Spirallike mappings and univalent subordination chains in Cn

Lemma 1.2. Let $h: B^n \to \mathbb{C}^n$ be a mapping such that $h \in \mathcal{N}$ , Dh(0) = A and m(A) > 0. Then $$||h(z)|| \le \frac{4r}{(1-r)^2} |V(A)|, ||z|| \le r < 1.$$

Theorem 2.1. Let $A:[0,\infty)\to L(\mathbb{C}^n,\mathbb{C}^n)$ be a measurable mapping which satisfies the assumptions of Definition 1.5. Also let $h=h(z,t):B^n\times[0,\infty)\to\mathbb{C}^n$ be a mapping which satisfies the following conditions: - (i) $h(\cdot, t) \in \mathcal{N}$ , Dh(0, t) = A(t) for t > 0; - (ii) $h(z, \cdot)$ is measurable on $[0, \infty)$ for $z \in B^n$ . Then for each $s \geq 0$ and $z \in B^n$ , the initial value problem $$\frac{\partial v}{\partial t} = -h(v, t) \ a.e. \ t \ge s, \ v(z, s, s) = z, \tag{2.1}$$ has a unique solution v = v(z, s, t) such that $v(\cdot, s, t)$ is a univalent Schwarz mapping, $v(z, s, \cdot)$ is Lipschitz continuous on $[s, \infty)$ locally uniformly with respect to $z \in B^n$ , $Dv(0, s, t) = \exp(-\int_s^t A(\tau)d\tau)$ for $t \ge s \ge 0$ , and the following relations hold: $$\frac{\|v(z,s,t)\|}{(1-\|v(z,s,t)\|)^2} \le e^{-\int_s^t m(A(\tau))d\tau} \frac{\|z\|}{(1-\|z\|)^2}, \ z \in B^n, \ t \ge s \ge 0.$$ (2.2) $$\frac{\|v(z,s,t)\|}{(1+\|v(z,s,t)\|)^2} \ge e^{-\int_s^t k(A(\tau))d\tau} \frac{\|z\|}{(1+\|z\|)^2}, \ z \in B^n, \ t \ge s \ge 0.$$ (2.3)

Theorem 2.3. Let $h = h(z, t) : B^n \times [0, \infty) \to \mathbb{C}^n$ and $A : [0, \infty) \to L(\mathbb{C}^n, \mathbb{C}^n)$ satisfy the assumptions of Theorem 2.1 and let v = v(z, s, t) be the Lipschitz continuous solution on $[s, \infty)$ of the initial value problem (2.1). Also, assume that the following condition holds: $$\sup_{s>0} \int_{s}^{\infty} \|e^{\int_{s}^{t} [A(\tau)-2m(A(\tau))I_{n}]d\tau}\|dt < \infty.$$ (2.6) Then the limit $$\lim_{t \to \infty} e^{\int_0^t A(\tau)d\tau} v(z, s, t) = f(z, s) \tag{2.7}$$ exists locally uniformly on $B^n$ for each $s \ge 0$ . Moreover, f(z,t) is a univalent subordination chain such that $Df(0,t) = e^{\int_0^t A(\tau)d\tau}, \{e^{-\int_0^t A(\tau)d\tau}f(\cdot,t)\}_{t\ge 0}$ is a normal family on $B^n$ and $f = f(\cdot,0)$ has generalized parametric representation with respect to A.

Corollary 2.4. Let $A: [0, \infty) \to L(\mathbb{C}^n, \mathbb{C}^n)$ be a measurable mapping which satisfies the assumptions of Definition 1.5 and the condition (2.6). Assume $f: B^n \to \mathbb{C}^n$ has generalized parametric representation with respect to A. Then there exists a univalent subordination chain f(z,t) such that $Df(0,t) = e^{\int_0^t A(\tau)d\tau}$ , $t \geq 0$ , $\{e^{-\int_0^t A(\tau)d\tau} f(\cdot,t)\}_{t\geq 0}$ is a normal family on $B^n$ and $f = f(\cdot,0)$ .

Theorem 2.5. Let $A:[0,\infty)\to L(\mathbb{C}^n,\mathbb{C}^n)$ be a measurable mapping which satisfies the assumptions of Definition 1.5 and the condition (2.6). Also let $f=f(z,t):B^n\times[0,\infty)\to\mathbb{C}^n$ be a mapping such that $f(\cdot,t)\in H(B^n)$ , f(0,t)=0, $Df(0,t)=e^{\int_0^t A(\tau)d\tau}$ for $t\geq 0$ , and $f(z,\cdot)$ is locally absolutely continuous on $[0,\infty)$ locally uniformly with respect to $z\in B^n$ . Let $h=h(z,t):B^n\times[0,\infty)\to\mathbb{C}^n$ satisfy the assumptions of Theorem 2.1. Assume f(z,t) satisfies the Loewner differential equation $$\frac{\partial f}{\partial t}(z,t) = Df(z,t)h(z,t) \text{ a.e. } t \ge 0, \ \forall z \in B^n.$$ (2.12) Also assume that $\{e^{-\int_0^t A(\tau)d\tau} f(\cdot,t)\}_{t\geq 0}$ is a normal family on $B^n$ . Then f(z,t) is a univalent subordination chain such that f(z,s)=f(v(z,s,t),t) and $$f(z,s) = \lim_{t \to \infty} e^{\int_0^t A(\tau)d\tau} v(z,s,t)$$ locally uniformly on $B^n$ for $s \ge 0$ , where v = v(z, s, t) is the unique locally Lipschitz continuous solution on $[s, \infty)$ of the initial value problem $$\frac{\partial v}{\partial t} = -h(v, t) \text{ a.e. } t \ge s, \ v(z, s, s) = z,$$ for all $s \ge 0$ and $z \in B^n$ . Moreover, $f(\cdot, 0)$ has generalized parametric representation with respect to A.

Theorem 2.6. Let $A:[0,\infty)\to L(\mathbb{C}^n,\mathbb{C}^n)$ be a measurable mapping which satisfies the assumptions of Definition 1.5 and the relation (2.6). Also let f(z,t) be a univalent subordination chain such that $Df(0,t)=e^{\int_0^t A(\tau)d\tau}$ for $t\geq 0$ , and let v = v(z, s, t) be the transition mapping associated with f(z, t). Assume $\{e^{-\int_0^t A(\tau)d\tau} f(\cdot, t)\}_{t\geq 0}$ is a normal family on $B^n$ . Then the limit $$\lim_{t \to \infty} e^{\int_0^t A(\tau)d\tau} v(z, s, t) = f(z, s)$$ exists locally uniformly on $B^n$ for $s \ge 0$ . Moreover, $f(\cdot, 0)$ has generalized parametric representation with respect to A.

Corollary 2.7. Let $A: [0, \infty) \to L(\mathbb{C}^n, \mathbb{C}^n)$ be a measurable mapping which satisfies the assumptions of Definition 1.5 and the condition (2.6). Also let $f \in S(B^n)$ . Then f has generalized parametric representation with respect to A if and only if there exists a univalent subordination chain f(z,t) such that $Df(0,t) = e^{\int_0^t A(\tau)d\tau}, \{e^{-\int_0^t A(\tau)d\tau}f(\cdot,t)\}_{t\geq 0}$ is a normal family on $B^n$ and $f = f(\cdot,0)$ . In view of (2.2) and (2.3), we deduce the following growth result for mappings in $\tilde{S}_A^0(B^n)$ , where A is a measurable diagonal matrix with its eigenvalues $a_1, \ldots, a_n$ , such that $\Re a_1(t) = \cdots = \Re a_n(t)$ for $t \geq 0$ (if $A = I_n$ , see [23] and [6]). Remark 2.8. Let $a_j: [0, \infty) \to \mathbb{C}$ be uniformly bounded measurable functions for $j=1,\ldots,n$ , such that $\Re a_1(t)=\cdots=\Re a_n(t)\geq \delta$ for some $\delta>0$ . If $A: [0,\infty)\to L(\mathbb{C}^n,\mathbb{C}^n)$ is a diagonal matrix with eigenvalues $a_1,\ldots,a_n$ and if $f\in \tilde{S}^0_A(B^n)$ , then $$\frac{\|z\|}{(1+\|z\|)^2} \le \|f(z)\| \le \frac{\|z\|}{(1-\|z\|)^2}, \ z \in B^n.$$

Theorem 3.3. Let $A: [0, \infty) \to L(\mathbb{C}^n, \mathbb{C}^n)$ be a locally Lebesgue integrable mapping such that m(A(t)) > 0 for $t \ge 0$ and the condition (1.1) holds. Also let $f \in H(B^n)$ be a normalized mapping. Then f is generalized spirallike with respect to A if and only if $f(z,t) = e^{\int_0^t A(\tau)d\tau} f(z)$ is a univalent subordination chain.

Corollary 3.4. Let $A:[0,\infty) \to L(\mathbb{C}^n,\mathbb{C}^n)$ be a measurable mapping which satisfies the assumptions of Definition 1.5 and the relation (2.6). Also let $f:B^n \to \mathbb{C}^n$ be a generalized spirallike mapping with respect to A. Then f has generalized parametric representation with respect to A.

Theorem 3.5. Let $A: [0, \infty) \to L(\mathbb{C}^n, \mathbb{C}^n)$ be a measurable mapping which satisfies the assumptions of Definition 1.5. Also let $f \in \mathcal{L}S_n$ . Then f is generalized spirallike with respect to A if and only if $$\Re\langle [Df(z)]^{-1}A(t)f(z),z\rangle>0,\ a.e.\ t\geq 0,\ \forall z\in B^n\setminus\{0\}. \eqno(3.1)$$

Corollary 3.7. Let $\delta \in (0, \pi/2)$ and $\alpha : [0, \infty) \to [-\delta, \delta]$ be a measurable function. Also let $f \in \mathcal{L}S_n$ and $A(t) = e^{-i\alpha(t)}I_n$ for $t \geq 0$ . Then f is generalized spirallike with respect to A if and only if $$\Re[e^{-i\alpha(t)}\langle [Df(z)]^{-1}f(z), z\rangle] > 0, \quad a.e. \quad t \ge 0, \ \forall z \in B^n \setminus \{0\}.$$ (3.3)

Theorem 4.7. Let $\alpha \in [0, 1]$ , $b, \eta > 0$ and $\lambda : [0, \infty) \to \mathbb{C}$ be a measurable function such that $|\lambda(t)| \leq b$ and $\Re \lambda(t) \geq \eta$ for $t \geq 0$ . Also let $f : U \to \mathbb{C}$ be a generalized spirallike function with respect to $\lambda$ and let $F_{\alpha} : B^{n} \to \mathbb{C}^{n}$ be given by $$F_{\alpha}(z) = \left(f(z_1), \tilde{z}\left(\frac{f(z_1)}{z_1}\right)^{\alpha}\right), \ z = (z_1, \tilde{z}) \in B^n.$$ Then $F_{\alpha}$ is generalized spirallike with respect to A, where $A(t) = \lambda(t)I_n$ .

Definition 1.1. A mapping $f: B^n \times [0, \infty) \to \mathbb{C}^n$ is called a univalent subordination chain if $f(\cdot, t)$ is biholomorphic on $B^n$ , f(0, t) = 0 for $t \ge 0$ , and $f(\cdot, s) \prec f(\cdot, t), 0 \le s \le t < \infty$ . The above subordination condition is equivalent to the existence of a unique Schwarz mapping v = v(z, s, t), called the transition mapping associated with f(z, t), such that $f(z, s) = f(v(z, s, t), t), z \in B^n, t \ge s \ge 0$ . If f(z, t) is a univalent subordination chain such that $Df(0, t) = e^t I_n$ for $t \ge 0$ , we say that f(z, t) is a normalized univalent subordination chain or a Loewner chain. The following class of mappings in $H(B^n)$ is important in our treatment: $$\mathcal{N} = \{ h \in H(B^n) : h(0) = 0, \Re \langle h(z), z \rangle > 0, z \in B^n \setminus \{0\} \}.$$ Several applications of this class in the study of biholomorphic mappings and univalent subordination chains on the unit ball in $\mathbb{C}^n$ may be found in [6, 9, 11, 19, 26, 29, 30]. For a given operator $A \in L(\mathbb{C}^n, \mathbb{C}^n)$ , let us denote by $$m(A) = \min\{\Re\langle A(z), z \rangle : ||z|| = 1\} \text{ and } k(A) = \max\{\Re\langle A(z), z \rangle : ||z|| = 1\}.$$ Also let $|V(A)| = \max_{\|z\|=1} |\langle A(z), z \rangle|$ be the numerical radius of the operator A. Then $|V(A)| \le \|A\| \le 2|V(A)|$ by [13, Theorem 1.3.1] (see also [1, Theorem 4.1]; cf. [16, 17]). Further, let $k_+(A) = \lim_{t \to \infty} \frac{\ln \|e^{tA}\|}{t}$ . It is well known that this limit exists, and for each $\omega > k_+(A)$ , there exists $\delta = \delta(\omega) > 0$ such that $\|e^{tA}\| \le \delta e^{\omega t}$ , $t \ge 0$ . In addition, $k_+(A) = \max\{\Re\lambda : \lambda \in \sigma(A)\}$ where $\sigma(A)$ is the spectrum of A (see [3, Theorem 4.1], [4]). To prove our results, we need the following lemma (see [8, Lemma 1.2]; cf. [6, Theorem 1.2]).

Definition 1.3. (cf. [12, 30]) Let $A \in L(\mathbb{C}^n, \mathbb{C}^n)$ be such that m(A) > 0. Also let $\Omega$ be a domain in $\mathbb{C}^n$ which contains the origin. We say that $\Omega$ is spirallike with respect to A if $e^{-tA}(w) \in \Omega$ , for all $w \in \Omega$ and t > 0. A mapping $f \in S(B^n)$ is called spirallike with respect to A if $f(B^n)$ is a spirallike domain with respect to A.

Definition 1.5. Let $A: [0, \infty) \to L(\mathbb{C}^n, \mathbb{C}^n)$ be a measurable mapping such that m(A(t)) > 0 for $t \ge 0$ and $\int_0^\infty m(A(t))dt = \infty$ . Moreover, assume that $||A(\cdot)||$ is uniformly bounded on $[0, \infty)$ and $$\int_{s}^{t} A(\tau)d\tau \circ \int_{r}^{s} A(\tau)d\tau = \int_{r}^{s} A(\tau)d\tau \circ \int_{s}^{t} A(\tau)d\tau, \ t \ge s \ge r \ge 0.$$ (1.1) Also, let $f \in H(B^n)$ be a normalized mapping. We say that f has generalized parametric representation with respect to A (and we denote the class of such f by $\tilde{S}_A^0(B^n)$ ) if there exists a mapping $h: B^n \times [0, \infty) \to \mathbb{C}^n$ such that $h(\cdot, t) \in \mathcal{N}$ , Dh(0, t) = A(t) for $t \in [0, \infty)$ , $h(z, \cdot)$ is measurable on $[0, \infty)$ for $z \in B^n$ , and $$f(z) = \lim_{t \to \infty} e^{\int_0^t A(\tau)d\tau} v(z, t)$$ (1.2) locally uniformly on $B^n$ , where v = v(z, t) is the unique locally absolutely continuous solution on $[0, \infty)$ of the initial value problem $$\frac{\partial v}{\partial t} = -h(v, t) \quad \text{a.e. } t \ge 0, \ v(z, 0) = z, \tag{1.3}$$ for all $z \in B^n$ .

Definition 3.1. Let $A: [0, \infty) \to L(\mathbb{C}^n, \mathbb{C}^n)$ be a locally Lebesgue integrable mapping such that m(A(t)) > 0 for $t \ge 0$ . Also let $\Omega$ be a domain in $\mathbb{C}^n$ which contains the origin. We say that $\Omega$ is generalized spirallike with respect to A if $e^{-\int_s^t A(\tau)d\tau}(w) \in \Omega$ , for all $w \in \Omega$ and $t \ge s \ge 0$ . A mapping $f \in S(B^n)$ is called generalized spirallike with respect to A if $f(B^n)$ is a generalized spirallike domain with respect to A. Remark 3.2. Clearly if A(t) is constant, we obtain the usual notion of spirallikeness with respect to a given constant operator A. In particular, if $A(t) \equiv I_n$ in Definition 3.1, then $\Omega$ is a starlike domain. We next obtain the following characterization of generalized spirallikeness with respect to a given measurable operator in terms of univalent subordination chains.