Functions regular in the unit circle

Abstract

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.

On a theorem of Cartwright in higher dimensions

“…where C depends only on p. This result was later refined and extended to more general weights by Cartwright herself [3] and Linden [6,7]. The works by Nikolskii [8] and Borichev [1] should also be mentioned, the latter in particular, where a very nice version of the reverse estimate u(z) −(1 + o(1))w(1 − |z|) was obtained for sufficiently fast growing weights (see [1,Section 1.3]); some estimates for the constant in the reverse inequality were also given earlier in [7].…”

“…We also show that the reverse estimate in this theorem is the best possible. Further results for other weights can be obtained by methods developed here; for example, some logarithmic factors can be added to the weight w 0 as in Theorem 2 of [6].…”

“…Theorem 3 contains another result of Cartwright from [2] (for n = 1 and w(y) = y −a , a < 1) and a theorem of Linden [6] (which corresponds to the case n = 1, weight w is regular and satisfies the integral condition above). In dimension 2 (n = 1), the statement also follows from results of Hayman and Korenblum [4].…”

Do some nontrivial closed z-invariant subspaces have the division property ?

“…In this section, we will use estimates of the growth of analytic functions in the disc which are the quotient of analytic functions satisfying themselves some growth conditions as |λ| → 1 − . Following the early works of Cartwright [10] and Linden [22], more sophisticated methods were developped in the seventies, see [20] and [27] for estimates of inverses of functions analytic in the disc. Concerning analytic functions of the form h = f /g, where g is allowed to have zeroes, the best results known to the author are due to Matsaev-Mogulskii [24], see also [26], who stated their results for functions holomorphic on a half-plane.…”

Regular Functions of Restricted Growth and Their Zeros in Tangential Regions

“…It has been shown [2] that the function <f> defined by the relation The inequality (2.10), together with (2.7) and (2.11), gives a contradiction. It follows that <f> is schlicht in H for some positive number r0.…”