A New Roper-Suffridge Extension Operator on a Reinhardt Domain

Abstract

We introduce a new Roper-Suffridge extension operator on the following Reinhardt domain Ω_{𝑛,𝑝2,…,𝑝𝑛}={𝑧∈ℂ^𝑛∶|𝑧₁|²+∑^𝑛_𝑗=2|𝑧_𝑗|^𝑝𝑗<1} given by 𝐹(𝑧)=(𝑓(𝑧₁)+𝑓′(𝑧₁)∑^𝑛_𝑗=2𝑎_𝑗𝑧^𝑝𝑗_𝑗,(𝑓′(𝑧₁))^1/𝑝2𝑧₂,…,(𝑓′(𝑧₁))^1/𝑝𝑛𝑧_𝑛), where 𝑓 is a normalized locally biholomorphic function on the unit disc 𝐷, 𝑝_𝑗 are positive integer, 𝑎_𝑗 are complex constants, and 𝑗=2,…,𝑛. Some conditions for 𝑎_𝑗 are found under which the operator preserves almost starlike mappings of order 𝛼 and starlike mappings of order 𝛼, respectively. In particular, our results reduce to many well-known results when all 𝛼_𝑗=0.

1. Introduction

In 1995, Roper and Suffridge [1] introduced an extension operator. This operator is defined as follows: Φ_𝑛𝑓(𝑧)=𝑓𝑧₁,𝑓^′𝑧₁𝑧^′₀^′,(1.1) where 𝑓 is a normalized locally biholomorphic function on the unit disk 𝐷 in ℂ, 𝑧=(𝑧₁,𝑧^′₀)^′ belonging to the unit ball ℬ^𝑛 in ℂ^𝑛, 𝑧₀=(𝑧₂,…,𝑧_𝑛)′∈ℂ^𝑛−1 and the branch of the square root is chosen such that 𝑓′(0)=1.

It is well known that the Roper-Suffridge extension operator has the following remarkable properties:(i)if 𝑓 is a normalized convex function on 𝐷, then Φ_𝑛(𝑓) is a normalized convex mapping on ℬ^𝑛;(ii)if 𝑓 is a normalized starlike function on 𝐷, then Φ_𝑛(𝑓) is a normalized starlike mapping on ℬ^𝑛;(iii)if 𝑓 is a normalized Bloch function on 𝐷, then Φ_𝑛(𝑓) is a normalized Bloch mapping on ℬ^𝑛.

The above result (i) was proved by Roper and Suffridge [1] and the result (ii) and (iii) was proved by Graham and Kohr [2, 3]. Until now, it is difficult to construct the concrete convex mappings, starlike mappings, and Bloch mappings on ℬ^𝑛. By making use of the Roper-Suffridge extension operator, we may easily give many concrete examples about these mappings. This is one important reason why people are interested in this extension operator.

In 2005, Muir [4] modified the Roper-Suffridge extension operator as follows: 𝐹(𝑧)=𝑓𝑧₁+𝑓^′𝑧₁𝑃𝑧₀,𝑓^′𝑧₁𝑧^′₀^′,(1.2) where 𝑃(𝑧₀) is a homogeneous polynomial of degree 2 with respect to 𝑧₀, and 𝑓,𝑧₁, and 𝑧₀ are defined as above. They proved that this operator preserves starlikeness and convexity if and only if ‖𝑃‖⩽1/4 and ‖𝑃‖⩽1/2, respectively. The modified operator plays a key role to study the extreme points of convex mappings on ℬ^𝑛 (see [5, 6]). Later, Kohr [7] and Muir [8] used the Loewner chain to study the modified Roper-Suffridge extension operator. Recently, the modified Roper-Suffridge extension operator on the unit ball is also studied by Wang and Liu [9] and Feng and Yu [10].

On the other hand, people also considered the generalized Roper-Suffridge extension operator on the general Reinhardt domains. For example, Gong and Liu [11, 12] induced the definition of 𝜀 starlike mappings and obtained that the operator Φ_{𝑛,(1/𝑝)}𝑓(𝑧)=𝑓𝑧₁,𝑓^′𝑧₁^1/𝑝𝑧^′₀^′(1.3) maps the 𝜀 starlike functions on 𝐷 to the 𝜀 starlike mappings on the Reinhardt domain Ω_𝑛,𝑝={𝑧∈ℂ^𝑛∶|𝑧₁|²+∑^𝑛_𝑗=2|𝑧_𝑗|^𝑝<1}, where 𝑝⩾1,𝑓,𝑧₁, and 𝑧₀ are defined as above. When 𝜀=0 and 𝜀=1, Φ_{𝑛,(1/𝑝)} maps the starlike function and convex function on 𝐷 to the starlike mapping and the convex mapping on Ω_𝑛,𝑝, respectively.

Furthermore, Gong and Liu [13] proved that the operator Φ_{𝑛,(1/𝑝2),…,1/𝑝𝑛}𝑓(𝑧)=𝑓𝑧₁,𝑓^′𝑧₁^1/𝑝2𝑧₂,…,𝑓^′𝑧₁^1/𝑝𝑛𝑧_𝑛^′(1.4) maps the 𝜀 starlike functions on 𝐷 to the 𝜀 starlike mappings on the domain Ω_{𝑛,𝑝2,…,𝑝𝑛}={𝑧∈ℂ^𝑛∶|𝑧₁|²+∑^𝑛_𝑗=2|𝑧_𝑗|^𝑝𝑗<1}, where 𝑝_𝑗⩾1,𝑗=2,…,𝑛,𝑓,𝑧₁, and 𝑧₀ are defined as above. Liu and Liu [14] proved that this operator preserves starlikeness of order 𝛼 on the domain Ω_{𝑛,𝑝2,…,𝑝𝑛}. On the other hand, Feng and Liu [15] proved that this operator preserves almost starlikeness of order 𝛼 on the domain Ω_{𝑛,𝑝2,…,𝑝𝑛}.

In contrast to the modified Roper-Suffridge extension operator in the unit ball, it is natural to ask if we can modify the Roper-Suffridge extension operator on the Reinhardt domains. In this paper, we will introduce the following modified operator: 𝐹(𝑧)=𝑓𝑧₁+𝑓^′𝑧₁^𝑛_𝑗=2𝑎_𝑗𝑧^𝑝𝑗_𝑗,𝑓^′𝑧₁^1/𝑝2𝑧₂,…,𝑓^′𝑧₁^1/𝑝𝑛𝑧_𝑛^′(1.5) on the Reinhardt domain Ω_{𝑛,𝑝2,…,𝑝𝑛}. We will give some sufficient conditions for 𝑎_𝑗 under which the above Roper-Suffridge operator preserves an almost starlike mappings of order 𝛼 and starlike mappings of order 𝛼, respectively.

In the following, we give some notation and definitions. Let ℂ^𝑛 be the space of 𝑛 complex variables 𝑧=(𝑧₁,…,𝑧_𝑛)′ with the Euclidean inner product ⟨𝑧,𝑤⟩=∑^𝑛_𝑖=1𝑧_𝑖𝑤_𝑖 and the Euclidean norm ‖𝑧‖=⟨𝑧,𝑧⟩^1/2, where 𝑧,𝑤∈ℂ^𝑛 and the symbol “′” means transpose. The unit ball of ℂ^𝑛 is the set ℬ^𝑛={𝑧∈ℂ^𝑛∶‖𝑧‖<1}, and the unit sphere is denoted by 𝜕ℬ^𝑛={𝑧∈ℂ^𝑛∶‖𝑧‖=1}. In the case of one complex variable, ℬ¹ is the unit disk, usually denoted by 𝐷. Let Ω be a domain in ℂ^𝑛. Denote 𝐻(Ω) by the space of all holomorphic mappings from Ω into ℂ^𝑛. A mapping 𝑓∈𝐻(ℬ^𝑛) is called normalized if 𝑓(0)=0 and 𝐽_𝑓(0)=𝐼_𝑛, where 𝐽_𝑓(0) is the complex Jacobian matrix of 𝑓 at the origin and 𝐼_𝑛 is the identity operator on ℂ^𝑛. A mapping 𝑓∈𝐻(ℬ^𝑛) is said to be locally biholomorphic if det𝐽_𝑓(𝑧)≠0 for every 𝑧∈ℬ^𝑛. A normalized mapping 𝑓∈𝐻(ℬ^𝑛) is said to be convex if 𝜆𝜔₁+(1−𝜆)𝜔₂∈𝑓(ℬ^𝑛) for arbitrary 𝜔₁,𝜔₂∈𝑓(ℬ^𝑛) and 0⩽𝜆⩽1. A normalized mapping 𝑓∈𝐻(ℬ^𝑛) is said to be starlike with respect to the origin if 𝜆𝑓(ℬ^𝑛)⊂𝑓(ℬ^𝑛),0⩽𝜆⩽1. A normalized mapping 𝑓∈𝐻(ℬ^𝑛) is said to be 𝜀 starlike if there exists a positive number 𝜀, 0⩽𝜀⩽1, such that 𝑓(ℬ^𝑛) is starlike with respect to every point in 𝜀𝑓(ℬ^𝑛).

A domain Ω is called a Reinhardt domain if (𝑒^𝑖𝜃1𝑧₁,𝑒^𝑖𝜃2𝑧₂,…,𝑒^𝑖𝜃𝑛𝑧_𝑛)^′∈Ω holds for any 𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑛)^′∈Ω and 𝜃₁,𝜃₂,…,𝜃_𝑛∈ℝ. A domain Ω is called a circular domain if 𝑒^𝑖𝜃𝑧∈Ω holds for any 𝑧∈Ω and 𝜃∈ℝ. The Minkowski functional 𝜌(𝑧) of the Reinhardt domain Ω_{𝑛,𝑝2,…,𝑝𝑛}=𝑧∈ℂ^𝑛∶𝑧₁²+^𝑛_𝑗=2𝑧_𝑗^𝑝𝑗<1,𝑝_𝑗⩾1,𝑗=2,…,𝑛(1.6) is defined as 𝜌(𝑧)𝑧=inf𝑡>0,𝑡∈Ω_{𝑛,𝑝2,…,𝑝𝑛},𝑧∈ℂ^𝑛.(1.7) Then, the Minkowski functional 𝜌(𝑧) is a norm of ℂ^𝑛 and Ω_{𝑛,𝑝2,⋯,𝑝𝑛} is the unit ball in the Banach space ℂ^𝑛 with respect to this norm. The Minkowski functional 𝜌(𝑧) is 𝐶¹ on Ω_{𝑛,𝑝2,…,𝑝𝑛} except for a lower-dimensional manifold Ω₀. Moreover, we give the following properties of the Minkowski functional 𝜌(𝑧) (see [16]): 2𝜕𝜌(𝑧)(𝑧)𝜕𝑧𝑧=𝜌,∀𝑧∈ℂ^𝑛⧵Ω₀,2𝜕𝜌𝑧𝜕𝑧𝑧=1,∀𝑧∈𝜕Ω_{𝑛,𝑝2,…,𝑝𝑛}⧵Ω₀,𝜕𝜌()=𝜕𝑧𝜆𝑧𝜕𝜌(𝑧)[)𝜕𝑧,∀𝜆∈0,∞,𝑧∈ℂ^𝑛⧵Ω₀,𝜕𝜌𝑒𝜕𝑧^𝑖𝜃𝑧=𝑒^−𝑖𝜃𝜕𝜌(𝑧)𝜕𝑧,∀𝑧∈ℂ^𝑛⧵Ω₀,𝜃∈ℝ.(1.8)

Definition 1.1 (see [17]). Suppose that Ω is a bounded starlike circular domain in ℂ^𝑛. Its Minkowski functional 𝜌(𝑧) is 𝐶¹ except for a lower-dimensional manifold. Let 0⩽𝛼<1. We say that a normalized locally biholomorphic mapping 𝑓∈𝐻(Ω) is an almost starlike mapping of order 𝛼 if the following condition holds: ℜ2𝜌(𝑧)𝜕𝜌(𝑧)𝐽𝜕𝑧^𝑓−1(𝑧)𝑓(𝑧)⩾𝛼,𝑧∈Ω⧵{0}.(1.9) When Ω=ℬ^𝑛, its Minkowski functional 𝜌(𝑧)=‖𝑧‖, the above inequality becomes ℜ𝑧^′𝐽_𝑓⁻¹(𝑧)𝑓(𝑧)⩾𝛼‖𝑧‖²,𝑧∈ℬ^𝑛.(1.10)
In particular, when 𝛼=0, 𝑓 reduces to a starlike mapping on Ω.

Definition 1.2 (see [18]). Suppose that Ω∈ℂ^𝑛 is a bounded starlike circular domain. Its Minkowski functional 𝜌(𝑧) is 𝐶¹ except for a lower-dimensional manifold. Let 0<𝛼<1. We say that a normalized locally biholomorphic mapping 𝑓∈𝐻(Ω) is a starlike mapping of order 𝛼 if the following condition holds: 2𝜌(𝑧)𝜕𝜌(𝑧)𝐽𝜕𝑧^𝑓−1(𝑧)𝑓(𝑧)−1<12𝛼{0}.2𝛼,𝑧∈Ω⧵(1.11) When Ω=ℬ^𝑛, the above inequality reduces to 1‖𝑧‖²𝑧^′𝐽_𝑓⁻¹(𝑧)𝑓(𝑧)−1<12𝛼2𝛼,𝑧∈ℬ^𝑛⧵{0}.(1.12)

2. Some Lemmas

In order to prove the main results, we need the following three lemmas.

Lemma 2.1 (see [19]). Let 𝑝 be a holomorphic function on 𝐷. If ℜ𝑝(𝑧)>0 and 𝑝(0)>0, then 𝑝^′(𝑧⩽(𝑧))2ℜ𝑝1−|𝑧|².(2.1)

Lemma 2.2 (see [19]). Let 𝑓 be a normalized biholomorphic function on 𝐷. Then, 1−|𝑧|²𝑓^′′(𝑧)𝑓^′(𝑧)−2𝑧⩽4(2.2) holds for all 𝑧∈𝐷.

Lemma 2.3 (see [20]). If 𝜌(𝑧) is a Minkowski function of the domain Ω_{𝑛,𝑝2,…,𝑝𝑛}, 𝑧≠0, then 𝜕𝜌𝜕𝑧₁(𝑧)=𝑧₁𝜌(𝑧)2𝑧₁/𝜌(𝑧)²+∑^𝑛_𝑗=2𝑝_𝑗𝑧_𝑗/𝜌(𝑧)^𝑝𝑗,𝜕𝜌𝜕𝑧_𝑗(𝑧)=𝑝_𝑗𝑧_𝑗𝑧_𝑗/𝜌(𝑧)^𝑝𝑗−22𝜌(𝑧)2𝑧₁/𝜌(𝑧)²+∑^𝑛_𝑗=2𝑝_𝑗𝑧_𝑗/𝜌(𝑧)^𝑝𝑗,𝑗=2,…,𝑛.(2.3)

3. Main Results

Theorem 3.1. Let 0⩽𝛼<1 and let 𝑓 be an almost starlike function of order 𝛼 on the unit disc 𝐷. If complex numbers 𝑎_𝑗 satisfy the condition |𝑎_𝑗|⩽(1−𝛼)/4,𝑗=2,…,𝑛, then 𝐹(𝑧)=𝑓𝑧₁+𝑓^′𝑧₁^𝑛_𝑗=2𝑎_𝑗𝑧^𝑝𝑗_𝑗,𝑓^′𝑧₁^1/𝑝2𝑧₂,…,𝑓^′𝑧₁^1/𝑝𝑛𝑧_𝑛^′(3.1) is an almost starlike mapping of order 𝛼 on the domain Ω_{𝑛,𝑝2,…,𝑝𝑛}, where 𝑝_𝑗 are positive integer and 𝑝_𝑗⩾2; the branches are chosen such that (𝑓′(𝑧₁))^1/𝑝𝑗∣_𝑧1=0=1.

Proof. By the definition of almost starlike mapping of order 𝛼, we need only to prove that the following inequality: ℜ2𝜌(𝑧)𝜕𝜌(𝑧)𝐽𝜕𝑧^𝐹−1(𝑧)𝐹(𝑧)⩾𝛼(3.2) holds for all 𝑧∈Ω_{𝑛,𝑝2,…,𝑝𝑛} and 𝑧≠0.
The case of 𝑧₀=0 is trivial. So, we need only consider that 𝑧=(𝑧₁,𝑧^′₀)′∈Ω_{𝑛,𝑝2,…,𝑝𝑛},𝑧₀≠0. Let 𝑧=𝜁𝑢=|𝜁|𝑒^𝑖𝜃𝑢,𝑢∈𝜕Ω_{𝑛,𝑝2,…,𝑝𝑛}, and 𝜁∈𝐷⧵{0}, then we have ℜ2𝜌(𝑧)𝜕𝜌(𝑧)𝐽𝜕𝑧^𝐹−1(𝑧)𝐹(𝑧)2⩾𝛼⟺ℜ𝜌𝜁𝑒^𝑖𝜃𝑢𝜕𝜌𝜁𝑒𝜕𝑧^𝑖𝜃𝑢𝐽_𝐹⁻¹𝜁𝑒^𝑖𝜃𝑢𝐹𝜁𝑒^𝑖𝜃𝑢2⩾𝛼⟺ℜ𝜁𝑒^−𝑖𝜃𝜕𝜌(𝑢)𝐽𝜕𝑧^𝐹−1𝜁𝑒^𝑖𝜃𝑢𝐹𝜁𝑒^𝑖𝜃𝑢⩾𝛼⟺ℜ2𝜕𝜌(𝑢)𝐽𝜕𝑧^𝐹−1(𝜁𝑢)𝐹(𝜁𝑢)𝜁⩾𝛼.(3.3) For a fixed 𝑢, the expression ℜ(2𝜕𝜌/𝜕𝑧)(𝑢)(𝐽_𝐹⁻¹(𝜁𝑢)𝐹(𝜁𝑢)/𝜁)−𝛼 is the real part of a holomorphic function with respect to 𝜁, so it is a harmonic function. By the minimum of harmonic function principle, we know that it attains its minimum on |𝜁|=1, so we need only to prove for all 𝑧∈𝜕Ω_{𝑛,𝑝2,…,𝑝𝑛} and 𝑧₀≠0. Hence, 𝜌(𝑧)=1 and inequality (3.2) becomes ℜ2𝜕𝜌(𝑧)𝐽𝜕𝑧^𝐹−1(𝑧)𝐹(𝑧)⩾𝛼,𝑧∈𝜕Ω_{𝑛,𝑝2,…,𝑝𝑛},𝑧₀≠0.(3.4)
In the following, we will prove inequality (3.4).
Since 𝐹(𝑧)=𝑓𝑧₁+𝑓^′𝑧₁^𝑛_𝑗=2𝑎_𝑗𝑧^𝑝𝑗_𝑗,𝑓^′𝑧₁^1/𝑝2𝑧₂,…,𝑓^′𝑧₁^1/𝑝𝑛𝑧_𝑛^′,(3.5) we have𝐽_𝐹(𝑧)=𝑓^′𝑧₁+𝑓^′′𝑧₁^𝑛_𝑗=2𝑎_𝑗𝑧^𝑝𝑗_𝑗𝑎₂𝑝₂𝑓^′𝑧₁𝑧^𝑝2₂⁻¹⋯𝑎_𝑛𝑝_𝑛𝑓^′𝑧₁𝑧^𝑝𝑛_𝑛⁻¹1𝑝₂𝑓^′𝑧₁^{(1/𝑝2)−1}𝑓^′′𝑧₁𝑧₂𝑓^′𝑧₁^1/𝑝21⋯0⋯⋯⋯⋯𝑝_𝑛𝑓^′𝑧₁^{(1/𝑝𝑛)−1}𝑓^′′𝑧₁𝑧_𝑛0⋯𝑓^′𝑧₁^1/𝑝𝑛.(3.6) Suppose that 𝐽_𝐹⁻¹(𝑧)𝐹(𝑧)=𝐴=(𝑥₁,𝑥₂,…,𝑥_𝑛)^′, then 𝐹(𝑧)=𝐽_𝐹(𝑧)𝐴; that is, 𝑥₁𝑓^′𝑧₁+𝑓^′′𝑧₁^𝑛_𝑗=2𝑎_𝑗𝑧^𝑝𝑗_𝑗+𝑓^′𝑧₁^𝑛_𝑗=2𝑎_𝑗𝑝_𝑗𝑥_𝑗𝑧^𝑝𝑗_𝑗⁻¹=𝑓𝑧₁+𝑓^′𝑧₁^𝑛_𝑗=2𝑎_𝑗𝑧^𝑝𝑗_𝑗,𝑥₁𝑓^′′𝑧₁𝑝₂𝑓^′𝑧₁𝑧₂+𝑥₂=𝑧₂,⋮𝑥₁𝑓^′′𝑧₁𝑝_𝑛𝑓^′𝑧_𝑛𝑧_𝑛+𝑥_𝑛=𝑧_𝑛.(3.7) Some computation shows that 𝑥₁=𝑓𝑧₁𝑓^′𝑧₁−^𝑛_𝑗=2𝑎_𝑗𝑝_𝑗−1𝑧^𝑝𝑗_𝑗,𝑥₂=𝑓𝑧1−¹𝑓^′′𝑧₁𝑝₂𝑓^′𝑧₁²+𝑓^′′𝑧₁𝑝₂𝑓^′𝑧₁^𝑛_𝑗=2𝑎_𝑗𝑝_𝑗−1𝑧^𝑝𝑗_𝑗𝑧₂,⋮𝑥_𝑛=𝑓𝑧1−¹𝑓^′′𝑧₁𝑝_𝑛𝑓^′𝑧₁²+𝑓^′′𝑧₁𝑝_𝑛𝑓^′𝑧₁^𝑛_𝑗=2𝑎_𝑗𝑝_𝑗−1𝑧^𝑝𝑗_𝑗𝑧_𝑛.(3.8) From Lemma 2.3, we obtain 𝜕𝜌𝜕𝑧₁(𝑧)=𝑧₁𝜌(𝑧)2𝑧₁/𝜌(𝑧)²+∑^𝑛_𝑗=2𝑝_𝑗𝑧_𝑗/𝜌(𝑧)^𝑝𝑗=𝑧₁2𝑧₁²+∑^𝑛_𝑗=2𝑝_𝑗𝑧_𝑗^𝑝𝑗,𝜕𝜌𝜕𝑧_𝑗(𝑧)=𝑝_𝑗𝑧_𝑗𝑧_𝑗^𝑝𝑗−22𝜌(𝑧)2𝑧₁/𝜌(𝑧)²+∑^𝑛_𝑗=2𝑝_𝑗𝑧_𝑗^𝑝𝑗=𝑝_𝑗𝑧_𝑗𝑧_𝑗^𝑝𝑗−222𝑧₁²+∑^𝑛_𝑗=2𝑝_𝑗𝑧_𝑗^𝑝𝑗.(3.9) In terms of (3.8) and (3.9), we obtain 2𝜕𝜌(𝑧)𝐽𝜕𝑧^𝐹−1(𝑧)𝐹(𝑧)=𝐺(𝑧)2𝑧₁²+∑^𝑛_𝑗=2𝑝_𝑗𝑧_𝑗^𝑝𝑗,(3.10) where 𝐺(𝑧)=2𝑧₁𝑓𝑧₁𝑓^′𝑧₁−^𝑛_𝑗=2𝑎_𝑗𝑝_𝑗−1𝑧^𝑝𝑗_𝑗+^𝑛_𝑗=2𝑝_𝑗𝑧_𝑗^𝑝𝑗𝑓𝑧1−¹𝑓^′′𝑧₁𝑝_𝑗𝑓^′𝑧₁²+𝑓^′′𝑧₁𝑝_𝑗𝑓^′𝑧₁^𝑛_𝑘=2𝑎_𝑘𝑝_𝑘−1𝑧^𝑝𝑘_𝑘=2𝑧₁²𝑓𝑧₁𝑧₁𝑓^′𝑧₁+^𝑛_𝑗=2𝑝_𝑗𝑧_𝑗^𝑝𝑗𝑓𝑧1−¹𝑓^′′𝑧₁𝑝_𝑗𝑓^′𝑧₁²+^𝑛_𝑘=2𝑎_𝑘𝑝_𝑘−1𝑧^𝑝𝑘_𝑘𝑓^′′𝑧₁𝑓^′𝑧₁^𝑛_𝑗=2𝑧_𝑗^𝑝𝑗−2𝑧₁.(3.11) By making use of the equality |𝑧₁|²+∑^𝑛_𝑗=2|𝑧_𝑗|^𝑝𝑗=1, we then get 𝐺(𝑧)=2𝑧₁²𝑓𝑧₁𝑧₁𝑓^′𝑧₁+^𝑛_𝑗=2𝑝_𝑗𝑧_𝑗^𝑝𝑗𝑓𝑧1−¹𝑓^′′𝑧₁𝑝_𝑗𝑓^′𝑧₁²+^𝑛_𝑗=2𝑎_𝑗𝑝_𝑗−1𝑧^𝑝𝑗_𝑗𝑓^′′𝑧₁𝑓^′𝑧₁1−𝑧₁²−2𝑧₁.(3.12) Let ℎ(𝑧₁)=(𝑓(𝑧₁)/𝑧₁𝑓′(𝑧₁))−𝛼. Notice that 𝑓 is an almost starlike function of order 𝛼 on the unit disc; hence, ℜℎ(𝑧₁)>0 and ℎ(0)=1−𝛼>0. By Lemma 2.1, we can obtain that ℎ^′(𝑧⩽(𝑧))2ℜℎ1−|𝑧|².(3.13) Furthermore, we get 𝑓𝑧₁𝑓^′′𝑧₁𝑓^′𝑧₁²=1−𝛼−ℎ𝑧₁−𝑧₁ℎ^′𝑧₁.(3.14) Substituting (3.14) into (3.12), we have 𝐺(𝑧)=2𝑧₁²ℎ𝑧₁+𝛼+^𝑛_𝑗=2𝑝_𝑗𝑧_𝑗^𝑝𝑗11−𝑝_𝑗+𝛼𝑝_𝑗+ℎ𝑧₁𝑝_𝑗+𝑧₁𝑝_𝑗ℎ^′𝑧₁+^𝑛_𝑗=2𝑎_𝑗𝑝_𝑗−1𝑧^𝑝𝑗_𝑗𝑓^′′𝑧₁𝑓^′𝑧₁1−𝑧₁²−2𝑧₁=ℎ𝑧₁2𝑧₁²+^𝑛_𝑗=2𝑧_𝑗^𝑝𝑗+2𝛼𝑧₁²+^𝑛_𝑗=2𝑝_𝑗−1+𝛼𝑧_𝑗^𝑝𝑗+^𝑛_𝑗=2𝑧₁𝑧_𝑗^𝑝𝑗ℎ^′𝑧₁+^𝑛_𝑗=2𝑎_𝑗𝑝_𝑗−1𝑧^𝑝𝑗_𝑗𝑓^′′𝑧₁𝑓^′𝑧₁1−𝑧₁²−2𝑧₁.(3.15) Hence, ℜ𝐺(𝑧)⩾2𝑧₁²+^𝑛_𝑗=2𝑧_𝑗^𝑝𝑗ℜℎ𝑧₁+2𝛼𝑧₁²+^𝑛_𝑗=2𝑝_𝑗−1+𝛼𝑧_𝑗^𝑝𝑗−^𝑛_𝑗=2𝑧_𝑗^𝑝𝑗𝑧₁ℎ^′𝑧₁−^𝑛_𝑗=2𝑎_𝑗𝑝_𝑗−1𝑧_𝑗^𝑝𝑗𝑓^′′𝑧₁𝑓′𝑧₁1−𝑧₁²−2𝑧₁.(3.16) By Lemma 2.2 and (3.13), we can get that ℜ𝐺(𝑧)⩾1+𝑧₁²ℜℎ𝑧₁+2𝛼𝑧₁²+^𝑛_𝑗=2𝑝_𝑗−1+𝛼𝑧_𝑗^𝑝𝑗−1−𝑧₁²2𝑧₁ℜℎ𝑧₁1−𝑧₁²−4^𝑛_𝑗=2𝑎_𝑗𝑝_𝑗−1𝑧_𝑗^𝑝𝑗=1+𝑧₁²ℜℎ𝑧₁+2𝛼𝑧₁²+^𝑛_𝑗=2𝑝_𝑗−1+𝛼𝑧_𝑗^𝑝𝑗−2𝑧₁ℜℎ𝑧₁−4^𝑛_𝑗=2𝑎_𝑗𝑝_𝑗−1𝑧_𝑗^𝑝𝑗=1−𝑧₁²ℜℎ𝑧₁+2𝛼𝑧₁²+^𝑛_𝑗=2𝑧_𝑗^𝑝𝑗𝛼+1−4𝑎_𝑗𝑝_𝑗−1.(3.17) Hence, when |𝑎_𝑗|⩽(1−𝛼)/4, we have ℜ𝐺(𝑧)⩾1−𝑧₁²ℜℎ𝑧₁+2𝛼𝑧₁²+𝛼^𝑛_𝑗=2𝑝_𝑗𝑧_𝑗^𝑝𝑗⩾2𝛼𝑧₁²+𝛼^𝑛_𝑗=2𝑝_𝑗𝑧_𝑗^𝑝𝑗.(3.18) In terms of (3.10) and (3.18), we obtain ℜ2𝜕𝜌(𝑧)𝐽𝜕𝑧^𝐹−1(𝑧)𝐹(𝑧)⩾𝛼,(3.19) which completes the proof of Theorem 3.1.

Remark 3.2. When 𝑎₂=𝑎₃=⋯=𝑎_𝑛=0, the result of Theorem 3.1 has been obtained by Liu and Liu [14].

Corollary 3.3. Let 𝑓 be a normalized biholomorphic starlike function on the unit disc 𝐷. If |𝑎_𝑗|⩽1/4,𝑗=2,…,𝑛, then 𝐹(𝑧)=𝑓𝑧₁+𝑓^′𝑧₁^𝑛_𝑗=2𝑎_𝑗𝑧^𝑝𝑗_𝑗,𝑓^′𝑧₁^1/𝑝2𝑧₂,…,𝑓^′𝑧₁^1/𝑝𝑛𝑧_𝑛^′(3.20) is a normalized biholomorphic starlike mapping on the domain Ω_{𝑛,𝑝2,…,𝑝𝑛}, where 𝑝_𝑗 are positive integer and 𝑝_𝑗⩾2; the branches are chosen such that (𝑓′(𝑧₁))^1/𝑝𝑗∣_𝑧1=0=1.

Theorem 3.4. Let 0<𝛼<1 and let 𝑓 be a starlike function of order 𝛼 on the unit disc 𝐷. If complex numbers 𝑎_𝑗 satisfy the condition |𝑎_𝑗|⩽(1−|2𝛼−1|)/8𝛼,𝑗=2,…,𝑛, then 𝐹(𝑧)=𝑓𝑧₁+𝑓^′𝑧₁^𝑛_𝑗=2𝑎_𝑗𝑧^𝑝𝑗_𝑗,𝑓^′𝑧₁^1/𝑝2𝑧₂,…,𝑓^′𝑧₁^1/𝑝𝑛𝑧_𝑛^′(3.21) is a starlike mapping of order 𝛼 on the domain Ω_{𝑛,𝑝2,…,𝑝𝑛}, where 𝑝_𝑗 are positive integer and 𝑝_𝑗⩾1; the branches are chosen such that (𝑓′(𝑧₁))^1/𝑝𝑗∣_𝑧1=0=1.

Proof. By the definition of starlike mapping of order 𝛼, we need only to prove that the following inequality: 2𝜌(𝑧)𝜕𝜌(𝑧)𝐽𝜕𝑧^𝐹−1(𝑧)𝐹(𝑧)−1<12𝛼2𝛼(3.22) holds for all 𝑧∈Ω_{𝑛,𝑝2,…,𝑝𝑛} and 𝑧₀≠0.
Similar to the proof of Theorem 3.1, we need only to prove that (3.22) holds for 𝜌(𝑧)=1 and 𝑧₀≠0 according to the maximum modulus theorem for analytic functions. So, it is suffice to show that 2𝜕𝜌(𝑧)𝐽𝜕𝑧^𝐹−1(𝑧)𝐹(𝑧)−1<12𝛼.2𝛼(3.23) From the proof of Theorem 3.1, we can get 𝜕𝜌(𝑧)𝐽𝜕𝑧^𝐹−1(𝑧)𝐹(𝑧)=122𝑧₁²+∑^𝑛_𝑗=2𝑝_𝑗𝑧_𝑗^𝑝𝑗2𝑧₁²𝑓𝑧₁𝑧₁𝑓′𝑧₁+^𝑛_𝑗=2𝑝_𝑗𝑧_𝑗^𝑝𝑗𝑓𝑧1−¹𝑓^′′𝑧₁𝑝_𝑗𝑓^′𝑧₁²+^𝑛_𝑗=2𝑎_𝑗𝑝_𝑗−1𝑧^𝑝𝑗_𝑗𝑓^′′𝑧₁𝑓^′𝑧₁1−𝑧₁²−2𝑧₁.(3.24) Hence, 2𝜕𝜌(𝑧)𝐽𝜕𝑧^𝐹−1(𝑧)𝐹(𝑧)−1=𝐻(𝑧)2𝛼2𝛼2𝑧₁²+∑^𝑛_𝑗=2𝑝_𝑗𝑧_𝑗^𝑝𝑗,(3.25) where 𝐻(𝑧)=2𝑧₁²𝑓𝑧2𝛼¹𝑧₁𝑓^′𝑧₁−1+2𝛼^𝑛_𝑗=2𝑝_𝑗𝑧_𝑗^𝑝𝑗11−−𝑓𝑧2𝛼¹𝑓^′′𝑧₁𝑝_𝑗𝑓^′𝑧₁²+2𝛼^𝑛_𝑗=2𝑎_𝑗𝑝_𝑗−1𝑧^𝑝𝑗_𝑗𝑓^′′𝑧₁𝑓^′𝑧₁1−𝑧₁²−2𝑧₁.(3.26) Let ℎ(𝑧₁)=2𝛼(𝑓(𝑧₁)/𝑧₁𝑓′(𝑧₁))−1. Then, |ℎ(𝑧₁)|<1 because 𝑓 is a starlike function of order 𝛼 on the unit disc 𝐷. By the Schwarz-Pick lemma, we obtain that ℎ′𝑧₁⩽1−ℎ(𝑧₁)²1−𝑧₁².(3.27) On the other hand, we can get 𝑓𝑧₁𝑓^′′𝑧₁𝑓^′𝑧₁²1=1−−ℎ𝑧2𝛼¹−𝑧2𝛼¹ℎ^′𝑧₁.2𝛼(3.28) Substituting (3.28) into (3.26), we have 𝐻(𝑧)=2𝑧₁²𝑓𝑧2𝛼¹𝑧₁𝑓^′𝑧₁−1+2𝛼^𝑛_𝑗=2𝑝_𝑗𝑧_𝑗^𝑝𝑗11−−12𝛼𝑝_𝑗−12𝛼𝑝_𝑗−ℎ𝑧₁2𝛼𝑝_𝑗−𝑧₁ℎ^′𝑧₁2𝛼𝑝_𝑗+2𝛼^𝑛_𝑗=2𝑎_𝑗𝑝_𝑗−1𝑧^𝑝𝑗_𝑗𝑓^′′𝑧₁𝑓^′𝑧₁1−𝑧₁²−2𝑧₁=2𝑧₁²ℎ𝑧₁+ℎ𝑧₁^𝑛_𝑗=2𝑧_𝑗^𝑝𝑗+^𝑛_𝑗=2𝑧₁ℎ^′𝑧₁𝑧_𝑗^𝑝𝑗+(2𝛼−1)^𝑛_𝑗=2𝑝_𝑗−1𝑧_𝑗^𝑝𝑗+2𝛼^𝑛_𝑗=2𝑎_𝑗𝑝_𝑗−1𝑧^𝑝𝑗_𝑗𝑓^′′𝑧₁𝑓^′𝑧₁1−𝑧₁²−2𝑧₁.(3.29) Hence, |𝐻(𝑧)|⩽1+𝑧₁²ℎ𝑧₁+^𝑛_𝑗=2𝑧₁ℎ′𝑧₁𝑧_𝑗^𝑝𝑗+|2𝛼−1|^𝑛_𝑗=2𝑝_𝑗−1𝑧_𝑗^𝑝𝑗+2𝛼^𝑛_𝑗=2𝑎_𝑗𝑝_𝑗−1𝑧_𝑗^𝑝𝑗𝑓^′′𝑧₁𝑓^′𝑧₁1−𝑧₁²−2𝑧₁.(3.30) By Lemma 2.2 and (3.27), we have |𝐻(𝑧)|⩽1+𝑧₁²ℎ𝑧₁+𝑧₁1−ℎ𝑧₁²1−𝑧₁²1−𝑧₁²+|2𝛼−1|^𝑛_𝑗=2𝑝_𝑗−1𝑧_𝑗^𝑝𝑗+8𝛼^𝑛_𝑗=2𝑎_𝑗𝑝_𝑗−1𝑧_𝑗^𝑝𝑗⩽1+𝑧₁²ℎ𝑧₁−1+1+𝑧₁²+2𝑧₁1−ℎ𝑧₁+|2𝛼−1|^𝑛_𝑗=2𝑝_𝑗−1𝑧_𝑗^𝑝𝑗+8𝛼^𝑛_𝑗=2𝑎_𝑗𝑝_𝑗−1𝑧_𝑗^𝑝𝑗⩽1+𝑧₁²+ℎ𝑧₁−11−𝑧₁²+^𝑛_𝑗=2|2𝛼−1|+8𝛼𝑎_𝑗𝑝_𝑗−1𝑧_𝑗^𝑝𝑗.(3.31) If |𝑎_𝑗|⩽(1−|2𝛼−1|)/8𝛼, then we obtain |𝐻(𝑧)|<1+𝑧₁²+||||2𝛼−1+8𝛼1−2𝛼−18𝛼^𝑛_𝑗=2𝑝_𝑗−1𝑧_𝑗^𝑝𝑗≤1+𝑧₁²+^𝑛_𝑗=2𝑝_𝑗−1𝑧_𝑗^𝑝𝑗=2𝑧₁²+^𝑛_𝑗=2𝑝_𝑗𝑧_𝑗^𝑝𝑗.(3.32) The equality (3.25) and (3.32) show that 2𝜕𝜌(𝑧)𝐽𝜕𝑧^𝐹−1(𝑧)𝐹(𝑧)−1<12𝛼,2𝛼(3.33) which completes the proof of Theorem 3.4.

4. Problem

In 2003, Gong and Liu [13] proved that the Roper-Suffridge extension operator Φ_{𝑛,(1/𝑝2),…,(1/𝑝𝑛)}𝑓(𝑧)=𝑓𝑧₁,𝑓^′𝑧₁^1/𝑝2𝑧₂,…,𝑓^′𝑧₁^1/𝑝𝑛𝑧_𝑛^′(4.1) does preserve convexity on Ω_𝑛,𝑝₂,…,𝑝_𝑛, which solved the open problem posed by Graham and Kohr [2]. Naturally, we will propose the following problem on the new Roper-Suffridge extension operator.

Problem 1. Let 𝑝_𝑗 be positive integer. Under what conditions for 𝑎_𝑗 such that if 𝑓 is a convex function in the disc 𝐷, then the mapping defined by the new Roper-Suffridge extension operator 𝐹(𝑧)=𝑓𝑧₁+𝑓^′𝑧₁^𝑛_𝑗=2𝑎_𝑗𝑧^𝑝𝑗_𝑗,𝑓^′𝑧₁^1/𝑝2𝑧₂,…,𝑓^′𝑧₁^1/𝑝𝑛𝑧_𝑛^′(4.2) is a convex mapping in the Reinhardt domain Ω_𝑛,𝑝₂,…,𝑝_𝑛?

Acknowledgments

The authors cordially thank the referees thorough reviewing with useful suggestions and comments made to the paper. The project was supported by the National Natural Science Foundation of China (Grant no. 11001246 and no. 11101139), the Natural Science Foundation of Zhejiang province (Grant no. Y6090694), and the Zhejiang Innovation Project (Grant no. T200905).