Distortion and Critical Values of the Finite Blaschke Product
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Distortion and Critical Values of the Finite Blaschke Product V. N. Dubinin1,2 Received: 11 March 2020 / Revised: 6 August 2020 / Accepted: 20 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We establish a sharp upper bound for the absolute value of the derivative of the finite Blaschke product, provided that the critical values of this product lie in a given disk. Keywords Distortion theorems · Blaschke product · Zolotarev fraction · Critical values · Symmetrization · Condenser capacity Mathematics Subject Classification 30C15 · 30C85
1 Introduction The inequalities for the absolute values of the derivative of a complex polynomial, taking into account its critical values, are of natural interest in the theory of multivalent functions. A certain influence on the study of such inequalities was made by the wellknown Erd˝os conjecture about the maximum modulus of the derivative on a connected lemniscate [9–11]. The impact of the critical values of the polynomial on various types of distortions was considered in [4,5]. In particular, the following result was obtained in [5]: if all critical values of the polynomial P(z) = c0 + c1 z + · · · + cn z n , cn = 0, n ≥ 2, lie in the disk |w| ≤ 1 then |P (z)| ≤ 2
1−n n
1
|cn | n Tn (Tn−1 (|P(z)|))
(1.1)
Communicated by Doron Lubinsky. The work is supported by the Russian Foundation for Basic Research (Grant No. 20-01-00018).
B
V. N. Dubinin [email protected]
1
Far Eastern Federal University, st. Sukhanov, 8, Vladivostok, Russia 690950
2
Institute for Applied Mathematics, FEBRAS, st. Radio, 7, Vladivostok, Russia 690041
123
Constructive Approximation
for every z. Equality holds in (1.1) for P = Tn and all real z, |z| ≥ cos(π/(2n)). Here Tn (z) = 2n−1 z n + . . . is the Chebyshev polynomial of the first kind of degree n, and Tn−1 (·) is the continuous branch of its inverse function defined on the ray [0, +∞] and taking this ray onto the ray [cos(π/(2n)), +∞] (see also [8]). The purpose of this article is to establish an inequality similar to (1.1) for the finite Blaschke products n z − zk B(z) = α , z ∈ U := {z : |z| < 1}, 1 − zk z k=1
|α| = 1, |z k | < 1, k = 1, . . . , n. These products and their applications have been studied by many authors (see, for example, [12,14–16] and the references therein). At the same time, problems associated with critical values have not been fully studied [16, 17, p. 365]. In a number of problems on finite Blaschke products, the extremal function is the so-called Chebyshev–Blaschke product [15]. We need a function Bnτ which, up to a linear fractional replacement of the argument, coincides with the Chebyshev– Blaschke product. For a fixed positive integer n > 1 and a number κ, 0 < κ < 1, consider the rational function Z ≡ Z n (ζ ; κ) defined parametrically as K(κ) Z n (sn(u; k); κ) := sn u ; κ , u ∈ C; K(k) the modulus k is determined from the condition K (k)K(κ) = nK (κ)K(k), 0 < k < 1, where K(·) and K (·) are complete elliptic integrals of the first kind [1]. The function
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