Bulletin of the Iranian Mathematical Society https://doi.org/10.1007/s41980-020-00456-9 ORIGINAL PAPER Starlike Functions Associated with Cosine Functions Khadija Bano 1 · Mohsan Raza 1 Received: 7 January 2020 / Revised: 19 June 2020 / Accepted: 27 August 2020 © Iranian Mathematical Society 2020 Abstract Let S ∗ cos denote the class of normalized analytic functions f such that zf ′ (z) f (z) ≺ cos(z ). For this class, we obtain structural formula, inclusion results, differential subordina- tions 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 A n denote the class of functions f of the form f (z ) = z + ∞  k =1 a n+k z n+k , which are analytic in the open unit disk D ={z :|z | < 1, z ∈ C}. It is clear that A 1 = 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 mohsan976@yahoo.com Khadija Bano khadijabano51@gmail.com 1 Department of Mathematics, Government College University, Faisalabad, Pakistan 123