Abstract
Let h be a nonvanishing analytic function in the open unit disc with h 0 = 1 . Consider the class consisting of normalized analytic functions f whose ratios f z / g z , g z / z p z , and p z are each subordinate to h for some analytic functions g and p . The radius of starlikeness of order α is obtained for this class when h is chosen to be either h z = 1 + z or h z = e z . Further, starlikeness radii are also obtained for each of these two classes, which include the radius of Janowski starlikeness, and the radius of parabolic starlikeness.
Highlights
To formulate a radius description for other entities besides starlikeness and convexity, consider in general two families G and M of A. e G-radius for the class M, denoted by RG(M), is the largest number R such that r− 1f(rz) ∈ G for every 0 < r ≤ R and f ∈ M. us, for example, an equivalent description of the radius of starlikeness for S is that the S∗-radius for the class S is RS∗ (S) tanh(π/4)
Among the very early studies in this direction is the class of close-to-convex functions introduced by Kaplan [2] and Reade’s class [3] of close-to-starlike functions
We examine two different subclasses of functions in A satisfying a certain subordination of ratios
Summary
To formulate a radius description for other entities besides starlikeness and convexity, consider in general two families G and M of A. e G-radius for the class M, denoted by RG(M), is the largest number R such that r− 1f(rz) ∈ G for every 0 < r ≤ R and f ∈ M. us, for example, an equivalent description of the radius of starlikeness for S is that the S∗-radius for the class S is RS∗ (S) tanh(π/4). Denoting by S the class of univalent functions f ∈ A, the number r0 tan h(π/4) is commonly referred to as the radius of starlikeness for the class S. For 0 ≤ α < 1, let S∗(α) denote the class of starlike functions of order α consisting of functions f ∈ A satisfying the subordination zf′(z) 1 f(z) ≺
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