Starlikeness associated with the sine hyperbolic function

Abstract

Let qλ (z) = 1 + λsinh(ς), 0 < λ < 1/sinh(1) be a non-vanishing analytic function in the open unit disk. We introduce a subclass \({{\cal S}^ * }({q_\lambda })\) (qλ) of starlike functions which contains the functions \(\mathfrak{f}\) such that \(z{\mathfrak{f}^\prime }/\mathfrak{f}\) is subordinated by qλ. We establish inclusion and radii results for the class \({{\cal S}^ * }\) (qλ) for several known classes of starlike functions. Furthermore, we obtain sharp coefficient bounds and sharp Hankel determinants of order two for the class \({{\cal S}^ * }\) (qλ). We also find a sharp bound for the third Hankel determinant for the case λ = 1/2.