Coefficient bounds for certain subclasses of starlike functions

Abstract

The conjecture proposed by Raina and Sokòł [Hacet. J. Math. Stat. 44(6):1427-1433 (2015)] for a sharp upper bound on the fourth coefficient has been settled in this manuscript. An example is constructed to show that their conjectures for the bound on the fifth coefficient and the bound related to the second Hankel determinant are false. However, the correct bound for the latter is stated and proved. Further, a sharp bound on the initial coefficients for normalized analytic function f such that zf (z)/f (z) ≺ √ 1 + λz, λ ∈ (0, 1], have also been obtained, which contain many existing results.

Fifth Coefficient Estimate for Certain Starlike Functions

“…For the interesting history and relevant work, see [20,29]. Many authors have computed upper bounds for the functional |J 2,3 | = |a 2 a 3 − a 4 |, a specific case of the generalized Zalcman coefficient functional J j,k , for various classes, see [4,9,10]. In section 3, we obtain sharp bound of |J 2,3 (f )| for the above specified classes.…”

“…V. Ravichandran and S. Verma [28] settled these conjectures for n = 5. For different choices of ϕ(z) the bound for |a 5 | is known [10,13,17], but not for the general case. Recently, O. S. Kwon et al [19] established fifth coefficient bound for functions belonging to Bazilevič class.…”

Hermitian–Toeplitz determinants for functions with bounded turning

“…In the last few years investigation of the Hankel determinant gained much attention and brief survey of those work until 2013 can be found in the introduction of the paper [24]. Much recent history of development in this direction can be found in [1,3,4,7,9,13,16,26,31]. Wide variety of applications of Toeplitz-plus-Hankel systems arise in linear filtering theory, discrete inverse scattering, and discretization of certain integral equations arising in mathematical physics [30].…”

Sharp estimation of Hermitian-Toeplitz determinants for Janowski type starlike and convex functions

“…The classes of Janowski type starlike and Janowski type convex functions are defined, respectively, by [14]). For more details, we refer the reader to [6,15]. In line of the investigation on Hankel determinants in recent years, Ali et al [1] investigated the sharp bound for the second and third order symmetric Toeplitz determinants.…”