On the growth of entire functions satisfying second order linear differential equations. (English) Zbl 0863.34007

The author studies the growth of the solutions \(f\not\equiv 0\) of the second order linear differential equation (1) \(f''+A(z)f'+ B(z)f=0\), where \(A(z)\) and \(B(z)\not\equiv 0\) are entire functions. It is well known that all solutions of (1) are entire functions, and that at least one of any two linearly independent solutions of (1) has infinite order \(\rho=\infty\) if \(A(z)\) is transzendental. The author gives a more precise estimation of the growth of the solutions of infinite order of (1) if \(\rho(A)< \rho(B)\) or \(\rho(B)< \rho(A)<1/2\).


34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D20 Entire functions of one complex variable (general theory)