Finite order solutions of linear differential equations in the unit disc

Necessary and sufficient conditions for the analytic coefficients of the complex linear differential equation(†) f ( k ) + a k − 1 ( z ) f ( k − 1 ) + ⋯ + a 1 ( z ) f ′ + a 0 ( z ) f = a k ( z ) are found such that all solutions satisfy σ ( f ) : = lim sup r → 1 − log + T ( r , f ) − log ( 1 − r ) ⩽ α . Moreover, estimates for the number of linearly independent solutions of maximal growth are found in terms of the growth of the coefficients. In addition, sufficient conditions for the coefficients such that the zero sequence { z n } of any non-trivial solution f of (†) with a k ≡ 0 satisfies ∑ ( 1 − | z n | ) α + 1 < ∞ are found. Several non-trivial examples are given in order to show that the established results are in a sense sharp.