Notes on the norm of pre-Schwarzian derivatives on bi-univalent functions of order $α$

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. 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,