On the Growth of Certain Meromorphic Solutions of Arbitrary Second Order Algebraic Differential Equations

Bank, Steven A.

doi:10.2307/2036752

Cited by 4 publications

(5 citation statements)

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“…Bank proved in [4] that if a meromorphic solution f of (1.3) satisfies N (r, a j , f) = O(e r c ) for a j , j = 1, 2 belongs to the extended complex planeĈ where c is some positive constant, then f satisfies (1.4). This result improved upon Bank's own result [3] where a weaker assumption that N (r, a j , f) = O(r c ) for a j , j = 1, 2 is assumed. In fact, Gol'dberg [7] proved a stronger result for a special subclass of (1.9).…”

Section: Theorem 12 (Steinmetz) Suppose That Insupporting

confidence: 57%

On the meromorphic solutions of an equation of Hayman

Chiang

Halburd

2003

Journal of Mathematical Analysis and Applications

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The behaviour of meromorphic solutions to differential equations has been the subject of much study. Research has concentrated on the value distribution of meromorphic solutions and their rates of growth. The purpose of the present paper is to show that a thorough search will yield a list of all meromorphic solutions to a multi-parameter ordinary differential equation introduced by Hayman. This equation does not appear to be integrable for generic choices of the parameters so we do not find all solutions -only those that are meromorphic. This is achieved by combining Wiman-Valiron theory and local series analysis. Hayman conjectured that all entire solutions of this equation are of finite order. All meromorphic solutions of this equation are shown to be either polynomials or entire functions of order one.2000 Mathematics Subject Classification. Primary 34A34, 34M05, 30D35; Secondary 34M35, 34M55.

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Section: Theorem 12 (Steinmetz) Suppose That Insupporting

confidence: 57%

On the meromorphic solutions of an equation of Hayman

Chiang

Halburd

2003

Journal of Mathematical Analysis and Applications

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“…In our first result of this paper (Theorem 1), we answer this question in the affirmative even for those meromorphic functions g for which T(r, g)/(r(log r)) tends to 0 on a sequence {/•"} -» +oo. An important tool used in the proof is a result of the author which was first proved in [1], and then improved in [2, §4]. This result sets forth an estimate for the growth of a meromorphic solution y(z) of an algebraic differential equation F = 0, having meromorphic coefficients, in terms of the growth of the coefficients and the counting functions for the distinct zeros and distinct poles of y(z), provided that y(z) is not a solution of some equation F q = 0, where F tl denotes the homogeneous part of F of total degree q in the indeterminates y, y ' , .…”

Section: T(ry) = R(logr)/7t+o(r)mentioning

confidence: 99%

“…Now if ^ = F / r , then it is easily verified [see 5, Lemma 3.5, p. 73], that for each 7=Sl, r O ) /r can be written as a polynomial in *&,"¥',... ," l ¥ ( ' 1) , with constant coefficients. Hence, if we divide both the numerator and denominator of the quotient on the right side of (14) by T q , we obtain the conclusion of Theorem 2.…”

Section: T M (Z)) (14)mentioning

confidence: 99%

Some results on the Gamma function and other hypertranscendental functions

Bank

1978

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…We are now able to complete the proof of the Lemma. Coming back to equation (1) we remember that F(z, w, w', . .…”

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A remark on meromorphic solutions of differential equations

Strelitz¹

1977

Proc. Amer. Math. Soc.

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The following assertion is proven. Suppose the functions F, P, Q of the differential equation F(z, w, w', w(',)) = P(z, w)/Q(z, w) to be polynomials of all their corresponding variables. If the considered equation has a transcendental meromorphic solution in \z\ < oo, then Q(z, w) does not depend on w. An example of possible applications is stated. We would like to recall the following problem which has been widely investigated in the analytic theory of differential equations: to find, if possible, all differential equations of a given form which have solutions of a given class; for example: to find all first order differential equations of the form>>' = f(x, y), where f(x,y) is a rational function of x and.y, and every solution of which is a transcendental meromorphic (entire) function (known complete result: the Riccati (linear) equation y' = y2 + p(x)y + q(x) (y' = p(x)y + q(x)) which alone has the required property [1], [4]). This note deals with a question of this kind. We consider the differential equation (1) F(z, w,w',..., wM) = P(z, w)/Qiz, w), where F, P and Q are polynomials of all their corresponding variables. Our main result is contained in the following assertion. Theorem. Suppose the polynomials Piz, w) and Qiz, w) to be mutually prime. If there exists a transcendental meromorphic solution of equation (1) in the finite z plane, then Qiz, w) does not depend on w. We would like to propose an application of this theorem: an equation of the form (2) win) = Piz,w)/Qiz,w) with polynomials P and Q can have a transcendental meromorphic solution only if Q iz, w) does not depend on w, that is, (2) must be of the form (3) w{n)=Piz,w) where P(z, w) is a polynomial of w and a rational function of z. Very simple considerations show that

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On the Growth of Certain Meromorphic Solutions of Arbitrary Second Order Algebraic Differential Equations

Cited by 4 publications

References 1 publication

On the meromorphic solutions of an equation of Hayman

On the meromorphic solutions of an equation of Hayman

Some results on the Gamma function and other hypertranscendental functions

A remark on meromorphic solutions of differential equations

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