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Keywords:
linear differential equations; differential polynomials; entire solutions; order of growth; exponent of convergence of zeros; exponent of convergence of distinct zeros
linear differential equations; differential polynomials; entire solutions; order of growth; exponent of convergence of zeros; exponent of convergence of distinct zeros
Summary:
In this paper, 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 $ab( a-b) \ne 0$ and $A_{j}( z) \lnot \equiv 0$ ($j=0,1$), $F(z) \lnot \equiv 0$ are entire functions such that $\max \left\lbrace \rho (A_j),\, j=0,1,\, \rho (F)\right\rbrace <1.$
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