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New estimates are obtained for the maximum modulus of the generalized logarithmic derivatives f (k) / f ( j) , where f is analytic and of finite order of growth in the unit disc, and k and j are integers satisfying k > j ≥ 0. These estimates are stated in terms of a fixed (Lindelöf) proximate order of f and are valid outside a possible exceptional set of arbitrarily small upper density. The results obtained are then used to study the growth of solutions of linear differential equations in thedoi:10.1017/s1446788710000029 fatcat:eclbhrzd6rbfhonrjy6vua5uwi