On the growth of solutions to the complex differential equation f″ + Af′ + Bf = 0

Abstract

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.

On a question of Gundersen concerning the growth of solutions of linear differential equations

“…Motivated by Theorem 1.1, many parallel results written thereafter focus on the case ρ(A) ≥ ρ(B) and B(z) is a transcendental entire function; see, for example, [3,16,18,19,24,25]. Regarding the case of a polynomial B(z), there are many results concerning the growth of solutions of the following special equation…”

On the relationship between the lower order of coefficients and the growth of solutions of differential equations

“…The concept of (usual) order has a wide range of applications including complex differential equations, see [14,15]. Some attempts to associate the lower order with complex differential equations exist, see [17,23,24].…”

On the growth of solutions of second order complex differential equation with meromorphic coefficients

“…There has been much work on this subject (cf. [3][4][5][6][7][8]). Furthermore, we also mention that if A(z) is entire with finite order having a finite deficient value, and B(z) is transcendental entire with μ(B) < 1 2 , then every solution f ≢ 0 of the Equation (1.1) has infinite order [8].…”

“…such that, for all R [t n , bt n ] \ F n , the arguments θ sets E v (R),(v = 1, 2, ..., p) and E ∞ (R) satisfying the following inequalities 8) where…”