On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings

Abstract

In this paper, we obtain the upper bounds to the third Hankel determinants for starlike functions of order α, convex functions of order α and bounded turning functions of order α. Furthermore, several relevant results on a new subclass of close-toconvex harmonic mappings are obtained. Connections of the results presented here to those that can be found in the literature are also discussed.

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:…”

Sharp bounds on Hankel determinants for Sakaguchi classes

“…Obviously, the case of the upper bounds on |H 3,1 (f )| is much more difficult than the cases of |H 2,1 (f )| and |H 2,2 (f )|. We refer to [4,7,10,37,39,41,45] for discussions on the upper bounds of the third Hankel determinants |H 3,1 (f )| for various classes of univalent functions. However, these results are far from sharpness.…”

Coefficient Inequalities for certain Starlike and Convex functions

“…Kowalczyk et al [17] established sharp inequality |H Recently, Kumar et al [22] improved certain existing bound on the third Hankel determinant for some classes of close-to-convex functions. For recent results on third Hankel determinant, see [10,24,25,39]. Hankel determinants are closely related to Hermitian-Toeplitz determinants [18,42].…”