Radius problems for functions associated with a nephroid domain
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Radius problems for functions associated with a nephroid domain Lateef Ahmad Wani1 · Anbhu Swaminathan1 Received: 1 June 2020 / Accepted: 23 July 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract Let S N∗ e be the collection of all analytic functions f (z) defined on the open unit disk D and satisfying the normalizations f (0) = f (0) − 1 = 0 such that the quantity z f (z)/ f (z) assumes values from the range of the function ϕ N e (z) := 1 + z − z 3 /3, z ∈ D, which is the interior of the nephroid given by 4 3 4v 2 (u − 1)2 + v 2 − − = 0. 9 3 In this work, we find sharp S N∗ e -radii for several geometrically defined function classes introduced in the recent past. In particular, S N∗ e -radius for the starlike class S ∗ is found to be 1/4. Moreover, radii problems related to the families defined in terms of ratio of functions are also discussed. Sharpness of certain radii estimates are illustrated graphically. Keywords Starlike functions · Subordination · Radius problem · Bernoulli and Booth lemniscates · Cardioid · Nephroid Mathematics Subject Classification 30C45 · 30C80
1 Introduction Let A be the class of all analytic functions satisfying the conditions f (0) = 0 and f (0) = 1 in the open unit disc D := {z : |z| < 1}. Clearly, for each f ∈ A, the function Q f (z) : D → C given by z f (z) Q f (z) := (1.1) f (z) is analytic and satisfies Q f (0) = 1. Let S ⊂ A be the family of univalent functions, and S ∗ (α) ⊂ S be the family of starlike functions of order α (0 ≤ α < 1) given by the analytic
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Anbhu Swaminathan [email protected]; [email protected] Lateef Ahmad Wani [email protected]
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Department of Mathematics, Indian Institute of Technology, Roorkee, Uttarakhand 247667, India 0123456789().: V,-vol
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characterization S ∗ (α) :=
f ∈ A : Re Q f (z) > α .
Further, let us define the class C (α) by the relation: f ∈ C (α) ⇐⇒ z f ∈ S ∗ (α). The functions in S ∗ := S ∗ (0) and C := C (0) are, respectively, starlike and convex in D. For functions f and g analytic on D, we say that f is subordinate to g, written f ≺ g, if there exists an analytic function w satisfying w(0) = 0 and |w(z)| < 1 such that f (z) = g(w(z)). Indeed, f ≺ g ⇒ f (0) = g(0) and f (D) ⊂ g(D). Furthermore, if the function g is univalent, then f ≺ g ⇐⇒ f (0) = g(0) and f (D) ⊂ g(D). By unifying several earlier results on subordination, Ma and Minda [15] introduced the function class S ∗ (ϕ) which, for brevity, we write as definition. Definition 1.1 Let S ∗ (ϕ) denote the class of functions characterized as S ∗ (ϕ) := f ∈ A : Q f (z) ≺ ϕ(z) ,
(1.2)
where the analytic function ϕ : D → C is required to satisfy the following conditions: (i) (ii) (iii) (vi)
ϕ(z) is univalent with Re(ϕ) > 0, ϕ(D) is starlike with respect to ϕ(0) = 1, ϕ(D) is symmetric about the real line, and ϕ (0) > 0.
Evidently, for every such ϕ, S ∗ (ϕ) is always a subclass of the class of starlike functions S ∗ , and S ∗ (ϕ) = S ∗ for ϕ(z) = (1 + z)/(1 − z). We call S ∗
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