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mathematics Article On the Boundary Dieudonné–Pick Lemma Olga Kudryavtseva 1,2,* and Aleksei Solodov 1 1 Moscow Center of Fundamental and Applied Mathematics, Lomonosov Moscow State University, 119991 Moscow, Russia; [email protected] 2 Department of Applied Mathematics, Volgograd State Technical University, 400005 Volgograd, Russia * Correspondence: [email protected] Abstract: The class of holomorphic self-maps of a disk with a boundary fixed point is studied. For this class of functions, the famous Julia–Carathéodory theorem gives a sharp estimate of the angular derivative at the boundary fixed point in terms of the image of the interior point. In the case when additional information about the value of the derivative at the interior point is known, a sharp estimate of the angular derivative at the boundary fixed point is obtained. As a consequence, the sharpness of the boundary Dieudonné–Pick lemma is established and the class of the extremal functions is identified. An unimprovable strengthening of the Osserman general boundary lemma is also obtained. Keywords: holomorphic map; fixed points; angular derivative; Schwarz lemma; Julia–Carathéodory theorem; boundary Dieudonné–Pick lemma; Osserman general boundary lemma MSC: 30C80 1. Introduction and Preliminaries Let be the set of holomorphic functions from the unit disk = fz 2 : jzj < 1g Citation: Kudryavtseva, O.; Solodov, B D C 2 A. On the Boundary Dieudonné–Pick into itself. Investigations of the properties of a map f B are closely connected with the Lemma. Mathematics 2021, 9, 1108. analysis of its fixed points. https://doi.org/10.3390/math9101108 An interior point z0, jz0j < 1, is a fixed point of f if f (z0) = z0. A boundary point z, jzj = 1, is a fixed point of f if \ limz!z f (z) = z. Academic Editor: Francesco Mainardi Note that, if f 2 B and f (z) 6≡ z, then f can have at most one fixed point in the interior of D. In the general case, f 2 B need not have fixed points in D. However, a Received: 19 April 2021 classical result known as the Denjoy–Wolff theorem (see [1]) states that if f 2 B is not Accepted: 10 May 2021 Möbius transformation, then there exists a unique point d, jdj 6 1, such that the sequence − Published: 13 May 2021 of iterates f n = f ◦ f n 1 of f = f 1, n = 2, 3, ..., converges to d locally uniformly in D. Moreover, if d 2 D, then f (d) = d. If d is a boundary point, then the angular limits Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in 0 f (z) − d 0 \ lim f (z) = f (d) and \ lim f (z) = \ lim = f (d) published maps and institutional affil- z!d z!d z!d z − d iations. exist at d; in addition, f (d) = d and the angular derivative f 0(d) 2 (0, 1]. In the literature, d is called the Denjoy–Wolff point of f . Let us define the subclasses of functions considered in our research. In B, we single out the subclass B[a ! b] of functions that map a point a, jaj < 1, to Copyright: © 2021 by the authors. a point b, jbj < 1: Licensee MDPI, Basel, Switzerland. B[a ! b] = f f 2 B : f (a) = bg. This article is an open access article distributed under the terms and For brevity, the standard class B[0 ! 0] of functions with fixed point z0 = 0 we conditions of the Creative Commons denote by B[0]. Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Mathematics 2021, 9, 1108. https://doi.org/10.3390/math9101108 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 1108 2 of 9 Let B[1] denote the set of functions f in B that fix the boundary point z = 1: B[1] = f 2 B : \ lim f (z) = 1 . z!1 The fundamental role of the Schwarz lemma and the Julia–Carathéodory theorem in complex analysis is well known (see [2,3]). These classical results characterize the behavior of the derivative on classes B[0] and B[1], respectively. The interest in developing these results stems from their applications in many directions, e.g., geometric function theory, hyperbolic geometry, complex dynamics, infinite-dimensional analysis, and the theory of composition operators (see [4–8]). The purpose of this paper is to solve the following extremal problem in the spirit of the Julia–Carathéodory theorem: in the class B[a ! b, 1] = B[a ! b] \ B[1] of functions with a given value of f 0(a) find the infinum of the angular derivative f 0(1). The solution of this problem allows us to find the extremal functions in the boundary Dieudonné–Pick lemma and obtain an unimprovable strengthening of the Osserman general boundary lemma. We also consider the class of functions with interior and boundary fixed points, B[0, 1] = B[0] \ B[1]. Note that classes of functions with several fixed points are studied in connection with various problems, e.g., the problem of fractional iteration of holomorphic function [9], the problem of finding domains of univalence [10–12], the problem of describing the Taylor coefficients [13,14], as well as estimating the angular derivatives [15] and the Schwarzian derivatives [16]. The paper is organized as follows. In Sections2 and3, we consider the extremal problems that correspond to various versions of the Schwarz lemma and Julia–Cara- théodory theorem. In Section4, we give a solution of the above-mentioned extremal problem. As a consequence, we identify the class of the extremal functions in the boundary Dieudonné–Pick lemma and formulate an unimprovable strengthening of the Osserman general boundary lemma. In Section5, we give proofs of the main results. 2. Schwarz–Pick and Dieudonné –Pick Lemmas In the class B[0], the Schwarz lemma is a result with deep sense and important consequences. In particular, it describes the character of the fixed point of the function f 2 B[0]: namely, excluding the case of the rotations of the unit disk, in the class B[0], the point z0 = 0 is attractive. Theorem 1 (Schwarz Lemma). Let f 2 B[0]. Then j f (z)j 6 jzj, 8z 2 D, (1) 0 j f (0)j 6 1. (2) If the equality in (2) is attained or the equality in (1) is attained for at least one z 2 D, z 6= 0, then f (z) ≡ eiqz, q 2 R. In this case, the equality in (1) is attained for all z 2 D. Inequality (2) can be interpreted as a solution of the following extremal problem: in the class B[0], find the supremum of j f 0(0)j. The solution of this problem, as follows from Theorem1, has the form sup j f 0(0)j = 1. f 2B[0] Moreover, the extremal functions are the rotations of the unit disk. The Schwarz lemma has an invariant form established by Pick. Let us give the necessary part of the formulation of the Pick lemma. Mathematics 2021, 9, 1108 3 of 9 Theorem 2 (Pick Lemma). Let f 2 B. Then for all z 2 D 1 − j f (z)j2 j f 0(z)j . (3) 6 1 − jzj2 If the equality in (3) is attained for at least one z 2 D , then f (z) ≡ eiq(z − q)/(1 − qz), q 2 D, q 2 R. In this case, the equality in (3) is attained for all z 2 D. The Pick lemma allows one to solve the following extremal problem: in the class B[a ! b], find the supremum of j f 0(a)j. As can be seen from Theorem2, the solution of this problem is as follows: 1 − jbj2 j 0( )j = sup f a 2 . f 2B[a!b] 1 − jaj Moreover, the extremal functions are Möbius transformations with f (a) = b. A further restriction of the class B[a ! b] leads to a refinement of the estimate of the derivative at the interior point. The following result is called the Dieudonné lemma (see [17]). Theorem 3 (Dieudonné Lemma). Let f 2 B[0]. Then for all z 2 D, z 6= 0, 2 2 0 f (z) jzj − j f (z)j f (z) − . z 6 jzj(1 − jzj2) Note that Theorem3 is a simple corollary of the sharpened version of the Schwarz lemma established by Mercer [18]. Let the mapping Tq be defined by q − z Tq(z) = , q 2 D. 1 − qz It takes the unit disk D onto itself with Tq(q) = 0 and Tq(0) = q. An application of Pick’s ideas to Theorem3 allows one to describe for the class B the set of the values of the derivative in terms of two points and their images under the mapping Tq (see also [19]). Theorem 4 (Dieudonné–Pick Lemma). Let f 2 B with f (z) = w and f (z1) = w1. Then 0 f (z) − c 6 r, where T (w ) 1 − jT (z )j2 1 − jwj2 jT (z )j2 − jT (w )j2 1 − jwj2 = w 1 z 1 = z 1 w 1 c 2 2 , r 2 2 2 . Tz(z1) 1 − jTw(w1)j 1 − jzj jTz(z1)j (1 − jTw(w1)j ) 1 − jzj 3. Julia–Carathéodory Theorem and Boundary Schwarz Lemma To formulate the Julia–Carathéodory theorem, consider the linear fractional transformation 1 − q z − q Lq(z) = , q 2 D. 1 − q 1 − qz It takes the unit disk D onto itself and satisfies the conditions Lq(q) = 0 and Lq(1) = 1. Theorem 5 (Julia–Carathéodory Theorem).
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