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Logarithmic derivatives of meromorphic or algebroid solutions of some homogeneous linear differential equations. (English) Zbl 0934.34077

Consider the linear differential operators \[ L_k(D)=D^k+\sum_{i=0}^{k-1}a_i(z)D^i,\quad L_l(D)=D^l+\sum_{j=0}^{l-1}b_j(z)D^j,\quad D=d/dz\quad (k>l), \] where \(a_i(z)\) and \(b_j(z)\) are entire functions. Let \(A(z)\) be a transcendental entire function such that \[ \sum^{k-1}_{i=0}T(r,a_i)+\sum^{l-1}_{j=0}T(r,b_j)= o(T(r,A))\quad \text{as }r\to +\infty,\quad r\in J, \] where \(J\) is an interval with infinite linear measure. The author treats an oscillation problem concerning the linear differential equation \(L_k(D)w=A(z)L_l(D)w.\) It is proved that, for each nontrivial solution, \[ m(r,Dw/w)\leq (2+o(1))(\overline{N}(r,1/w)+\overline{N} (r,1/A)) \text{ as }r\to +\infty, \quad r\in J', \] where \(J'\subset J\) is some interval such that the linear measure of \(J- J'\) is finite, unless \(L_l(D)w=0\) or both \(A(z)\) and \(w(z)\) are zero-free functions. For such zero-free solutions their properties are examined. Furthermore, in the case where \(A(z)\) is a certain algebroid function, it is shown that an analogous estimate is valid for algebroid solutions.

MSC:

34M25 Formal solutions and transform techniques for ordinary differential equations in the complex domain
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
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