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Growth of solutions and oscillation of differential polynomials generated by some complex linear differential equations. (English) Zbl 1201.34136
Summary: This paper is devoted to studying the growth and the oscillation of solutions of the second order non-homogeneous linear differential equation
\[ f^{\prime \prime } + A_{1} (z) e^{P(z)}f^{\prime } + A_{0} (z) e^{Q(z)}f =F, \] where \(P (z)\), \(Q (z)\) are nonconstant polynomials such that \(\deg P=\deg Q=n\) and \(A_{j} (z)( \not\equiv 0 )\) \((j=0,1), F\not\equiv 0\) are entire functions with \(\rho ( A_{j} ) < n\) \(( j=0,1 )\). We also investigate the relationship between small functions and differential polynomials \(g_{f} (z)=d_{2}f^{\prime \prime } + d_{1}f^{\prime } + d_{0}f\), where \(d_{0} (z)\), \(d_{1} (z)\), \(d_{2} (z)\) are entire functions that are not all equal to zero with \(\rho ( d_{j} ) < n\) \(( j=0,1,2 )\) generated by solutions of the above equation.

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
34M03 Linear ordinary differential equations and systems in the complex domain
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