Sharp bounds on the third Hankel determinant for the Ozaki close-to-convex and convex functions

Our main purpose in this paper is to obtain certain sharp estimates of the third Hankel determinant for the class F of Ozaki close-to-convex functions. This class was introduced by Ozaki in 1941. Functions in F are not necessarily starlike but are convex in one direction and so are close-to-convex. We prove that the sharp bounds of Script capital H3,1(f) and Script capital H3,1(f-1) for f is an element of F are all equal to 1/16. We also calculate the sharp bounds of the third Hankel determinant with entry of coefficients on the inverse of convex functions.