\documentclass[10pt]{studiamnew} \usepackage{graphicx} \usepackage{amsmath} \usepackage{amssymb} \sloppy \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{question}[theorem]{Question} \theoremstyle{definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{problem}[theorem]{Problem} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{solution}[theorem]{Solution} \newtheorem{notation}[theorem]{Notation} \renewcommand{\theequation}{\thesection.\arabic{equation}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\B}{\mathbb{B}^n} \newcommand{\Btwo}{\mathbb{B}^2} \newcommand{\Bthree}{\mathbb{B}^3} \newcommand{\BB}{\mathbb{B}^{n+1}} \newcommand{\p}{\mathcal{P}} \newcommand{\M}{\mathcal{M}} \newcommand{\A}{\mathcal{A}} \newcommand{\LS}{\mathcal{L}S} \newcommand{\ov}{\overline} \numberwithin{equation}{section} \begin{document} % \setcounter{page}{1} \setcounter{firstpage}{1} \setcounter{lastpage}{4} \renewcommand{\currentvolume}{??} \renewcommand{\currentyear}{??} \renewcommand{\currentissue}{??} % \title{Extension operators and Janowski starlikeness with complex coefficients} \author{Andra Manu} \address{``Babe\c{s}-Bolyai'' University, \\ Faculty of Mathematics and Computer Sciences\\ 1, Kog\u{a}lniceanu Street,\\ 400084 Cluj-Napoca,\\ Romania} \email{andra.manu@math.ubbcluj.ro} % \subjclass{Primary: 32H02, Secondary: 30C45.} \keywords{$g$-Loewner chain, $g$-parametric representation, $g$-starlikeness, Janowski starlikeness, Janowski almost starlikeness, extension operator} \begin{abstract} In this paper, we obtain certain generalizations of some results from \cite{Manu1} and \cite{Manu2}. Let $\Phi_{n, \alpha, \beta}$ be the extension operator introduced in \cite{GrahamHamadaKohrSuffridge} and let $\Phi_{n, Q}$ be the extension operator introduced in \cite{Muir0}. Let $a \in \C$, $b \in \R$ be such that $|1-a| < b \leq {\rm Re}\ a$. We consider the Janowski classes $S^*(a,b, \B)$ and $\A S^*(a,b, \B)$ with complex coefficients introduced in \cite{PCurt4}. In the case $n=1$, we denote $S^*(a,b, \mathbb{B}^1)$ by $S^*(a,b)$ and $\A S^*(a,b, \mathbb{B}^1)$ by $\A S^*(a,b)$. We shall prove that the following preservation properties concerning the extension operator $\Phi_{n, \alpha, \beta}$ hold: $\Phi_{n, \alpha, \beta} (S^*(a,b)) \subseteq S^*(a,b, \B)$, $\Phi_{n, \alpha, \beta} (\A S^*(a,b)) \subseteq \A S^*(a,b, \B)$. Also, we prove similar results for the extension operator $\Phi_{n, Q}$ : $\Phi_{n, Q}(S^*(a,b)) \subseteq S^*(a,b, \B)$, $\Phi_{n, Q}(\A S^*(a,b)) \subseteq \A S^*(a,b, \B)$). \end{abstract} \maketitle \section{Preliminaries} Let $\C^n$ be the space of $n$ complex variables equipped with the Euclidean inner product $\langle \cdot, \cdot \rangle$ and the Euclidean norm $\| \cdot \|$. Let $\B$ be the open unit ball in $\C^n$ and let $U$ be the unit disc in $\C$. Also, let $H(\B)$ be the set of holomorphic mappings from $\B$ into $\C^n$. A mapping $f \in H(\B)$ is said to be normalized if $f(0)=0$ and $Df(0)=I_n$. Let $J_f(z)$ be the complex Jacobian determinant of the Fr\' echet derivative $Df(z)$, i.e. $J_f(z)=det Df(z)$. A mapping $f \in H(\B)$ is locally biholomorphic mapping on $\B$ if $J_f(z) \neq 0$ for all $z \in \B$. We denote by $\LS_n$ the set of normalized locally biholomorphic mappings on the unit ball $\B$. In the case $n=1$, we use the notation $\LS$ instead of $\LS_1$. Let $S(\B)$ be the set of normalized biholomorphic mappings on $\B$ and let $S$ be the set of normalized univalent functions on $U$. Also, let $S^*(\B)$ be the set of normalized starlike mappings on $\B$. Let $f,g \in H(\B)$. Then we say that $f \prec g$ if there exists a Schwarz mapping $\varphi$ (i.e. $\varphi \in H(\B)$, $\|\varphi(z)\| \leq \|z\|$, $z \in \B$) such that $f = g \circ \varphi$ on $\B$. Moreover, if $g$ is biholomorphic on $\B$, then the subordination condition $f \prec g$ is equivalent with $f(0)=g(0)$ and $f(\B) \subseteq g(\B)$. We recall that $f : \B \times [0,\infty) \to \C^n$ is a Loewner chain if $f(\cdot, t)$ is biholomorphic on $\B$, $f(0,t)=0$, $Df(0,t)=e^t I_n$ for $t \geq 0$ and $f(\cdot, s) \prec f(\cdot,t)$ with $0 \leq s \leq t < \infty$ (see \cite{Pfaltzgraff1}, \cite{GrahamKohr}). The subordination condition $f(\cdot, s) \prec f(\cdot,t)$ is equivalent to the following statement: there is a unique biholomorphic Schwarz mapping $v=v(z,s,t)$ such that $ f(z,s) = f(v(z,s,t),t),\ z \in \B$, $0 \leq s \leq t$. The mapping $v = v(z,s,t)$ is called the \emph{transition mapping} associated to $f(z,t)$ and satisfies the semigroup property: $v(z, s, u) = v(v(z, s, t), t, u)$, for all $z \in \B$, $0 \leq s \leq t \leq u$. In addition, $Dv(0,s,t)=e^{s-t} I_n$, $0 \leq s \leq t$ (see \cite{Pfaltzgraff1}, \cite{GrahamKohr}). We recall that the following class of holomorphic mappings (see \cite{Pfaltzgraff1}, \cite{Suffridge2}; see also \cite{GrahamKohr}): $$ \M =\{ h \in H(\B) : h(0)=0, Dh(0)=I_n, {\rm Re} \ \langle h(z),z \rangle >0, z \in \B \backslash \{0\} \} $$ is the generalization to higher dimensions ($n \geq 2$) of the Carath\' edory class of functions with positive real part on $U$. We next give the definition of parametric representation on the unit ball in $\C^n$ (see \cite{GrahamHamadaKohr2}, \cite{GrahamKohr}). \begin{definition} We say that a mapping $f \in S(\B)$ has \emph{parametric representation} if there exists a Loewner chain $f(z,t)$ such that $f$ can be embedded as the first element of $f(z,t)$ and the family $\{e^{-t}f(\cdot,t)\}_{t\geq 0}$ is normal on $\B$. \end{definition} Let $S^0(\B)$ be the family of mappings with parametric representation. This set has been introduced by Graham, Hamada and Kohr in \cite{GrahamHamadaKohr2}. Various results regarding this class can be found in \cite{GrahamHamadaKohr2}, \cite{GrahamKohrKohr}, \cite{GKohr3} and the references therein. In the following we consider a function $g:U\to \C$ which satisfies the following conditions (see \cite{GrahamHamadaKohrKohr}): \begin{assumption}\label{definition-function-g} Let $g: U \to \C$ be such that $g$ is a univalent (i.e. holomorphic and injective) function on $U$, $g(0)=1$ and $g$ has positive real part on $U$. \end{assumption} For example, the function $g: U \to \C$ given by $g(\zeta)=\frac{1+\zeta}{1-\zeta}$, $\zeta \in U$, satisfies the requirements of Assumption \ref{definition-function-g}. In the following, let $g :U \to \C$ be an arbitrary function which satisfies the conditions of Assumption \ref{definition-function-g}. Let $\M_g$ be the following nonempty subset of $\M$ introduced by Graham, Hamada, Kohr and Kohr in \cite{GrahamHamadaKohrKohr} (see also \cite{GrahamHamadaKohr2}, where the function $g$ satisfies in addition the relation $g(\ov{\zeta}) = \ov{g(\zeta)}$, $z \in U$, and other conditions): \begin{equation*}\label{m-g} \M_g =\left \{ h \in H(\B) : h(0)=0, Dh(0)=I_n, \left\langle h(z),\frac{z}{\|z\|^2} \right\rangle \in g(U), z \in \B \backslash \{0\} \right \}. \end{equation*} For $g(\zeta) = \frac{1-\zeta}{1 + \zeta}$, $\zeta \in U$, we have that $\M_g = \M$. Next, we recall the definition of a $g$-Loewner chain (see \cite{GrahamHamadaKohrKohr}; see also \cite{GrahamHamadaKohr2} and \cite{GrahamKohrKohr}, for $g(\zeta) = \frac{1-\zeta}{1 + \zeta}$, $\zeta \in U$). \begin{definition} Let $f(z,t) : \B \times [0, \infty) \to \C^n$. We say that $f(z,t)$ is a $g$-\emph{Loewner chain} if $f(z,t)$ is a Loewner chain such that the family $\{e^{-t} f(\cdot, t)\}_{t \geq 0}$ is normal on $\B$ and the mapping $h(z,t)$ which occurs in the following Loewner differential equation: \begin{equation*} \frac{\partial f}{\partial t} = D f(z,t) h(z,t), \text{ a.e. } t \geq 0,\ \forall z \in \B, \end{equation*} has the property $h(\cdot, t) \in \M_g$, for a.e. $t \geq 0$. \end{definition} We remark that a normalized holomorphic mapping $f : \B \to \C^n$ has $g$-parametric representation if and only if there exists a $g$-Loewner chain $f(z,t)$ such that $f$ can be embedded as the first element of the $g$-Loewner chain (see \cite{GrahamHamadaKohrKohr}; see also \cite{GrahamHamadaKohr2}). Let $S^0_g(\B)$ be the set of mappings with $g$-parametric representation on $\B$. Then $S^0_g(\B) \subseteq S^0(\B)$ (see \cite{GrahamHamadaKohrKohr}). If $g(\zeta) = \frac{1-\zeta}{1 + \zeta}$, $\zeta \in U$, then any $g$-Loewner chain is a Loewner chain and the set $S^0_g(\B)$ becomes $S^0(\B)$ (see \cite{GrahamHamadaKohrKohr}; see also \cite{GrahamHamadaKohr2}). In the case $n \geq 2$, there exists Loewner chains that are not $g$-Loewner chains when $g(\zeta) = \frac{1-\zeta}{1 + \zeta}$, $\zeta \in U$. For example, when $n=2$, the mapping $p(z,t) : \Btwo \times [0, \infty) \to \C^n$ given by $$ p(z,t) = \left(\frac{e^t z_1}{(1-z_1)^2}, \frac{e^t z_2}{(1-z_2)^2} + \frac{e^{2t} z_1^2}{(1-z_1)^4} \right), \ z=(z_1,z_2) \in \Btwo, \ t\geq 0, $$ is a Loewner chain, but the family $\{e^{-t} p(\cdot, t)\}_{t \geq 0}$ is not normal on $\Btwo$. Thus, $p(\cdot,t)$ is not a $g$-Loewner chain for $g(\zeta) = \frac{1-\zeta}{1 + \zeta}$, $\zeta \in U$ (see \cite{GrahamHamadaKohr2}). In the next part, we shall refer to the following univalent function $g$ on $U$ with $g(0)=1$ and positive real part on $U$: \begin{assumption}\label{function-g-complex} Let $g : U \to \C$ be a holomorphic function on $U$ given by \begin{equation} g(\zeta) = \frac{1 + A\zeta}{1+B\zeta}, \ \zeta \in U, \end{equation} where $A,B \in \C$, $A \neq B$ and $g$ has positive real part on $U$. \end{assumption} This function was considered in \cite{PCurt4}. Imposing the condition that the function $g$ given by Assumption \ref{function-g-complex} to have positive real part implies certain conditions on the complex parameters $A$ and $B$. These conditions are illustrated in the following remark due to Curt \cite{PCurt4}. \begin{remark}\label{cond-on-param-complex}{\bf \cite{PCurt4}} Let $g: U \to \C$ be a function described by Assumption \ref{function-g-complex}. Then one of the following two conditions holds: \begin{equation}\label{first-cond} |B| < 1,\ |A| \leq 1 \text{ and } {\rm Re}(1 - A\ov{B}) \geq |A-B|, \end{equation} or \begin{equation}\label{second-cond} |B| = 1, \ |A| \leq 1 \text{ and } -1 \leq A\ov{B} < 1. \end{equation} \end{remark} In this context, we remark that the function $g$ maps the unit disc onto the open disc of center $ a := \frac{1- A\ov{B}}{1-|B|^2}$ and radius $b := \frac{|A-B|}{1-|B|^2}$, for $|B|<1$. It is immediate that $|1-a| < b \leq {\rm Re}\ a$. If $|B|=1$ then $g$ maps the unit disc onto the half-plane $\{z \in \C: {\rm Re} z > \frac{1+ A\ov{B}}{2}\}$. Moreover, we have that $g$ is convex on $U$. Next, we present the following subclasses of starlike mappings on $\B$ introduced by Curt \cite{PCurt4}: \begin{definition}\label{Janowski-complex} Let $a \in \C$, $b \in \R$ be such that $|1-a|< b \leq {\rm Re}\ a$. Let \begin{align*} S^*(a,b,\B) = \left \{ f\in \LS_n: \left | \frac{\|z\|^2}{\langle [Df(z)]^{-1}f(z), z\rangle} -a \right | 0$), then we have that $$ \A S^*\left(a,a,\B\right)=S^*_{\frac{1}{2a}}(\B) \text{ and } S^*\left(a,a,\B\right)=\A S^*_{\frac{1}{2a}}(\B).$$ \end{itemize} \end{remark} Note that the functions mentioned in Remark \ref{charact.janowski.classes.with.g.starl}(i), (ii) satisfy the conditions of Assumption \ref{function-g-complex}. Next, we consider the following extension operator introduced by Graham, Hamada, Kohr and Suffridge in \cite{GrahamHamadaKohrSuffridge}. \begin{definition}\label{definition-phi-n-alpha-beta-extension-operator} Let $\alpha \geq 0$, $\beta \geq 0$ and $n \geq 2$. Let $\Phi_{n, \alpha, \beta} : \LS \to \LS_n$ be given by \begin{equation}\label{phi-n-alpha-beta-extension-operator} \Phi_{n, \alpha, \beta} (f)(z) = \left( f(z_1), \tilde{z} \left( \frac{f(z_1)}{z_1} \right)^{\alpha} (f'(z_1))^{\beta} \right), \ z =(z_1, \tilde{z}) \in \B, \end{equation} where $$ \left( \frac{f(z_1)}{z_1} \right)^{\alpha} \Big\rvert_{z_1=0} = 1, \ (f'(z_1))^{\beta}\big\rvert_{z_1=0}=1.$$ \end{definition} For $\alpha=0$ and $\beta = 1/2$, the extension operator $\Phi_{n, \alpha, \beta}$ reduces to Roper-Suffridge extension operator $\Phi_{n} : \LS \to \LS_n$ given by (see \cite{RoperSuffridge}) $$\Phi_n(f)(z) = \left(f(z_1), \tilde{z} \sqrt{f'(z_1)} \right), \ z=(z_1, \tilde{z}) \in \B,$$ where the branch of the square root is chosen such that $\sqrt{f'(z_1)}\rvert_{z_1=0} = 1$. The extension operator $\Phi_{n, \alpha, \beta}$ satisfies important preservation properties for $\alpha \in [0,1]$, $\beta \in [0, 1/2]$, $\alpha + \beta \leq 1$. In \cite{GrahamHamadaKohrSuffridge}, it was shown that $\Phi_{n, \alpha, \beta}(f) (S) \subseteq S^0(\B)$ and $\Phi_{n, \alpha, \beta}(f) (S^*) \subseteq S^*(\B)$. In the same paper, the authors proved that $\Phi_{n, \alpha, \beta}$ conserves convexity only if $(\alpha, \beta)=(0, 1/2)$. Also, $\Phi_{n, \alpha, \beta}$ conserves starlikeness of order $\gamma \in (0,1)$ (see \cite{Liu}), spirallikeness of type $\gamma \in (-\pi/2, \pi/2)$ and order $\delta \in (0,1)$ (see \cite{LiuLiu}; see also \cite{TChirila}) and almost starlikeness of type $\gamma \in (0,1)$ and order $\delta \in [0,1)$ (see \cite{TChirila}). More recent preservation results regarding this extension operator and Bloch mappings, in the case of complex Banach spaces, are obtained in \cite{GrahamHamadaKohrKohr}. We next present the definition of the Muir extension operator $\Phi_{n, Q}$ (see \cite{Muir0}). \begin{definition}\label{definition-muir-extension-operator} Assume that $Q: \C^{n-1} \to \C$ is a homogeneous polynomial of degree $2$ and $n \geq 2$. Let $\Phi_{n, Q} : \LS \to \LS_n$ be such that \begin{equation} \Phi_{n, Q} (f)(z) = (f(z_1) + Q(\tilde{z}) f'(z_1), \tilde{z} \sqrt{f'(z_1)}), \ z = (z_1, \tilde{z}) \in \B, \end{equation} where $\sqrt{f'(z_1)}\rvert_{z_1=0}=1$. \end{definition} For $Q \equiv 0$, the extension operator $\Phi_{n, Q}$ reduces to the extension operator $\Phi_{n}$. The extension operator $\Phi_{n, Q}$ preserves parametric representation and starlikeness if $\|Q\| \leq 1/4$ (see \cite{GKohr3}), convexity if $\|Q\| \leq 1/2$ ( see \cite{Muir0}) and starlikeness of order $\alpha \in (0,1)$ if $\|Q\| \leq \frac{1 - |2 \alpha - 1|}{8 \alpha}$ (see \cite{WangLiu}; see also \cite{TChirila3}). In a recent study, there has been investigated results concerning extended Loewner chains and this extension operator, as well as other preservation results (see \cite{Muir2}). Also, modifications of the Muir extension operator were considered in \cite{GrahamHamadaKohrKohr}. \bigskip Assume that $a \in \C$, $b \in \R$ such that $|1-a|< b \leq {\rm Re}\ a$. In the next part, we aim to show that the extension operators $\Phi_{n, \alpha, \beta}$ and $\Phi_{n, Q}$ map a function $f \in S^*(a,b)$ into a mapping from $S^*(a,b, \B)$. Also, $\Phi_{n, \alpha, \beta}$ and $\Phi_{n, Q}$ map a function $f \in \A S^*(a,b)$ into a mapping from $\A S^*(a,b, \B)$. Therefore, the extension operators $\Phi_{n, \alpha, \beta}$ and $\Phi_{n, Q}$ preserve the Janowski starlikeness and Janowski almost starlikeness with complex coefficients from the case of one complex variable to several complex variables. \section{Main results} In \cite{GrahamHamadaKohrKohr}, I. Graham, H. Hamada, G. Kohr and M. Kohr proved that $g$-parametric presentation and $g$-starlikeness is preserved through the extension operators $\Phi_{n, \alpha, \beta}$ and $\Phi_{n, Q}$, when the function $g$ is convex on $U$ and satisfies the conditions of Assumption \ref{definition-function-g}. This result was obtained in a more general case, namely on the unit ball of a complex Banach space. All along this section we assume that $n\geq 2$. We state in the next two results the preservation of $g$-starlikeness under $\Phi_{n, \alpha, \beta}$ and $\Phi_{n, Q}$, when the function $g$ is convex on $U$ satisfying Assumption \ref{definition-function-g}. \begin{theorem}\label{op-1-g-star}{\bf \cite{GrahamHamadaKohrKohr}} Let $g :U \to \C$ be a univalent holomorphic function on $U$, with $g(0)=1$, ${\rm Re} g(\zeta) > 0$, $\zeta \in U$, and $g$ is convex on $U$. Also, let $\alpha \in [0,1]$, $\beta \in [0, 1/2]$, $\alpha + \beta \leq 1$. If $f \in S^*_g$ then $F = \Phi_{n, \alpha, \beta} (f) \in S^*_g(\B)$. \end{theorem} In the next result, let be the distance from $1$ to $\partial g(U)$, denoted by $d(1, \partial g(U))$, and equal to $\inf_{\zeta \in \partial g(U)} |\zeta -1|$. \begin{theorem}\label{op-2-g-star}{\bf \cite{GrahamHamadaKohrKohr}} Let $g :U \to \C$ be a univalent function on $U$, with $g(0)=1$, ${\rm Re} g(\zeta) > 0$, $\zeta \in U$, and $g$ is convex on $U$. Also, let $\|Q\| \leq d(1, \partial g(U))/4$, where $Q$ is a homogeneous polynomial of degree $2$ from $\C^{n-1}$ to $\C$. If $f \in S^*_g$ then $F = \Phi_{n, Q} (f) \in S^*_g(\B)$. \end{theorem} It is clear that, for the function $g$ defined by Assumption \ref{function-g-complex}, the above statements hold. In addition, we have the following result. \begin{remark} Let $g$ be a function satisfying the conditions from Assumption \ref{function-g-complex}. Then $$d(1, \partial g(U)) = \frac{|A-B|}{1+|B|}.$$ \end{remark} \begin{proof} Since the function $g$ satisfies the requirements of Assumption \ref{function-g-complex}, then, in view of Remark \ref{cond-on-param-complex}, the complex coefficients $A$ and $B$ satisfy one of the following two relations: \begin{equation*} |B| < 1, \ |A| \leq 1 \text{ and } {\rm Re}(1 - A\ov{B}) \geq |A-B|, \end{equation*} or \begin{equation*} |B| = 1, \ |A| \leq 1 \text{ and } -1 \leq A\ov{B} < 1. \end{equation*} We shall analyze the above two cases. $\bullet$ Assume that $|B| = 1, \ |A| \leq 1 \text{ and } {\rm Re}(1 - A\ov{B}) \geq |A-B|$. In this case, we have $g(U) = \{z \in \C: {\rm Re}\ z > \frac{1+ A\ov{B}}{2}\}$. Thus, $$\partial g(U) = \{z \in \C : z = \frac{1+ A\ov{B}}{2} + i y, \ y \in \R\}.$$ Let $\zeta \in \partial g(U)$. Then $\zeta = \frac{1+ A\ov{B}}{2} + i y$, where $y \in \R$. We have that $$ |\zeta - 1 | = \left| \frac{1+ A\ov{B}}{2} + i y - 1\right| = \left|\frac{- 1 + A\ov{B}}{2} + i y \right|.$$ Using the above relation and the fact that $-1 \leq A\ov{B} < 1$, we have that $$ \inf_{\zeta \in \partial g(U)} |\zeta -1| = \inf_{y \in \R} \left|\frac{- 1 + A\ov{B}}{2} + i y \right| = \inf_{y \in \R} \sqrt{\left(\frac{1- A\ov{B}}{2}\right)^2 + y^2} = \frac{1- A\ov{B}}{2}. $$ Note that, for $|B| = 1$ and since $-1 \leq A\ov{B} < 1$, we have the following equivalence: $$ \frac{1- A\ov{B}}{2} = \frac{|1- A\ov{B}|}{2} = \frac{\Big||B|^2- A\ov{B}\Big|}{ 1 + |B|} = \frac{|\ov{B}|\cdot |A - B|}{1 + |B|} = \frac{|A-B|}{1 + |B|}.$$ $\bullet$ Assume that $|B| = 1, \ |A| \leq 1 \text{ and } -1 \leq A\ov{B} < 1$. Then $g(U) = U\left(\frac{1- A\ov{B}}{1-|B|^2}, \frac{|A-B|}{1-|B|^2}\right)$. Thus, $$\partial g(U) = \left \{ z \in C: z = \frac{1- A\ov{B}}{1-|B|^2} + \lambda \frac{|A-B|}{1-|B|^2}, \ |\lambda| =1 \right \}. $$ Let $\zeta \in \partial g(U)$. Then there exists $\lambda \in C$ with $|\lambda| = 1$ such that $\zeta = \frac{1- A\ov{B}}{1-|B|^2} + \lambda \frac{|A-B|}{1-|B|^2}$. Further, an elementary computation implies that: \begin{align*} |\zeta - 1| &= \Big|\frac{1- A\ov{B}}{1-|B|^2} + \lambda \frac{|A-B|}{1-|B|^2} -1 \Big| = \frac{\Big| |B|^2 - A\ov{B} + \lambda |A-B| \Big| }{1-|B|^2} \\ &= \frac{\Big| \lambda |A-B| - \ov{B} (A-B) \Big| }{1-|B|^2} \\ &\geq \frac{\Big| |A-B| - |\ov{B}| \cdot |A-B| \Big|}{1-|B|^2} = \frac{|A-B| \cdot |1 - |\ov{B}||}{1-|B|^2} \\ &= \frac{|A-B| \cdot |1 - |B||}{1-|B|^2} = \frac{|A-B|}{1 + |B|}. \end{align*} Note that the equality is attained in the above inequality when $\lambda_0 = \frac{\ov{B}(A-B)}{|\ov{B}(A-B)|}$ ( $|\lambda_0| = 1$). In this case, we get $$ \inf_{\zeta \in \partial g(U)} |\zeta -1| = \inf_{|\lambda|=1}\Big|\frac{1- A\ov{B}}{1-|B|^2} + \lambda \frac{|A-B|}{1-|B|^2} -1 \Big| = \frac{|A-B|}{1 + |B|}. $$ Taking into account the both cases analyzed above, we conclude that $$d(1, \partial g(U)) = \inf_{\zeta \in \partial g(U)} |\zeta -1| = \frac{|A-B|}{1+|B|}.$$ \end{proof} In view of Theorem \ref{op-1-g-star} and Remark \ref{charact.janowski.classes.with.g.starl}, we deduce the following consequence. \begin{theorem}\label{main-result-1} Let $a \in \C$, $b \in \R$ be such that $|1-a| < b \leq {\rm Re}\ a$. Also, let $\alpha \in [0,1]$, $\beta \in [0, 1/2]$, $\alpha + \beta \leq 1$. Then the following properties hold: \begin{itemize} \item[(i)] if $f \in S^*(a,b)$ then $\Phi_{n, \alpha, \beta}(f) \in S^*(a,b, \B)$, \item[(ii)] if $f \in \A S^*(a,b)$ then $\Phi_{n, \alpha, \beta}(f) \in \A S^*(a,b, \B)$. \end{itemize} \end{theorem} \begin{proof} \begin{itemize} \item[(i)] If we take the function $g$ as in Remark \ref{charact.janowski.classes.with.g.starl} (i), then $S^*_g = S^*(a,b)$ and $S^*_g(\B) = S^*(a,b,\B)$. Therefore, in view of Theorem \ref{op-1-g-star}, we deduce that $$\Phi_{n, \alpha, \beta}(S^*(a,b)) \subseteq S^*(a,b, \B).$$ \item[(ii)] Let the function $g$ be given as in Remark \ref{charact.janowski.classes.with.g.starl} (ii). In this case, we have that $S^*_g = \A S^*(a,b)$ and $S^*_g(\B) = \A S^*(a,b,\B)$. From Theorem \ref{op-1-g-star}, we obtain that $$\Phi_{n, \alpha, \beta}(\A S^*(a,b)) \subseteq \A S^*(a,b, \B).$$ \end{itemize} This completes the proof. \end{proof} In the case $a,b \in \R$ with $|1 - a| < b \leq a = {\rm Re} \ a$, the above result was obtained in \cite{Manu1}. The next two results are consequences of Theorem \ref{op-2-g-star} and Remark \ref{charact.janowski.classes.with.g.starl}. \begin{theorem} Let $a \in \C$, $b \in \R$ be such that $|1-a| < b \leq {\rm Re} \ a$. Let $Q: \C^{n-1} \to \C$ be a homogeneous polynomial of degree $2$, such that $$\|Q\| \leq \frac{b^2 - (1-a)(1-\ov{a})}{4(b + ||a|^2 - b^2 - a|)}.$$ If $f \in S^*(a,b)$, then $\Phi_{n, Q}(f) \in S^*(a,b, \B)$. \end{theorem} \begin{proof} Let $g$ be the function from Remark \ref{charact.janowski.classes.with.g.starl} (i). Thus, we get that $S^*_g$ becomes $S^*(a,b)$ and $S^*_g(\B)$ becomes $S^*(a,b,\B)$. Then the asserted property of the Muir extension operator $\Phi_{n, Q}$ follows from Theorem \ref{op-1-g-star}, i.e. \begin{equation}\label{proof-muir-1} \Phi_{n, Q}(S^*(a,b)) \subseteq S^*(a,b, \B). \end{equation} The function $g$ has the form from Assumption \ref{function-g-complex}, where $A = \frac{\ov{a}-1}{b}$ and $B = \frac{|a|^2 - b^2 - a}{b}$. Moreover, we have that: \begin{align*} \frac{|A-B|}{4(1 + |B|)} & = \frac{\Big| \ov{a} - 1 - |a|^2 + b^{2} + a \Big | }{ 4 \Big| b + | |a|^2 - b^2 - a| \Big| } = \frac{ | b^2 - ( |a|^2 - 2 {\rm Re}\ a + 1) | }{ 4 (b + ||a|^2 - b^2 - a|) } \\ & = \frac{ | b^2 - (1-a)(1-\ov{a}) | }{ 4 (b + ||a|^2 - b^2 - a|) } = \frac{ b^2 - (1-a)(1-\ov{a}) }{ 4(b + ||a|^2 - b^2 - a|) }, \end{align*} since $|a|^2 - 2 {\rm Re}\ a + 1 = (1-a)(1-\ov{a}) \in \R$ and $b > | 1 - a | = | 1 - \ov{a} |$. Therefore, the assumption $\|Q\| \leq \frac{b^2 - (1-a)(1-\ov{a})}{4(b + ||a|^2 - b^2 - a|)}$ shows that the relation \eqref{proof-muir-1} holds, as asserted. \end{proof} If we assume that $a \in \R$ in the hypothesis of the above result, then we deduce the preservation property concerning the extension operator $\Phi_{n, Q}$ and the class $S^*(a,b)$ with real coefficients obtained in \cite{Manu2}. Let us now refer to the Muir extension operator $\Phi_{n, Q}$ and state the following property. \begin{theorem} Let $a \in \C$, $b \in \R$ be such that $|1-a| < b \leq {\rm Re}\ a$. Let $Q: \C^{n-1} \to \C$ be a homogeneous polynomial of degree $2$, such that $$\|Q\| \leq \frac{b^2 - (1-a)(1-\ov{a})}{4(b + |1-\ov{a}|)}.$$ If $f \in \A S^*(a,b)$ then $\Phi_{n, Q}(f) \in \A S^*(a,b, \B)$. \end{theorem} \begin{proof} We consider the function $g$ as in Remark \ref{charact.janowski.classes.with.g.starl} (ii). It is clear that $S^*_g = \A S^*(a,b)$ and $S^*_g(\B) = \A S^*(a,b,\B)$. Taking into account Theorem \ref{op-1-g-star}, we deduce that the following relation is true: \begin{equation}\label{proof-muir-2} \Phi_{n, Q}(\A S^*(a,b)) \subseteq \A S^*(a,b, \B). \end{equation} The function $g$ can be also written in the form given in Assumption \eqref{function-g-complex}, where $A = \frac{a - |a|^2 + b^2}{b}$ and $B = \frac{1 - \ov{a}}{b}$. Next, we evaluate the following quantity: \begin{align*} \frac{|A-B|}{4(1 + |B|)} &= \frac{\Big|a - |a|^2 + b^2 - 1 + \ov{a}\Big|}{4 |b + |1 - \ov{a}| |} = \frac{|b^2 - ( |a|^2 - 2 {\rm Re}\ a + 1)|}{4 (b + |1 - \ov{a}|)} \\ &= \frac{|b^2 - (1-a)(1-\ov{a})|}{4 (b + |1 - \ov{a}|)} = \frac{b^2 - (1-a)(1-\ov{a})}{4 (b + |1 - \ov{a}|)}, \end{align*} using the fact that $|a|^2 - 2 {\rm Re}\ a + 1 = (1-a)(1-\ov{a}) \in \R$ and $b > | 1 - a | = | 1 - \ov{a} |$. Consequently, the condition $\|Q\| \leq \frac{b^2 - (1-a)(1-\ov{a})}{4(b + |1-\ov{a}|)}$ implies that the relation \eqref{proof-muir-2} holds, as asserted. \end{proof} For $a,b \in \R$ where $|1 - a| < b \leq a = {\rm Re} \ a$, the above property was obtained in \cite{Manu2}. \begin{question} Assume that $n \geq 2$. Let $\Psi_{n} : \LS_n \to \LS_{n+1}$ be the Pfaltzgraff-Suffridge extension operator given by (see \cite{PfaltzgraffSuffridge1}): \begin{equation*} \Psi_{n}(f)(z) = \left(f(\tilde{z}), z_{n+1} [J_f(\tilde{z})]^{\frac{1}{n+1}}\right), \ z=(\tilde{z}, z_{n+1}) \in \BB, \end{equation*} were $ [J_f(\tilde{z})]^{\frac{1}{n+1}}\Big\rvert_{\tilde{z}=0} = 1$. We wonder if it is possible that Janowski (almost) starlikeness with complex coefficients to be preserved under the extension operator $\Psi_{n}$ from the unit ball $\B$ to the unit ball $\BB$. If it is true, under which conditions does this property hold? \end{question} \bigskip {\bf Conclusions:} In this paper, we have considered $g$-parametric representation and $g$-starlikeness on the Euclidean unit ball $\B$, when the function $g : U \to \C$ is univalent on $U$, $g(0)=1$ and has positive real part on $U$ (see \cite{GrahamHamadaKohrKohr}). Then we have referred to the property of preservation of $g$-starlikeness under the extension operator $\Phi_{n, \alpha, \beta}$, when $g$ is convex on $U$ and $\alpha \in [0,1]$, $\beta \in [0, 1/2]$, $\alpha + \beta \leq 1$ (see \cite{GrahamHamadaKohrKohr}). For the same conditions imposed on $g$, we have stated that the Muir extension operator $\Phi_{n, Q}$ preserves $g$-starlikeness when $\|Q\| \leq d(1, \partial g(U))/4$ (see \cite{GrahamHamadaKohrKohr}). Assume $a \in \C$, $b \in \R$ such that $|1-a| < b \leq {\rm Re}\ a$. Using the connection between the Janowski classes $S^*(a,b)$, $\A S^*(,b)$ and $g$-starlikeness, for a particular choice of $g$ depending on the parameters $a, b$, we have proved that $\Phi_{n, \alpha, \beta}$ preserves these classes for $\alpha \in [0,1]$, $\beta \in [0, 1/2]$, $\alpha + \beta \leq 1$. By making use of the same idea, we also prove that $\Phi_{n, Q}$ conserves these classes when $\|Q\| \leq M(a,b)$, where $M(a,b)$ is a constant depending on the parameters $a$ and $b$. These results generalize the properties obtained in \cite{Manu1, Manu2}, for the Janowski classes with real parameters. \begin{thebibliography}{99} \bibitem{TChirila} Chiril\u a, T., \emph{An extension operator associated with certain $g$-Loewner chains}, Taiwanese J. Math., \textbf{17}(2013), no. 5, 1819-1837. \bibitem{TChirila3} Chiril\u a, T., \emph{Analytic and geometric properties associated with some extension operators}, Complex Var. 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