Upper Bound of Second Hankel determinant for generalized Sak

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Abstract

In this paper, the authors introduce a generalized Sakaguchi type spiral-like function class S(λ, β, s, t) and obtain sharp upper bound to the second Hankel determinant |H2(1)| for the function f in the above class. Relevances of the main result are also brieﬂy indicated. Key Words: Analytic functions, Starlike functions, Sakaguchi type functions, λ-spiral-like functions, Second Hankel determinant, Toeplitz determinants Contents 1

Results & Lemmas (3)

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Lemma 2.1. Lemma 2.1. (see [6]) If p ∈P, then |cn| ≤2, for each n ≥1 and the inequality is sharp for the function 1+z 1−z.

Lemma 2.1. (see [6]) If p ∈P, then |cn| ≤2, for each n ≥1 and the inequality is sharp for the function 1+z 1−z.

Lemma 2.2. Lemma 2.2. ([16], also see [17, p. 254]) Let the function p ∈P be given by the power series (2.1). Then 2c2 = c2 1 + x(4 −c2 1), (2.2) and…

Lemma 2.2. ([16], also see [17, p. 254]) Let the function p ∈P be given by the power series (2.1). Then 2c2 = c2 1 + x(4 −c2 1), (2.2) and 4c3 = c3 1 + 2(4 −c2 1)c1x −(4 −c2 1)c1x2 + 2(4 −c2 1)(1 −|x|2)y (2.3) for some complex numbers x, y satisfying |x| ≤1 and |y| ≤1. 3. Main Result

Theorem 3.1. Theorem 3.1. Let the function f given by (1.1) be in the class S(λ, β, s, t). Then |a2a4 −a2 3| ≤4(1 −β)2cos2λ (2s2 −st −t2)2. (3.1) The…

Theorem 3.1. Let the function f given by (1.1) be in the class S(λ, β, s, t). Then |a2a4 −a2 3| ≤4(1 −β)2cos2λ (2s2 −st −t2)2 . (3.1) The estimate in (3.1) is sharp.

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

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