Abstract
Generalizing the Zalcman conjecture given by $\vert a_n^2 - a_{2n-1}\vert \leq (n-1)^2$, Ma proposed and proved that the inequality
$$\vert a_n a_m-a_{n+m-1}\vert \leq (n-1)(m-1), \quad m,n \in \mathbb{N},$$
holds for functions $f(z)=z+a_2z^2 +a_3 z^3 +\cdots\in \mathcal{S}^*$, the class of starlike functions in the open unit disk. In this work, we extend this problem to several complex variables for $m=2$ and $n=3$, considering the class of starlike mappings defined on the unit ball in a co
Results & Lemmas (6)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 1
Lemma 1. [17] Let be a normalized locally biholomorphic mapping. The mapping f is said to be starlike on if and only if The class of all…
Lemma 1. [17] Let $f : \mathbb{B} \to X$ be a normalized locally biholomorphic mapping. The mapping f is said to be starlike on $\mathbb{B}$ if and only if
$$\operatorname{Re}(l_z[Df(z)]^{-1}f(z)) > 0, \quad x \in \mathbb{B} \setminus \{0\}, \quad l_z \in T_z.$$
The class of all such mappings on $\mathbb{B}$ is denoted by $\mathcal{S}^*(\mathbb{B})$ .
Lemma 2
Lemma 2. [13] is said to be a bounded starlike circular domain if and only if there exists a unique continuous function, called the…
Lemma 2. [13] $\Omega \subset \mathbb{C}^n$ is said to be a bounded starlike circular domain if and only if there exists a unique continuous function $\rho : \mathbb{C}^n \to \mathbb{R}$ , called the Minkowski functional of $\Omega$ , such that
(i)
$$\rho(z) \ge 0$$
, $z \in \mathbb{C}^n$ ; $\rho(z) = 0 \Leftrightarrow z = 0$ ,
(ii)
$$\rho(tz) = |t|, \rho(z), t \in \mathbb{C}, z \in \mathbb{C}^n,$$
(iii)
$$\Omega = \{z \in \mathbb{C}^n : \rho(z) < 1\}.$$
Furthermore, if $\rho(z)$ , $z \in \Omega$ , belongs to $\mathcal{C}^1$ except on some lower-dimensional manifold $E \subset \mathbb{C}^n$ , then $\rho(z)$ satisfies the following properties:
<span id="page-2-2"></span>
$$2\frac{\partial \rho(z)}{\partial z}z = \rho(z), \quad z \in \mathbb{C}^n \setminus E,$$
$$2\frac{\partial \rho(z)}{\partial z}\Big|_{z=z_0} = 1, \quad z_0 \in \partial\Omega \setminus E,$$
$$\frac{\partial \rho(\lambda z)}{\partial z} = \frac{\partial \rho(z)}{\partial z}, \quad \lambda \in (0, \infty), \ z \in \mathbb{C}^n \setminus E,$$
$$\frac{\partial \rho(e^{i\theta}z)}{\partial z} = e^{-i\theta}\frac{\partial \rho(z)}{\partial z}, \quad \theta \in \mathbb{R}, \ z \in \mathbb{C}^n \setminus E,$$
$$(1)$$
where $\frac{\partial \rho(z)}{\partial z} = \left(\frac{\partial \rho(z)}{\partial z_1}, \dots, \frac{\partial \rho(z)}{\partial z_n}\right)$ .
Lemma 3
Lemma 3. [13] Let be a bounded starlike circular domain with, whose Minkowski functional belongs to except on some lower-dimensional…
Lemma 3. [13] Let $\Omega \subset \mathbb{C}^n$ be a bounded starlike circular domain with $0 \in \Omega$ , whose Minkowski functional $\rho(z)$ belongs to $\mathcal{C}^1$ except on some lower-dimensional manifolds $E \subset \mathbb{C}^n$ . Let $f: \Omega \to \mathbb{C}^n$ be a normalized locally biholomorphic mapping. Then f is starlike on $\Omega$ if and only if
$$\operatorname{Re}\left(\frac{\partial \rho(z)}{\partial z} (Df(z))^{-1} f(z)\right) > 0, \quad z \in \Omega \setminus E.$$
The class of all starlike mappings on $\Omega$ is denoted by $\mathcal{S}^*(\Omega)$ .
Lemma 4
Lemma 4. [2] Let. Then the following holds:, and.
Lemma 4. [2] Let $p(z) = 1 + \sum_{n=1}^{\infty} p_n z^n \in \mathcal{P}$ . Then the following holds:
$$|p_n| \le 2$$
, $|p_2 - p_1^2| \le 2$ and $|p_3 - p_1p_2| \le 2$ .
Theorem 5
Theorem 5. Let with f(0) = 1 and suppose that F(z) = zf(z). If, then The bound is sharp.
Theorem 5. Let $f \in \mathcal{H}(\mathbb{B}, \mathbb{C})$ with f(0) = 1 and suppose that F(z) = zf(z). If $F(z) \in \mathcal{S}^*(\mathbb{B})$ , then
$$\left| \left( \frac{l_z(D^2F(0)(z^2))}{2!||z||^2} \right) \left( \frac{l_z(D^3F(0)(z^3))}{3!||z||^3} \right) - \left( \frac{l_z(D^4F(0)(z^4))}{4!||z||^4} \right) \right| \leq 2.$$
The bound is sharp.
Theorem 6
Theorem 6. Let with f(0) = 1 and suppose that F(z) = zf(z). If, then The bound is sharp.
Theorem 6. Let $f \in \mathcal{H}(\Omega, \mathbb{C})$ with f(0) = 1 and suppose that F(z) = zf(z). If $F(z) \in \mathcal{S}^*(\Omega)$ , then
$$\left| \left( 2 \frac{\partial \rho(z)}{\partial z} \frac{D^2 F(0)(z^2)}{2! \rho^2(z)} \right) \left( 2 \frac{\partial \rho(z)}{\partial z} \frac{D^3 F(0)(z^3)}{3! \rho^3(z)} \right) - 2 \frac{\partial \rho(z)}{\partial z} \frac{D^4 F(0)(z^4)}{4! \rho^4(z)} \right| \leq 2, \quad z \in \Omega \setminus E.$$
The bound is sharp.
Function classes studied:
Related Papers