An Extension Operator Associated With Certain $G$-Loewner Chains

Abstract

In this paper we are concerned with an extension operator Φ n,α,β that provides a way of extending a locally univalent function f on the unit disc U to a locally biholomorphic mapping F ∈ H(B n ). By using the method of Loewner chains, we prove that if f can be embedded as the first element of a g-Loewner chain on the unit disc, whereAlso, if f is spirallike of type δ and order γ on U , where δ ∈ (−π/2, π/2) and γ ∈ (0, 1), then F = Φ n,α,β (f) is spirallike of type δ and order γ on B n . We also obtain a subordination preserving result under the operator Φ n,α,β and we consider some radius problems associated with this operator.

Extension Operators Preserving Janowski Classes of Univalent Functions

“…The following result was obtained in [8, Theorem 2.1] (see also [11] for α = 0), in the case that g(ζ) = (1 + ζ)/(1 − ζ), ζ ∈ U . Also, Theorem 2.1 was recently obtained by Chirilȃ (see [1]) when the function g is given by g…”

“…In this paper, we continue the work in [1,2,7,8] concerning extension operators and g-Loewner chains in C n . We consider the operator Φ n,α,β and the set S 0 g (U ) of normalized holomorphic functions on unit disk U that have g-parametric representation, where g(ζ)…”

“…) reduces to the set S 0 (B n ) of mappings which have parametric representation, hence any Loewner chain f (z, t) on B n , such that {e −t f ( • , t)} t≥0 is a normal family on B n , is a g-Loewner chain on B n (see [7]). Also, the family S 0 g (B n ) where the function g is given by g(ζ) = 1+ζ 1+(2γ−1)ζ , ζ ∈ U and γ ∈ (0, 1) was studied by Chirilȃ in [1]. Remark 1.18.…”

“…Various properties of the operator Φ n,α,β were investigated in [11,21], in the case α = 0 and β ∈ [0, 1/2] (see also [9], in the case β = 0). Also, the operator Φ n,α,β was studied in [1,8,10].…”

“…As a consequence, the operator Φ n,α,β preserves the notions of Janowski starlikeness and Janowski almost starlikeness from the unit disk into the unit ball B n . Particular cases from [1,2] will be also mentioned.…”