### A note on the oscillation theory of certain second order differential equations.

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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...$

We give some growth properties for solutions of linear complex differential equations which are closely related to the Brück Conjecture. We also prove that the Brück Conjecture holds when certain proximity functions are relatively small.

In this paper, we investigate the growth of solutions of a certain class of linear differential equation where the coefficients are analytic functions in the closed complex plane except at a finite singular point. For that, we will use the value distribution theory of meromorphic functions developed by Rolf Nevanlinna with adapted definitions.

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.

The main purpose of this paper is to partly answer a question of L. Z. Yang [Israel J. Math. 147 (2005), 359-370] by proving that every entire solution f of the differential equation ${f}^{\text{'}}-{e}^{P\left(z\right)}f=1$ has infinite order and its hyperorder is a positive integer or infinity, where P is a nonconstant entire function of order less than 1/2. As an application, we obtain a uniqueness theorem for entire functions related to a conjecture of Brück [Results Math. 30 (1996), 21-24].