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. In this note our aim is to point out that certain inequalities for modified Bessel functions of the first and second kind, deduced recently by Laforgia and Natalini, are in fact equivalent to the corresponding Turán type inequalities for these functions. Moreover, we present some new Turán type inequalities for the aforementioned functions and we show that their product is decreasing as a function of the order, which has an application in the study of stability of radially symmetric solutions in a generalized FitzHugh-Nagumo equation in two spatial dimensions. At the end of this note an open problem is posed, which may be of interest for further research.
Ramanujan's work on the asymptotic behaviour of the hypergeometric function has been recently refined to the zero‐balanced Gaussian hypergeometric function F(a, b; a + b; x) as x→1. We extend these results for F(a, b; c; x) when a, b, c>0 and c
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.
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