Second Hankel Determinant for a Class of Analytic Functions Defined by Fractional Derivative

Abstract

By making use of the fractional differential operator Ω λ z due to Owa and Srivastava, a class of analytic functions R λ α, ρ 0 ≤ ρ ≤ 1, 0 ≤ λ < 1, |α| < π/2 is introduced. The sharp bound for the nonlinear functional |a 2 a 4 − a 2 3 | is found. Several basic properties such as inclusion, subordination, integral transform, Hadamard product are also studied.

The bounds of some determinants for functions of bounded turning of order alpha

“…This lemma was also applied in [3] and [4] to find sharp bounds analogous as in Theorem 2.1 for the class of starlike functions of order α and of strongly starlike functions, respectively. Let now remark that the result of Theorem 2.1 can be achieved in a simple way by using Theorem 2.4 of [8] which particularly produces the following result: if p ∈ P is of the form (1.4) and µ ∈ [0, 1], then for n, m ∈ N, m < n, the following sharp estimate holds Theorem 2.4 can be found in [14] as Corollary 3.2. We reprove it by using the result below for the class R shown in [9].…”

“…For fixed q and n the growth problem is reduced to find the bound of the Hankel determinant over selected compact subclasses of A. Recently many authors examined the Hankel determinant H 2,2 (f ) of order 2 as well as the Hankel determinant H 3,1 (f ) of order 3 (see e.g., [9], [14], [11], [2] Given α ∈ [0, 1), by R(α) we denote a subclass of A of functions f such that…”

On Coefficients Problems for Typically Real Functions Related to Gegenbauer Polynomials

“…They discussed the second Hankel determinants for various classes of univalent functions. Some results in this direction can be found in [3][4][5]10,11]. …”

“…The boundary of Ω 2 (T (λ)) consists of points (a 2 , a 3 , a 4 ) that correspond to the following functions:…”

Upper Bound of Second Hankel determinant for generalized Sakaguchi type spiral-like functions

“…Second Hankel determinant for various subclasses of analytic functions were obtained by various authors. For details, (see [1,2,11,12,13,14,15,18]). Following the techniques devised by Libera and Zlotkiewicz (see [16,17]), in the present paper, the authors determine a sharp upper bound of the second Hankel determinant |H 2 (1)| for the function f belonging to the class S(λ, β, s, t).…”