Hankel Determinant H<sub>2</sub>(3) for Certain Subclasses of Univalent Functions

Let S to be the class of functions which are analytic, normalized and univalent in the unit disk U = {z : |z| < 1}.The main subclasses of S are starlike functions, convex functions, close-to-convex functions, quasiconvex functions, starlike functions with respect to (w.r.t.) symmetric points and convex functions w.r.t.symmetric points which are denoted by S * , K, C, C * , S * S , and K S respectively.In recent past, a lot of mathematicians studied about Hankel determinant for numerous classes of functions contained in S. The qth Hankel determinant for q ≥ 1 and n ≥ 0 is defined by H q (n).H 2 (1) = a 3 -a 2 2 is greatly familiar so called Fekete-Szegö functional.It has been discussed since 1930's.Mathematicians still have lots of interest to this, especially in an altered version of a 3 -µa 2 2 .Indeed, there are many papers explore the determinants H 2 (2) and H 3 (1).From the explicit form of the functional H 3 (1), it holds H 2 (k) provided k from 1-3.Exceptionally, one of the determinant that is H 2 (3) = a 3 a 5 -a 4 2 has not been discussed in many times yet.In this article, we deal with this Hankel determinant H 2 (3) = a 3 a 5 -a 4 2 .From this determinant, it consists of coefficients of function f which belongs to the classes S * S and K S so we may find the bounds of |H 2 (3)| for these classes.Likewise, we got the sharp results for S * S and K S for which a 2 = 0 are obtained.