Starlike Functions Associated with Cosine Functions
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Starlike Functions Associated with Cosine Functions Khadija Bano1 · Mohsan Raza1 Received: 7 January 2020 / Revised: 19 June 2020 / Accepted: 27 August 2020 © Iranian Mathematical Society 2020
Abstract ∗ denote the class of normalized analytic functions f such that z f (z) ≺ cos(z). Let Scos f (z) For this class, we obtain structural formula, inclusion results, differential subordinations and some radii problems such as radius of convexity, radius for the class of Janowski starlike functions and radius for some other subclasses of starlike functions. Keywords Analytic functions · Cosine functions · Radii problems Mathematics Subject Classification 30C45 · 30C50
1 Introduction Let An denote the class of functions f of the form f (z) = z +
∞ an+k z n+k , k=1
which are analytic in the open unit disk D = {z : |z| < 1, z ∈ C}. It is clear that A1 = A is the class of normalized analytic functions. Also let S denote the subclass of analytic functions A which are univalent in D. A function f is said to be subordinate to a function g written as f ≺ g, if there exists a Schwarz function w with w (0) = 0 and |w(z)| < 1 such that f (z) = g (w(z)). In particular, if g is univalent in D and f (0) = g (0), then f (D) ⊂ g ( D). Let S ∗ (β) , C (β) and SS ∗ (β) denote the classes of starlike, convex and strongly starlike functions of order β, respectively, and are analytically defined as
Communicated by Hamid Reza Ebrahimi Vishki.
B
Mohsan Raza [email protected] Khadija Bano [email protected]
1
Department of Mathematics, Government College University, Faisalabad, Pakistan
123
Bulletin of the Iranian Mathematical Society
S ∗ (β) =
C (β) = SS ∗ (β) =
z f (z) > β, z ∈ D,β ∈ [0, 1) , f (z) z f (z) : f ∈ A and Re 1 + > β, z ∈ D,β ∈ [0, 1) , f (z) βπ z f (z) < . , z ∈ D,β ∈ 1] : f ∈ A and arg (0, f (z) 2
f : f ∈ A and Re f f
It is clear that SS ∗ (1) = S ∗ (0) = S ∗ , and C (0) = C, where S ∗ and C are usual classes of starlike and convex functions, respectively. The class M (β) (β > 1) is the class of functions f such that Rez f (z)/ f (z) < β. Let ϕ be analytic and univalent function in D such that ϕ (D) is convex with ϕ (0) = 1 and Re ϕ (z) > 0, z ∈ D. Then a function f ∈ A is in the class S ∗ (ϕ), if it satisfies z f (z)/ f (z) ≺ ϕ(z), z ∈ D . The class S ∗ (ϕ) was introduced by Ma and Minda [8]. For particular choices of function ϕ, we obtain several classes of analytic and univalent functions. For some ∗ details, see [2,4,6,7,9,10,12,15–17]. Recently, Tang et al. [18] introduced the class Scos ∗ of starlike functions related to cosine functions. The class Scos of starlike functions is defined as z f (z) ∗ ≺ cos(z) . Scos := f ∈ A : f (z) ∗ is that, for any z ∈ D, the ratio The geometrical interpretation of the fact f ∈ Scos z f (z) cosh(1)+cos(1) f (z) lies in the interior of the domain bounded by the disc with centre 2
∗ if and radius cosh(1)−cos(1) . It is clear that a function f is said to be in the class Scos 2 there exists an analytic f
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