An Investigation of the Third Hankel Determinant Problem for Certain Subfamilies of Univalent Functions Involving the Exponential Function

Abstract

In the current article, we consider certain subfamilies S e ∗ and C e of univalent functions associated with exponential functions which are symmetric along real axis in the region of open unit disk. For these classes our aim is to find the bounds of Hankel determinant of order three. Further, the estimate of third Hankel determinant for the family S e ∗ in this work improve the bounds which was investigated recently. Moreover, the same bounds have been investigated for 2-fold symmetric and 3-fold symmetric functions.

Hankel Determinants for Univalent Functions Related to the Exponential Function

“…. .In the next theorem we improve the bound of H 2,2 obtained by Shi et al (see, Theorem 4 in[7]). If g ∈ K e is given by (27) and |b2 | = p/4, p ∈ [0, 2], then |b 2 b 4 − b 3 2 | ≤ 1 4608 (128 + 8p 2 − 9p 4 ) .…”

“…In fact, the second coefficient of f ∈ S * e or g ∈ K e can be replaced by the coefficient p 1 of a corresponding function P with a positive real part. This idea leads to better estimates than those in [6,7]. Moreover, the new bounds of a 2 a 3 − a 4 and H 2,2 are sharp.…”

“…Now, expanding both sides of (11) in the Taylor series and comparing coefficients at z k , k = 1, 2, 3, 4, we obtain (see also Formulae (15)-(18) in [7]),…”

“…In their two following papers Zhang et al [6] and Shi et al [7] broadened the range of discussed coefficient problems. They found the coefficient estimates for a n , n = 2, 3, 4, 5 and the bounds of the following functionals: a 2 a 3 − a 4 , H 2,1 , H 2,2 and, as a consequence, H 3,1 .…”

Upper Bound of the Third Hankel Determinant for a Subclass of Close-to-Convex Functions Associated with the Lemniscate of Bernoulli

“…The Hankel determinant plays a vital role in the theory of singularities [17] and is useful in the study of power series with integer coefficients (see [18][19][20]). Noteworthy, several authors obtained the sharp upper bounds on H 2 (2) (see, for example, [5,[21][22][23][24][25][26][27][28][29]) for various classes of functions. It is a well-known fact for the Fekete-Szegö functional that:…”