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Some properties of solutions of second-order linear differential equations. (English) Zbl 1318.30042

Let \(A\) be an entire function of finite order. Suppose \(f_1\) and \(f_2\) are linearly independent solutions of the equation \(f''+ A(z)f= 0\). If \(d_1\) and \(d_2\) are entire functions of finite order which do not vanish identically and for which \(d_1/cd_2\) for a complex number \(c\), the authors study the growth and oscillatory behavior of \(g= d_1 f_1+ d_2 f_2\).
For example, they prove that if the maximum of the orders of \(d_1\) and \(d_2\) is strictly less than the order of \(A\), then the hyper-order of \(g\) equals that of \(f_1\), \(f_2\) and \(A\). Similar results are obtained for the hyper-exponent of convergence of the zeros sequence of \(g-\phi\) where \(\phi\) is an appropriately chosen entire function not identically zero.

MSC:

30D15 Special classes of entire functions of one complex variable and growth estimates
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
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