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Starlike functions and higher order differential subordinations S. Sivaprasad Kumar1 · Priyanka Goel1 Received: 11 March 2020 / Accepted: 14 August 2020 © The Royal Academy of Sciences, Madrid 2020

Abstract In the present investigation, a third order differential subordination result of Antonino and Miller (Complex Var Ellipt Equ 56(5):439–454, 2011) has been modified to accommodate Ma–Minda functions so that new avenues could be explored related to them. Further as ∗ = { f ∈ A : z f  (z)/ f (z) ≺ an application, many alluring results pertaining to S SG −z 2/(1 + e )}, the class of Sigmoid starlike functions, involving higher order differential subordinations are also obtained. Keywords Starlike functions · Sigmoid function · Subordination Mathematics Subject Classification 30C45 · 30C50

1 Introduction Let D = {z ∈ C : |z| < 1} be the unit disk and A be the class of functions analytic on D with the usual normalization f (0) = f  (0) − 1 = 0. Let S denote the subclass of A consisting of the univalent functions and S ∗ denotes the subclass of S consisting of the starlike functions. For any two members f and F in A , we say f is subordinate to F, denoted by f ≺ F if f (z) = F(ω(z)), where ω : D → D satisfies ω(0) = 0. Furthermore, if F is univalent, then f ≺ F if and only if f (0) = F(0) and f (D) ⊂ F(D). Using the concept of subordination, Ma and Minda [11] defined a general subclass of S ∗ denoted by S ∗ (φ) consisting of all the functions f in A for which z f  (z)/ f (z) ≺ φ(z). Here φ ∈ Φ where Φ is the class of all functions φ which are analytic in D with φ  (0) > 0 and maps D onto a domain lying in the right half plane, symmetric with respect to real axis and starlike with respect to 1. For different choices of φ, S ∗ (φ) reduces to several well known subclasses of S ∗ . In our present work, we need to recall the following subclasses of S ∗ (φ). For −1 ≤ B < A ≤ 1, Janowski [9] defined S ∗ [A, B] = S ∗ ((1 + Az)/(1 + Bz)), known

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Priyanka Goel [email protected] S. Sivaprasad Kumar [email protected]

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Delhi Technological University, Delhi 110042, India 0123456789().: V,-vol

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as the class of Janowski starlike functions. In 1996, Sokół [22] defined √ the class of starlike functions associated with lemniscate of Bernoulli as S L ∗ = S ∗ ( 1 + z). By taking φ(z) as exponential function, Mendiratta et al. [13] introduced√the class Se∗ = S ∗ (e z ). In 2015, Raina and Sokół [19] introduced the class Sq∗ = S ∗ (z + 1 + z 2 ) associated with a crescent shaped region. In [6], authors introduced and studied the class S S∗ = S ∗ (1+sin z). Recently ∗ of Sigmoid starlike functions by taking φ(z) to be the in [8], we introduced the class S SG modified Sigmoid function given by 2/(1 + e−z ). Geometrically, we say that a function ∗ if and only if z f  (z)/ f (z) lies in the region Δ f belongs to the class S SG SG = {w ∈ C : | log(w/(2 − w))| < 1}. In this paper, we established several inclusion results, radius estimates, coefficient estimates and first order d