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Growth of solutions of second order linear differential equations. (English) Zbl 1151.34069
The authors study the growth of solutions of the linear differential equation
$f''+ A_1(z) e^{az}f'+ A_0(z) e^{bz} f= H(z),\tag{2}$ where $$A_1(z)$$, $$A_0(z)$$ and $$H(z)$$ are entire functions of order less than one, and $$a,b\in\mathbb{C}$$.
The authors prove the following theorems
Theorem 1.1. Suppose that $$A_0\not\equiv 0$$, $$A_1\not\equiv 0$$, $$H$$ are entire functions of order less than one, and the complex constants $$a$$, $$b$$ satisfy $$ab\neq 0$$ and $$a\neq b$$. Then every nontrivial solution $$f$$ of (2) is of infinite order.
Theorem 1.3. Suppose that $$A_0\not\equiv 0$$, $$A_1\not\equiv 0$$, $$D_0$$, $$D_1$$, $$H$$ are entire functions of order less than one, and the complex constants $$a$$, $$b$$ satisfy $$ab\neq 0$$ and $$b/a< 1$$. Then every nontrivial solution $$f$$ of equation $f''+ (A_1(z) e^{a\cdot z}+ D_1(z)) f'+ (A_0(z) e^{b\cdot z}+ D_0(z)) f= H(z)$ is of infinite order.

##### MSC:
 34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
##### Keywords:
linear differential equations; entire solutions; growth
Full Text:
##### References:
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