Complex oscillation and nonoscillation results

Exponential polynomials in the oscillation theory

Supposing that $A(z)$ is an exponential polynomial of the form 
$$ 
A(z)=H_0(z)+H_1(z)e^{\zeta_1z^n}+\cdots +H_m(z)e^{\zeta_mz^n}, 
$$ where $H_j$'s are entire and of order $<n$, it is demonstrated that the function $H_0(z)$ and the geometric location of the leading coefficients $\zeta_1,\ldots,\zeta_m$ play a key role in the oscillation of solutions of the differential equation $f''+A(z)f=0$. The key tools consist of value distribution properties of exponential polynomials, and elementary properties of the Phragmen-Lindelof indicator function. In addition to results in the whole complex plane, results on sectorial oscillation are proved.

In this paper, we investigate the exponent of convergence of the zero-sequence of solutions of the second order linear differential equation $$f''+\left(\sum_{j=1}^{l}Q_j(z)e^{P_j(z)}\right)f=0,$$ where Pj (z) (j= 1,2, ..., l ≥ 3) are polynomials of degree n ≥ 1, Qj (z) are entire functions of order less than n, and obtain some results which improve and generalize the previous results in [8, 9, 13].

/pdf/on-the-zeros-of-solutions-of-a-class-of-second-order-linear-28yln2uhtg.pdf

Jin Tu

Papers

On the zeros of solutions of a class of second order linear differential equations