The zero and order of some second order linear differential equations with transcendental meromorphic coefficients (Q1804215)

The author considers meromorphic solutions of the differential equation \(f''+ B_ 1 f'+ B_ 0 f= F\), where \(B_ 0\), \(B_ 1\), \(F\not\equiv 0\) are meromorphic functions of finite order with finitely many poles. Denote by \(\sigma(f)\) the order and by \(\overline\lambda(f)\) the exponent of convergence of the distinct zeros of \(f\). It is shown that if \(B_ 0/F\) is nonconstant and if there exist \(\alpha\), \(\beta>0\) such that \(\sigma(B_ 0)< \beta\), \(\sigma(F)< \beta\) and \(| B_ 1(z)|\geq \exp(\alpha| z|^ \beta)\) for all large \(| z|\) outside certain (arbitrarily small) sectors, then any meromorphic solution \(f\) of the above differential equation satisfies \(\overline\lambda(f)= \sigma(f)= \infty\). A second result deals with the case that \(\sigma(F)\geq \beta\).