The sharp bound of the third Hankel determinant of the kth-r

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Abstract

The objective of this paper is to estimate the sharp bound of the third Hankel determinant for the kth-root transformation to the class of functions whose derivative has a positive real part satisfying the normalized conditions f(0) = 0 and f ′(0) = 1 in the open unit disk D := {z ∈C : |z| < 1}.

Results & Lemmas (3)

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Lemma 1.1 Lemma 1.1 ([14], p. 41). If g ∈P, then |ct| ≤2 for t ∈N; the equality occurs for the function g(z) = 1+z 1−z, z ∈D.

Lemma 1.1 ([14], p. 41). If g ∈P, then |ct| ≤2 for t ∈N; the equality occurs for the function g(z) = 1+z 1−z, z ∈D.

Lemma 1.2 Lemma 1.2 ([11, 12, 13]). If g ∈P and c1 ≥0, then 2c2 = c2 1 + tζ, 4c3 = c3 1 + 2c1tζ −c1tζ2 + 2t 1 −|ζ|2 η, and 8c4 = c4 1 + 3c2 1tζ +…

Lemma 1.2 ([11, 12, 13]). If g ∈P and c1 ≥0, then 2c2 = c2 1 + tζ, 4c3 = c3 1 + 2c1tζ −c1tζ2 + 2t 1 −|ζ|2 η, and 8c4 = c4 1 + 3c2 1tζ + 4 −3c2 1 tζ2 + c2

Theorem 1. Theorem 1. If f ∈ℜ, then H3,1,k(f) ≤ 1 4k2, and the result is sharp for p(z):= (1 + z3)/(1 −z3).

Theorem 1. If f ∈ℜ, then H3,1,k(f) ≤ 1 4k2 , and the result is sharp for p(z) := (1 + z3)/(1 −z3).

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The sharp bound of the third Hankel determinant of the kth-root transformation for bounded turning functions

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