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In this paper, we obtain upper bounds on initial coefficients and third Hankel determinant H3,1( f ) = ∣ ∣ ∣ ∣ ∣ ∣ a1 a2 a3 a2 a3 a4 a3 a4 a5 ∣ ∣ ∣ ∣ ∣ ∣ of the coefficients of analytic function f (z) = z + a2z + a3z + · · · , belonging to the class S ∗(q) in the open unit disk D , which satisfies the subordination condition that z f ′(z)/ f (z) ≺ q(z) (z ∈ D), where q(z) = √ 1+ z2 +z . Several results are presented exhibiting improvement in earlier work. Mathematics subject classification (2010): 30C45.

Bounds on coefficients and third Hankel determinant for a class of analytic functions related with certain conic domain