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Complex oscillation theory of differential polynomials. (English) Zbl 1244.34108
Summary: We 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 \(\rho (d_j)<1\) \((j=0,1,2) \) generated by solutions of the differential equation \(f^{\prime \prime }+A_{1}(z) e^{az}f^{\prime }+A_{0}(z) e^{bz}f=F\), where \(a,b\) are complex numbers that satisfy \(ab( a-b) \neq 0\) and \(A_{j}( z) \lnot \equiv 0\) (\(j=0,1\)), \(F(z) \lnot \equiv 0\) are entire functions such that \(\max \left\{ \rho (A_j),\, j=0,1,\, \rho (F)\right\} <1\).
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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References:
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