### Oscillation of fast growing solutions of linear differential equations in the unit disc.

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This paper is devoted to considering the iterated order and the fixed points of some differential polynomials generated by solutions of the differential equation $${f}^{{}^{\text{'}\text{'}}}+{A}_{1}\left(z\right){f}^{{}^{\text{'}}}+{A}_{0}\left(z\right)f=F,$$ where ${A}_{1}\left(z\right)$, ${A}_{0}\left(z\right)$ $(\neg \equiv 0)$, $F$ are meromorphic functions of finite iterated $p$-order.

We investigate the growth and fixed points of meromorphic solutions of higher order linear differential equations with meromorphic coefficients and their derivatives. Our results extend the previous results due to Peng and Chen.

This paper is devoted to considering the complex oscillation of differential polynomials generated by meromorphic solutions of the differential equation $${f}^{\left(k\right)}+{A}_{k-1}\left(z\right){f}^{(k-1)}+\cdots +{A}_{1}\left(z\right){f}^{\text{'}}+{A}_{0}\left(z\right)f=0,$$ where ${A}_{i}\left(z\right)$ $(i=0,1,\cdots ,k-1)$ are meromorphic functions of finite order in the complex plane.

In this paper, we investigate the relationship between small functions and differential polynomials ${g}_{f}\left(z\right)={d}_{2}{f}^{\text{'}\text{'}}+{d}_{1}{f}^{\text{'}}+{d}_{0}f$, where ${d}_{0}\left(z\right)$, ${d}_{1}\left(z\right)$, ${d}_{2}\left(z\right)$ are entire functions that are not all equal to zero with $\rho \left({d}_{j}\right)<1$ $(j=0,1,2)$ generated by solutions of the differential equation ${f}^{\text{'}\text{'}}+{A}_{1}\left(z\right){e}^{az}{f}^{\text{'}}+{A}_{0}\left(z\right){e}^{bz}f=F$, where $a,b$ are complex numbers that satisfy $ab(a-b)\ne 0$ and ${A}_{j}\left(z\right)\neg \equiv 0$ ($j=0,1$), $F\left(z\right)\neg \equiv 0$ are entire functions such that $max\left\{\rho \left({A}_{j}\right),\phantom{\rule{0.166667em}{0ex}}j=0,1,\phantom{\rule{0.166667em}{0ex}}\rho \left(F\right)\right\}<1.$

In this paper we discuss the growth of solutions of the higher order nonhomogeneous linear differential equation $$\begin{array}{cc}& {f}^{\left(k\right)}+{A}_{k-1}{f}^{(k-1)}+\cdots +{A}_{2}{f}^{\text{'}\text{'}}+({D}_{1}\left(z\right)+{A}_{1}\left(z\right){\mathrm{e}}^{az}){f}^{\text{'}}\hfill \\ & +({D}_{0}\left(z\right)+{A}_{0}\left(z\right){\mathrm{e}}^{bz})f=F\phantom{\rule{1.0em}{0ex}}(k\ge 2),\hfill \end{array}$$ where $a$, $b$ are complex constants that satisfy $ab(a-b)\ne 0$ and ${A}_{j}\left(z\right)$ $(j=0,1,\cdots ,k-1)$, ${D}_{j}\left(z\right)$ $(j=0,1)$, $F\left(z\right)$ are entire functions with $max\{\rho \left({A}_{j}\right)\phantom{\rule{4pt}{0ex}}(j=0,1,\cdots ,k-1),\phantom{\rule{4pt}{0ex}}\rho \left({D}_{j}\right)$ $(j=0,1)\}<1$. We also investigate the relationship between small functions and the solutions of the above equation.

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