The Sharp Bound of the Third Hankel Determinant for Starlike Functions of Order 1/2

Hankel determinants for starlike functions associated with a cardioid domain

“…of order 3 is a more difficult problem, and non-sharp bounds for |H 3 (1) (f )| have been found by various authors, e.g., [1,3,18,25]. We note that using a result in [11], some authors [2,8,9,10,12,17,19] have obtained sharp bounds for |H 3 (1) (f )| for certain subclasses of univalent functions.…”

The sharp bound of the third Hankel determinants for inverse of starlike functions with respect to symmetric points

“…In recent years, research on the estimation of an upper bound of the second and third order Hankel determinant is investigated by many authors. Particularly, the problem of estimating H 3,1 (f ) is technically much more difficult [2,4,6,7,11,14,16], and only few sharp bounds have been obtained. The class of starlike functions with respect to symmetric points is introduced by Sakaguchi [12] and is denoted as S * s .…”

“…Every such a function is called the Carathéodory function. In view of ( 2) and ( 3), the coefficients of functions in S * s have suitable representation expressed by coefficients of functions in P. Hence, to estimate the upper bound of |H 3,1 (f −1 )|, we build our computation on the well known formulas on coefficients c 2 (see [9, p. 166] ), c 3 (see [8]) and c 4 can be found in [11].…”

A Conjecture on $H_3(1)$ For Certain Starlike Functions

“…authors in [1] have obtained the sharp estimates for |H 3 (1)| ≤ 1/36 for the choice ϕ(z) = √ 1 + z. Since, a proper, careful and infact precise re-arrangement of terms is highly required to obtain the best possible bound and it results in few research articles, which are available in this area for the sharp bound of H 3 (1), see [1,8,20]. The most important step in obtaining the sharp bound of H 3 (1) was to rewrite equation (1.1) [11], which is stated below in the form of a lemma:…”

“…In the recent times, it has been observed that finding the sharp estimates of third Hankel determinant for general Ma-Minda class is not feasible till now. But for some classes, sharp estimates have been obtained, for instance, see[1,8,20] and now including Theorem 2.1 as well which motivated us to settle the Conjecture 1.2. Further looking at the difficulty of the general class, we restrict ourselves to answer the problem for the n-fold symmetric functions.…”