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Abstract
Counterexamples show that many results in the geometric function theory of one complex variable are not applicable for several complex variables. In this paper, we obtain sharp bounds for the Zalcman functional for $n=3$ associated with the starlike mappings defined on the unit ball in a complex Banach space and on the unit polydisk in $\mathbb{C}^n$. These results confirm the validity of the Zalcman conjecture in higher dimensions for $n=3$.
Results & Lemmas (3)
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 · coeff
Lemma 1. [9] Let. Then and,,. This following results provide sharp bound of the Zalcman functional for the class of starlike mappings in…
Lemma 1. [9] Let $p(z) = 1 + \sum_{n=1}^{\infty} p_n z^n \in \mathcal{P}$ . Then
$$|p_n| \le 2$$
and $|p_n - p_m p_{n-m}| \le 2$ , $n \ge 2$ , $1 \le m \le n - 1$ .
This following results provide sharp bound of the Zalcman functional for the class of starlike mappings in higher dimensions.
Theorem 2
Theorem 2. Let with f(0) = 1 and. Then The bound is sharp.
Theorem 2. Let $f \in \mathcal{H}(\mathbb{B},\mathbb{C})$ with f(0) = 1 and $F(z) = zf(z) \in \mathcal{S}^*(\mathbb{B})$ . Then
$$\left| \left( \frac{l_z(D^3 F(0)(z^3))}{3!||z||^3} \right)^2 - \left( \frac{l_z(D^5 F(0)(z^5))}{5!||z||^5} \right) \right| \le 4.$$
The bound is sharp.
Theorem 3
Theorem 3. Let with f(0) = 1 and. Then (7) The inequality is sharp.
Theorem 3. Let $f \in \mathcal{H}(\mathbb{U}^n, \mathbb{C})$ with f(0) = 1 and $F(z) = zf(z) \in \mathcal{S}^*(\mathbb{U}^n)$ . Then
$$\left\| \frac{1}{3!} D^3 F(0) \left( z^2, \frac{D^3 F(0)(z^3)}{3!} \right) - \frac{D^5 F(0)(z^5)}{5!} \right\| \le 4 \|z\|^5, \quad z \in \mathbb{U}^n.$$
(7)
The inequality is sharp.
Function classes studied:
Coefficient bounds & claims (5)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
|a3^2 - a5| ≤ 4 for class S* (sharp) [Theorem A (cited from Brown-Tsao 1986)]
coefficient_bound
|lz(D3F(0)(z3))/(3!||z||^3)|^2 - |lz(D5F(0)(z5))/(5!||z||^5)| ≤ 4 for class S*(B) (sharp) [Theorem 2]
coefficient_bound
|D3F(0)(z2, D3F(0)(z3)/3!) / 3! - D5F(0)(z5)/5!| ≤ 4*||z||**5 for class S*(Un) (sharp) [Theorem 3]
function_family
Class S*(B): Normalized biholomorphic starlike mappings on the unit ball B in a complex Banach space X
function_family
Class S*(Un): Starlike mappings on the unit polydisk Un in Cn
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