A coefficient inequality for a subclass of the Carathéodory functions defined using conical domains

For 0 ≤ k < ∞, let Ωk be the conical domain in the complex plane C defined by Ωk = {w ∈ C : w = u + iv, u2 > k2((u − 1)2 + v2), u > 0}. Let qk(z) be the Riemann map of U := {z ∈ C : |z| < 1} onto Ωk satisfying qk(0) = 1, q′k(0) > 0. Let P (qk) be the class of analytic functions h(z) subordinate in U to qk(z) and represented by h(z) = 1 + b1z + b2z + · · ·, (z ∈ U). Sharp estimates for |b2 − ub1| (−∞ < u < ∞) are found in this note. This result improves upon an estimate of Kanas in terms of both bounds and ranges of the parameter u [S. Kanas, Coefficient estimates in subclasses of the Carathéodory class related to conical domains, Acta Math. Univ. Comenianae LXXIV 2 (2005) 149–161]. © 2011 Elsevier Ltd. All rights reserved.