Abstract
In the present paper we estimate the norm of the pre-Schwarzian derivative of bi-starlike functions of order $α$ where $α\in[0,1)$. Initially this problem was handled by Rahmatan et al. in [Bull Iran Math Soc {\bf43}: 1037-1043, 2017]. We pointed out that the proofs and bounds by Rahmatan et al. are incorrect and present correct proofs and bounds.
Results & Lemmas (2)
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Theorem 2.3
Theorem 2.3]. Therefore every function f in the class U has an inverse f −1 which satisfies the following conditions: f −1(f(z)) = z (z ∈∆)…
Theorem 2.3]. Therefore every function f in the class U has an inverse f −1 which satisfies the following conditions: f −1(f(z)) = z (z ∈∆) and f(f −1(w)) = w (|w| < r0(f); r0(f) ≥1/4), where (1.2) f −1(w) = w −a2w2 + (2a2 2 −a3)w3 −(5a3 2 −5a2a3 + a4)w4 + · · · =: g(w). 2010 Mathematics Subject Classification. 30C45. Key words and phrases. univalent, locally univalent, bi-univalent, bi-starlike, subordination, pre-Schwarzian.
Theorem 2.1.
Theorem 2.1. Let α ∈[0, 1). If the function f of the form (1.1) is in the class S∗ Σ(α), then (2.1) ||f|| ≤ 6,
Theorem 2.1. Let α ∈[0, 1). If the function f of the form (1.1) is in the class S∗ Σ(α), then (2.1) ||f|| ≤ 6,
Function classes studied:
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