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On the growth of solutions of $$f''+gf'+hf=0$$. (English) Zbl 0719.34011
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 $$\rho (h)<\rho (g)=1$$.

##### 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
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