an:00120168
Zbl 0765.34004
Gundersen, Gary G.; Steinbart, Enid M.
Finite order solutions of nonhomogeneous linear differential equations
EN
Ann. Acad. Sci. Fenn., Ser. A I, Math. 17, No. 2, 327-341 (1992).
00011310
1992
j
34M99 34A30 34C11
order of entire function; Phragmen-Lindel??f theory; upper density
Consider the differential equation (1) \(f^{(n)}+A_{n-1}(z)f^{(n- 1)}+\cdots+A_ 1(z)f'+A_ 0(z)f=H(z)\), where \(A_ i(z)\), \(i=0,\dots,n- 1\) and \(H(z)\) are entire functions. In this paper the authors give the answers to the following two questions: 1. What conditions on \(A_ i(z)\), \(i=0,\dots,n-1\) and \(H(z)\) will guarantee that every solution of (1) has infinite order? 2. If (1) possesses a solution \(f\) of finite order, then how do the properties of \(A_ i(z)\) and \(H(z)\) affect the properties of \(f\)? For example, if in (1) the \(\max\{\rho(A_ 1),\dots,\rho(A_{n-1}),\;\rho(H)\}<\rho(A_ 0)<1/2\), then every solution of (1) has infinite order. Several examples are given to illustrate the results. \((\rho(A)\) denote the order of \(A(z)\)).
A.Kl???? (Praha)