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For a function /, regular in the unit disc, integral logarithmic means are defined by the formulae 2π ι t Γ 1 ί 2π rJ) = \Έt] ' l0g lf{reiθ for 0 < p < oo. These are related to Moo(r,/) = sup|log|/(z)|| (0 1 the orders , / Λ .. logΛf»(r,/) ^(/) = ll ?i" P log 1/(1-r) are continuous at infinity in the sense that a property which does not generally hold when λoo(f) < 1. It transpires that in the extreme cases λoo(f) = λi(/) + 1, and λoo{f) = λ\(f) > 1, λ p (f) is uniquely determined for 1 < p < oo. 1. Introduction, For a given function /, regular in the unit disc Z)(0,l) = {z: \z\< 1} let

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Letbe a function regular for | z | < 1. With the hypotheses f(0) = 0 andfor some positive constant α, Cartwright(1) has deduced upper bounds for |f(z) | in the unit circle. Three cases have arisen and according as (1) holds with α < 1, α = 1 or α > 1, the bounds on each circle | z | = r are given respectively byK(α) being a constant which depends only on the corresponding value of α which occurs in (1). We shall always use the symbols K and A to represent constants dependent on certain parameters such as α, not necessarily having the same value at each occurrence.