On Hankel Determinant Inequalities

: This article aims to obtain the second Hankel determinant inequalities for the inverse of the well-known classes of univalent functions, namely, starlike and convex functions.

The inverse  −1 of every function  ∈ ∆ , defined by  −1 (()) = , is analytic in || < (), (() ≥ ) and has Maclaurin's series expansion (2) Early in 1923 Löwner invented the famous parametric method to find sharp bounds on all the coefficients for the inverse functions in ∆ (or ∆ ⋆ ).Thus if  ∈ ∆(∆ ⋆ ) is given by (2) then with equality for every  for the inverse of the Koebe function

Preliminaries
Let the function  given by the power series () = 1 +  1  +  2  2 +⋅⋅⋅ be analytic in a neighborhood of the origin.For a real number  define the function ℎ by Thus   () denotes the  ℎ coefficient in Maclaurin's series expansion of the  ℎ of the function ().

Main Result
The result is sharp.
Proof.We use the fact that Now for fixed  we write .
Further,  3 ≥ 0 is equivalent to and this provides the result; for some , || ≤ 1.

) 2 + 3 ( 4 −
2 )( − 2)( − 4) > 0Thus (y) is an increasing function that attained its maximum at y = 1.The upper bound for the above corresponds to y = 1 and  = 1,in which case |18 1 In addition, the upper bound for the functional | 2  4 − 3  2| for functions  belongs to the class ∆ ⋆ and   .The objective of this study is to obtain the upper bounds for the functional | 2  4 −  3 2 | for the inverse function  −1 given by (2) if  belonging to ∆ ⋆ and   , respectively.