The Sharp Bound of the Hankel Determinant of the Third Kind for Starlike Functions with Real Coefficients.

Let SR * be the class of starlike functions with real coefficients, i.e., the class of analytic functions f which satisfy the condition f (0) = 0 = f ′ (0) − 1 , Re { z f ′ (z) / f (z) } > 0 , for z ∈ D : = { z ∈ C : | z | < 1 } and a n : = f (n) (0) / n ! is real for all n ∈ N . In the present paper, it is obtained that the sharp inequalities − 4 / 9 ≤ H 3 , 1 (f) ≤ 3 / 9 hold for f ∈ SR * , where H 3 , 1 (f) is the third Hankel determinant of order 3 defined by H 3 , 1 (f) = a 3 (a 2 a 4 − a 3 2) − a 4 (a 4 − a 2 a 3) + a 5 (a 3 − a 2 2) . [ABSTRACT FROM AUTHOR]