The Sharp Bound of the Hankel Determinant of the Third Kind

Abstract

In the present paper, we proved the sharp inequality |H3,1( f )| ≤1/9 for analytic functions f with an := f (n)(0)/n!, n ∈N, a1 := 1, such that Re zf ′(z) f (z) > 1 2, z ∈D := {z ∈C : |z| < 1}, where H3,1( f ) := a1 a2 a3 a2 a3 a4 a3 a4 a5 is the third Hankel determinant.

Results & Lemmas (2)

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Lemma 2.1 Lemma 2.1 If p ∈P is of the form (1.6) with c1 ≥0, then 2c2 = c2 1 + (4 −c2 1)ζ, (2.1) 4c3 = c3 1 + (4 −c2 1)c1ζ(2 −ζ) + 2(4 −c2 1)(1…

Lemma 2.1 If p ∈P is of the form (1.6) with c1 ≥0, then 2c2 = c2 1 + (4 −c2 1)ζ, (2.1) 4c3 = c3 1 + (4 −c2 1)c1ζ(2 −ζ) + 2(4 −c2 1)(1 −|ζ|2)η (2.2)

Theorem 2.2 Theorem 2.2 max

Theorem 2.2 max

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Results & Lemmas (2)

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