Abstract
In this present investigation, we first give a survey of the work done so far in this area of Hankel determinant for univalent functions. Then the upper bounds of the second Hankel determinant|a2a4−a32|for functions belonging to the subclassesS(α,β),K(α,β),Ss∗(α,β), andKs(α,β)of analytic functions are studied. Some of the results, presented in this paper, would extend the corresponding results of earlier authors.
Highlights
Let A denote the class of functions of the form ∞f (z) = z + ∑akzk (1)k=2 which are analytic in the unit disc U = {z : |z| < 1}, and let S denote the subclass of A that is univalent in U
Suppose that f and g are analytic functions in U; we say that f is subordinate to g, written f ≺ g, if there exists a Schwarz function ω, which is analytic in U with ω(0) = 0 and |ω(z)| < 1 for all z ∈ U, such that f(z) = g(ω(z)), z ∈ U
Janteng et al [26] have considered the functional |H2(2)| and found a sharp bound, the subclass of S denoted by R, defined as R{f(z)} > 0. They have shown that if f ∈ R, |H2(2)| ≤ 4/9. These authors [27, 28] studied the second Hankel determinant and sharp bound for the classes of starlike and convex functions, close-to-starlike and close-to-convex functions with respect to symmetric points denoted by S∗, K, Sc∗, and Kc and have shown that |H2(2)| ≤ 1, |H2(2)| ≤ 1/8, |H2(2)| ≤ 1, and |H2(2)| ≤ 1/9, respectively
Summary
Ss∗(α, the upper β), and Ks bounds (α, β) of of the second Hankel determinant |a2 analytic functions are studied. Let K(α, β) denote the class of functions f in A satisfying the following inequality: Let Ss∗(α, β) denote the class of functions f in A satisfying the following inequality:
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