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Relation between differential polynomials and small functions. (English) Zbl 1203.34148
The authors discuss the growth of solutions of the second-order non-homogeneous differential equation
\[ f'' +A_1(z)e^{az} f' +A_0(z)e^{bz} f=F, \] where \(a,b\) are complex numbers and \(A_j(z)\not\equiv 0\) \((j=0,1)\), and \(F\not\equiv 0\) are entire functions such that \(\max \{ \rho(A_0), \rho(A_1), \rho(F)\}<1 \). Slight improvements of the results of I. Laine and J. Wang [J. Math. Anal. Appl. 342, 39–51 (2008; Zbl 1151.34069)], and Z. X. Chen [Sci. China Ser. A 45, No. 3, 290–300 (2002; Zbl 1054.34139)] are obtained. Relations between small functions and some differential polynomials generated by solutions of the equation are studied.

MSC:
34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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