable
Schr\"odinger equation generated by first order operators such that the second
order operators possess a finite-dimensional invariant subspace in a Lie
algebra of
In this paper, we consider the differential equation
, where
and
are meromorphic functions,
is a non-constant polynomial. Assume that
has an infinite deficient value and finitely many Borel directions. We give some conditions on
which guarantee that every solution
of the equation has infinite order.
MSC:
34AD20, 30D35.
We consider the differential equation f'' + Af' + Bf = 0 where A(z) and B(z) ≢ 0 are mero-morphic functions. Assume that A(z) belongs to the Edrei-Fuchs class and B(z) has a deficient value ∞, if f ≢ 0 is a meromorphic solution of the equation, then f must have infinite order.
Mathematical Subject Classification 2000: 34M10; 30D35.
Purpose
We consider a pair of homogeneous linear differential equations with transcendental entire coefficients of finite order, and the question of when solutions of these equations can have the same zeros or nearly the same zeros.
Method
We apply the Nevanlinna theory and properties of entire solutions of linear differential equations.
Conclusion
The results determine all pairs of such equations having solutions with the same zeros, or nearly the same zeros.
Two perturbation results on the linear differential function f″ + Π(z)A(z)f = 0 are obtained, where Π(z) and A(z) are periodic entire functions with period 2πi and σ
e(Π) < σ
e(A).
In this paper we discuss the growth of solutions of the higher order nonhomogeneous linear differential equation where a, b are complex constants that satisfy ab(a − b) ≠ 0 and A
j
(z) (j = 0, 1, …, k − 1), D
j
(z) (j = 0, 1), F(z) are entire functions with max{ϱ(A
j
) (j = 0, 1, …, k − 1), ϱ(D
j
) (j = 0, 1) < 1. We also investigate the relationship between small functions and the solutions of the above equation.
Suppose that a homogeneous linear differential equation has entire coefficients of finite order and a fundamental set of solutions each having zeros with finite exponent of convergence. Upper bounds are given for the number of zeros of these solutions in small discs in a neighbourhood of infinity.
We consider two non-homogeneous first order differential equations and we use Nevanlinna theory to determine when the solutions of these differential equations can have the same zeros or nearly the same zeros.
A Bank-Laine function is an entire function E such that E(z) = 0 implies that E’(z) = ±1. Such functions arise as the product of linearly independent solutions of a second order linear differential equation ω″ + A(z)ω = 0 with A entire. Suppose that
where R is a rational function, g is a polynomial, and the αj and βj,k are non-zero complex numbers, and that E’(z) = ±1 at all but finally many zeros z of E. Then the quotients αj/αj′ are all rational numbers and E is a Bank-Laine function and reduces to the form E(z) = P0 (eαz) eQ0(z) with α a non-zero complex number and P0 and Q0 polynomials.
We consider the class of differential equations
are constants, k ≥ 3 and where A0(z) is a non-constant periodic entire function, which is a rational function of e
z. In this paper we develop a method that enables us to decide if this equation can have solutions with few zeros, and we also present the construction of these solutions.
Purpose
Two problems are discussed in this paper. In the first problem, we consider one homogeneous and one non-homogeneous differential equations and study when the solutions of these differential equations can have (nearly) the same zeros. In the second problem, we consider two linear second-order differential equations and investigate when the solutions of these differential equations can take the value 0 and a non-zero value at (nearly) the same points.
Method
We apply the Nevanlinna theory and properties of entire solutions of linear differential equations.
Conclusion
In the first problem, the results determine all pairs of such equations having solutions with the same zeros or nearly the same zeros. Regarding the second problem, the results also show all pairs of such equations having solutions taking the value 0 and a non-zero value at (nearly) the same points.
In this paper, we consider the differential equation f″ + Af′ + Bf = 0, where A(z) and B(z) ≠ 0 are entire functions. Assume that A(z) has a finite deficient value, then we will give some conditions on B(z) which can guarantee that every solution f ≠ 0 of the equation has infinite order.
This paper is devoted to studying the growth problem, the zeros and fixed points distribution of the solutions of linear differential
equations f″+e
−z
f′+Q(z)f = F(z), where Q(z) ≡ h(z)e
cz
and c ∊ ℝ.
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.
1 Basic Nevanlinna theory.- 2 Unicity of functions of finite (lower) order.- 3 Five-value, multiple value and uniqueness.- 4 The four-value theorem.- 5 Functions sharing three common values.- 6 Three-value sets of meromorphic functions.- 7 Functions sharing one or two values.- 8 Functions sharing values with their derivatives.- 9 Two functions whose derivatives share values.- 10 Meromorphic functions sharing sets.
Synopsis
We show that if B(z) is either (i) a transcendental entire function with order (B) ≠1, or (ii) a polynomial of odd degree, then every solution f ≠0 to the equation f ″ + e −z f ′ + B(z)f = 0 has infinite order. We obtain a partial result in the case when B(z) is an even degree polynomial. Our method of proof and lemmas for case (i) of the above result have independent interest.
This paper investigates the growth of solutions of the equation f″ + e-z f′ + Q(z)f = 0 where the order (Q) = 1. When Q(z) = h(z)ebz, h(z) is nonzero polynomial, b ≠ - 1 is a complex constant, every solution of the above equation has infinite order and the
hyper-order 1. We improve the results of M. Frei, M. Ozawa, G. Gundersen and J. K. Langley.
It is shown that if Q(z) is a non-constant polynomial, then all non-trivial solutions of y"+ (ez + Q(z))y = 0 have zeros with infinite exponent of convergence. Similar methods are used to settle a problem of M. Ozawa: if P(z) is a non-constant polynomial, all non-trivial solutions of y"+e-zy' + P(z)y = 0 have infinite order.
Ozawa Non-existence of finite order solutions of Hokkaido Math
- I Amemiya
- I Amemiya
Non-existence of finite order solutions of w&Prime;+e - zw'+Q(z)w=0, Hokkaido math
- I Amemiya
- M Ozawa
- I Amemiya
- M Ozawa
On a solution of w&Prime;+e - zw'+(az+b)w=0, Kodai math
- M Ozawa
- M Ozawa