Hankel Determinants for Univalent Functions Related to the Exponential Function

Abstract

Recently, two classes of univalent functions S e * and K e were introduced and studied. A function f is in S e * if it is analytic in the unit disk, f ( 0 ) = f ′ ( 0 ) − 1 = 0 and z f ′ ( z ) f ( z ) ≺ e z . On the other hand, g ∈ K e if and only if z g ′ ∈ S e * . Both classes are symmetric, or invariant, under rotations. In this paper, we solve a few problems connected with the coefficients of functions in these classes. We find, among other things, the estimates of Hankel determinants: H 2 , 1 , H 2 , 2 , H 3 , 1 . All these estimates improve the known results. Moreover, almost all new bounds are sharp. The main idea used in the paper is based on expressing the discussed functionals depending on the fixed second coefficient of a function in a given class.

Third Hankel Determinant For A Class Of Starlike Functions Associated With Exponential Function

“…This class was introduced by Mendiratta et al [16]. Initially, Zhang et.al [29] estimated the bound of H 3 (1) as 0.565, then in an attempt to improve the result Shi et.al [23] and Zaprawa [28] estimated the bounds of H 3 (1) as 0.5004 and 0.385 respectively . Although they were improving the result but still couldn't achieve the sharp bound.…”

Initial successive coefficients for certain classes of univalent functions involving the exponential function

“…By choosing λ = 1, we obtain the families S * e and K * e which were introduced and investigated by Mediratta et al [11] and were later studied by many authors, see [4,5,12,17,18,22,23] and the references cited therein. Clearly, for 0 < ζ ≤ 1 and 1 ≤ η ≤ π 2 , we have S * ζe ⊆ S * e ⊆ S * ηe .…”