We consider the class of all sense‐preserving harmonic mappings f=h+g¯ of the unit disk double-struckD, where h and g are analytic with g(0)=0, and determine the Bohr radius if any one of the following conditions holds:
1.h is bounded in double-struckD.
2.h satisfies the condition Re h(z)≤1 in double-struckD with h(0)>0.
3.both h and g are bounded in double-struckD.
4.h is bounded and g′false(0false)=0.
We also consider the problem of determining the Bohr radius when the supremum of the modulus of the dilatation of f in double-struckD is strictly less than 1. In addition, we determine the Bohr radius for the space scriptB of analytic Bloch functions and the space BH of harmonic Bloch functions. The paper concludes with two conjectures.
Abstract. We determine the Bohr radius for the class of odd functions f satisfying |f (z)| ≤ 1 for all |z| < 1, settling the recent conjecture of Ali, Barnard and Solynin [9]. In fact, we solve this problem in a more general setting. Then we discuss Bohr's radius for the class of analytic functions g, when g is subordinate to a member of the class of odd univalent functions.
Preliminaries and Main ResultsLet A denote the space of all functions analytic in the unit disk D := {z ∈ C : |z| < 1} equipped with the topology of uniform convergence on compact subsets of D. Then the classical Bohr's inequality [14] states that if a power series f (z) = ∞ n=0 a n z n belongs to A and |f (z)| < 1 for all z ∈ D, then M f (r) := ∞ n=0 |a n |r n ≤ 1 for all |z| = r ≤ 1/3 and the constant 1/3 cannot be improved. The constant r 0 = 1/3 is known as Bohr's radius. Bohr actually obtained the inequality for r ≤ 1/6, but subsequently later, Wiener, Riesz and Schur, independently established the sharp inequality for |z| ≤ 1/3. For a detailed account of the development, we refer to the recent survey article on this topic [8] [1-3, 6, 7].The present investigation is motivated by the following problem of Ali, Barnard and Solynin [9]. In [9, Lemma 2.2], it was shown that M f (r) ≤ 1 for all |z| = r ≤ r * , where r * is a solution of the equation 5r 4 + 4r 3 − 2r 2 − 4r + 1 = 0,2000 Mathematics Subject Classification. Primary 30A10, 30H05, 30C35; Secondary 30C45.
In this paper, we introduce the study of the Bohr phenomenon for a quasisubordination family of functions, and establish the classical Bohr’s inequality for the class of quasisubordinate functions. As a consequence, we improve and obtain the exact version of the classical Bohr’s inequality for bounded analytic functions and also for
K
K
-quasiconformal harmonic mappings by replacing the constant term by the absolute value of the analytic part of the given function. We also obtain the Bohr radius for the subordination family of odd analytic functions.
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