On the growth of solutions of \(f''+gf'+hf=0\). (English) Zbl 0719.34011

Hellerstein, Simon; Miles, Joseph; Rossi, John

Trans. Am. Math. Soc. 324, No. 2, 693-706 (1991).

Let g and h be entire functions satisfying \(\rho (h)<\rho (g)\leq\) where \(\rho\) (\(\cdot)\) denotes the order. It is proved that any nonconstant solution f of the differential equation \(f''+gf'+hf=0\) has infinite order. This had been proved by G. G. Gundersen [Trans. Am. Math. Soc. 305, 415-429 (1988; Zbl 0634.34004)] if \(\rho (h)<\rho (g)<\) and by M. Ozawa [Kodai Math. J. 3, 295-309 (1980; Zbl 0463.34028)] if \(\rho (g)<\) and h is a polynomial. The proof uses, among other things, results of cos \(\pi\rho\)-type, delicate estimations of the logarithmic derivative, and growth lemmas for real functions. It is also noted that a modification of the argument shows that the conclusion holds if \(\rho (h)<\mu (g)\leq\) where \(\mu\) (\(\cdot)\) denotes the lower order. On the other hand, it remains open whether the conclusion holds if \(\rho (h)<\rho (g)<1\) and \(\rho (g)>\). As shown by an example, it need not hold if \(\rho (h)<\rho (g)=1\).

Reviewer: W.Bergweiler (Aachen)

MSC:

34M99	Ordinary differential equations in the complex domain
30D05	Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
30D20	Entire functions of one complex variable (general theory)
30D35	Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

Keywords:

Nevanlinna characteristic; entire functions; differential equation; infinite order

Citations:

Zbl 0634.34004; Zbl 0463.34028