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- Title
On the Infinite Growth of Solutions to Second Order Complex Differential Equation.
- Authors
Guowei Zhang
- Abstract
By a well known result of G. Gundersen, if the complex differential equation f'' + A(z)f' + B(z)f = 0 with the coefficients satisfying ρ(A) < ρ(B), where ρ(*) denotes the growth order of an entire function *, then the nontrivial solutions of this equation are of infinite order. Moreover, there exist examples which show that if ρ(A) = ρ(B), then this equation can have a finite order solution. In this paper we discuss the remained case, if ρ(A) > ρ(B) whether the solutions have infinite order. In fact, we prove that if ρ(A) > ρ(B) > 0 and A(z) satisfies three conditions respectively, then every nontrivial solution of this equation has infinite growth order. The three conditions are (1) A(z) is a nontrivial solution of w'' + P(z)w = 0, where P(z) is a polynomial; (2) A(z) is an entire function of exponential growth; (3) A(z) is a completely regular growth entire function.
- Subjects
DIFFERENTIAL equations; EXPONENTIAL functions; EQUATIONS
- Publication
IAENG International Journal of Applied Mathematics, 2023, Vol 53, Issue 3, p956
- ISSN
1992-9978
- Publication type
Article