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)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
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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