GROWTH AND ZEROS OF MEROMORPHIC SOLUTIONS TO SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS

The main purpose of this article is to investigate the growth of meromorphic solutions to homogeneous and non-homogeneous second order linear differential equations f '' + Af' + Bf = F, where A (z), B (z) and F (z) are meromorphic functions with finite order having only finitely many poles. We show that, if there exist a positive constants sigma > 0, alpha > 0 such that vertical bar A (z)vertical bar >= e(alpha vertical bar z vertical bar sigma) as vertical bar z vertical bar -> +infinity, z is an element of H, where (dens) over bar{vertical bar z vertical bar: z is an element of H} > 0 and rho = max {p(B), p(F)} < sigma, then every transcendental meromorphic solution f has an infinite order. Further, we give some estimates of their hyper-order, exponent and hyper-exponent of convergence of distinct zeros.