ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2019, Vol. 40, No. 9, pp. 1319–1323. c  Pleiades Publishing, Ltd., 2019. Coefficient Inequalities for Bloch Functions I. R. Kayumov 1* and K.-J. Wirths 2** (Submitted by F. G. Avkhadiev) 1 Kazan (Volga Region) Federal University, Kazan, Tatarstan, 420008 Russia 2 Technische Universit ¨ at Braunschweig, Institut f ¨ ur Analysis und Algebra, Universit ¨ atsplatz 2, 38106 Braunschweig, Germany Received April 3, 2019; revised April 10, 2019; accepted April 18, 2019 Abstract—In this article we derive new estimates for the moduli of the Taylor coefficients of Bloch functions. We use one of these estimates to prove an inequality of an area type for such functions. DOI: 10.1134/S1995080219090117 Keywords and phrases: Bloch functions, Taylor coefficients, area functional, bounded analytic functions. 1. INTRODUCTION In the present little note we consider functions F holomorphic in the unit disc D = {z ||z| < 1} such that the inequality |F ′ (z)|≤ 1 1 −|z| 2 , z ∈ D, (1) is valid. Let us denote this family of functions by B, as functions of this type play an important role in the history of the Bloch constant, compare f.i. [2]. Let the Taylor expansion of F ∈B be given by F (z)= ∞  k=0 b k z k . (2) It is clear that |b 1 |≤ 1. The sharp bounds for the other coefficients have been found in [9] as |b k |≤ k +1 2k  k +1 k − 1  k−1 2 =: B k , k ∈ N \{1}. (3) Further it has been proved in [6] that equality in (3) is attained if and only if F ′ (z)= kB k z k−1 or a rotation of this function. In [8], Problem N, the problem to find the coefficient body for the family B was posed. A second step in this direction was the determination of {(b 1 ,b 2 ) | F ∈ B} in [9] and [2] as follows. Theorem A. Let F ∈B and x ∈ [0, 1/ √ 3] be chosen such that |b 1 | =3 √ 3x(1 − x 2 )/2. Then this equation and the inequality |b 2 |≤ 3 √ 3(1 − 3x 2 )(1 − x 2 )/4 describe the coefficient region {(b 1 ,b 2 ) | F ∈ B}. Some generalizations and applications of Theorem A have been demonstrated in [1]. Further, Bonk [2] found for sufficiently small values of |b 1 | the max{|b n || F ∈B, |b 1 |fixed}. * E-mail: ikayumov@kpfu.ru ** E-mail: kjwirths@tu-bs.de 1319