KYUNGPOOK Math. J. 57(2017), 473-492 https://doi.org/10.5666/KMJ.2017.57.3.473 pISSN 1225-6951 eISSN 0454-8124 c ⃝ Kyungpook Mathematical Journal Radius of Starlikeness for Analytic Functions with Fixed Sec- ond Coefficient Rosihan M. Ali School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia e-mail : rosihan@usm.my Virendra Kumar Department of Mathematics, Central University of Haryana, Mahendergarh–123029 Haryana, India e-mail : vktmaths@yahoo.in V. Ravichandran ∗ Department of Mathematics, University of Delhi, Delhi–110007, India e-mail : vravi@maths.du.ac.in; vravi68@gmail.com Shanmugam Sivaprasad Kumar Department of Applied Mathematics, Delhi Technological University, Delhi–110042, India e-mail : spkumar@dce.ac.in Abstract. Sharp radius constants for certain classes of normalized analytic functions with fixed second coefficient, to be in the classes of starlike functions of positive order, parabolic starlike functions, and Sok´o l-Stankiewicz starlike functions are obtained. Our results extend several earlier works. 1. Introduction Let A denote the class of analytic functions f defined on D := {z ∈ C : |z| < 1}, which are normalized by the conditions f (0) = 0, and f ′ (0) = 1 and let S denote its subclass consisting of univalent functions. The well-known Bieberbach theorem states that the second coefficient in the Maclaurin series of functions in S is bounded by two. This estimate for the second coefficient plays an important role in the study * Corresponding Author. Received August 20, 2015; accepted February 26, 2016. 2010 Mathematics Subject Classification: 30C45, 30C80. Key words and phrases: Starlike functions, Sok´ol-Stankiewicz starlike functions, parabolic starlike functions, convex functions, radius constants. 473