Second-Order Toeplitz Determinant for Quasi-Convex Mappings

Lemma 1. [5] If $$\omega(z) = \sum_{n=1}^{\infty} c_n z^n \in \mathcal{B}_0$$ , then for any $\lambda \in \mathbb{C}$ , $$|c_2 + \lambda c_1^2| \le \max\{1, |\lambda|\}.$$

Lemma 2. [14] If $\omega(z) = \sum_{n=1}^{\infty} c_n z^n \in \mathcal{B}_0 \text{ and } (\nu_1, \nu_2) \in \bigcup_{i=1}^{7} \Theta_i, \text{ then}$ $$|c_3 + \nu_1 c_1 c_2 + \nu_2 c_1^3| \le |\nu_2|,$$ where $$\begin{split} \Theta_1 &= \left\{ (\nu_1, \nu_2) : |\nu_1| \leq \frac{1}{2}, |\nu_2| \leq 1 \right\}, \\ \Theta_2 &= \left\{ (\nu_1, \nu_2) : \frac{1}{2} \leq |\nu_1| \leq 2, \; \frac{4}{27} (|\nu_1| + 1)^3 - (|\nu_1| + 1) \leq \nu_2 \leq 1 \right\}, \\ \Theta_3 &= \left\{ (\nu_1, \nu_2) : |\nu_1| \leq \frac{1}{2}, \nu_2 \leq -1 \right\}, \; \; \Theta_4 = \left\{ (\nu_1, \nu_2) : |\nu_1| \geq \frac{1}{2}, \nu_2 \leq -\frac{2}{3} (|\nu_1| + 1) \right\}, \\ \Theta_5 &= \left\{ (\nu_1, \nu_2) : |\nu_1| \leq 2, \; \nu_2 \geq 1 \right\}, \; \; \Theta_6 = \left\{ (\nu_1, \nu_2) : 2 \leq |\nu_1| \leq 4, \; \nu_2 \geq \frac{1}{12} (\nu_1^2 + 8) \right\}, \end{split}$$ and $$\Theta_7 = \left\{ (\nu_1, \nu_2) : |\nu_1| \ge 4, \ \nu_2 \ge \frac{2}{3} (|\nu_1| - 1) \right\}.$$

Lemma 3. Let $g \in \mathcal{H}(\mathbb{U})$ be a biholomorphic function such that $g(0) = \Psi(0)$ and $g(\mathbb{U}) \subset \Psi(\mathbb{U})$ . If $(r_1, r_2) \in \bigcup_{i=1}^7 \Theta_i$ , then $$\left| 6(g'(0))^3 + 9g'(0)g''(0) + 2g'''(0) \right| \le \left| 6(\Psi'(0))^3 + 9\Psi'(0)\Psi''(0) + 2\Psi'''(0) \right|,$$ where $$r_1 = \frac{3(\Psi'(0))^2 + 2\Psi''(0)}{2\Psi'(0)}$$ and $r_2 = \frac{6(\Psi'(0))^3 + 9\Psi'(0)\Psi''(0) + 2\Psi'''(0)}{12\Psi'(0)}$

Theorem 4. If $f \in C(\Psi)$ such that $|\Psi''(0) + 2(\Psi'(0))^2| \ge 2\Psi'(0)$ and $(r_1, r_2) \in \bigcup_{i=1}^7 \Theta_i$ , then $$|T_{2,3}(f)| \le \frac{1}{144} \left( 2(\Psi'(0))^2 + \Psi''(0) \right)^2 + \frac{1}{576} \left( (\Psi'(0))^3 + \frac{3\Psi'(0)\Psi''(0)}{2} + \frac{\Psi'''(0)}{3} \right)^2,$$ where $$r_1 = \frac{1}{2\Psi'(0)} \left( 3(\Psi'(0))^2 + 2\Psi''(0) \right), \quad r_2 = \frac{1}{2\Psi'(0)} \left( (\Psi'(0))^3 + \frac{3\Psi'(0)\Psi''(0)}{2} + \frac{\Psi'''(0)}{3} \right).$$ The estimate is sharp. Proof. Let f ∈ C(Ψ) be of the form [\(1\)](#page-0-0), then there exists a Schwarz function ω(z) = P^∞ ⁿ=1 cnz ⁿ ∈ B⁰ such that $$1 + \frac{zf''(z)}{f'(z)} = \Psi(\omega(z)).$$ Comparison of same powers of z after Taylor series expansions of f, Ψ and ω yields $$a_3 = \frac{\Psi'(0)}{6} \left( c_2 + \left( \Psi'(0) + \frac{\Psi''(0)}{2\Psi'(0)} \right) c_1^2 \right) \tag{4}$$ and $$a_4 = \frac{\Psi'(0)}{12} \left( c_3 + r_1 c_1 c_2 + r_2 c_1^3 \right). \tag{5}$$ According to the premises, Ψ satisfies |Ψ′′(0) + 2(Ψ′ (0))² | ≥ 2Ψ′ (0) and (r1, r2) ∈ S7 ⁱ=1 Θⁱ . Consequently, by applying Lemma [1](#page-3-1) and Lemma [2](#page-3-0) to [\(4\)](#page-5-0) and [\(5\)](#page-5-1), respectively, we obtain $$|a_3| \le \frac{2(\Psi'(0))^2 + \Psi''(0)}{12} \text{ and } |a_4| \le \frac{1}{24} \left( (\Psi'(0))^3 + \frac{3\Psi'(0)\Psi''(0)}{2} + \frac{\Psi'''(0)}{3} \right).$$ (6) From [\(2\)](#page-1-0), it follows that $$|T_{2,3}(f)| \le |a_2|^2 + |a_3|^2. (7)$$ Using the estimates for |a2| and |a3| from [\(6\)](#page-5-2) in [\(7\)](#page-5-3), we get $$|T_{2,3}(f)| \le \frac{(2(\Psi'(0))^2 + \Psi''(0))^2}{144} + \frac{1}{576} \left( (\Psi'(0))^3 + \frac{3\Psi'(0)\Psi''(0)}{2} + \frac{\Psi'''(0)}{3} \right)^2.$$ (8) To show the sharpness, we consider the function f^Ψ : U → C, given by $$1+\frac{f_\Psi^{\prime\prime}(z)}{f_\Psi^\prime(z)}=\Psi(iz).$$ Clearly f^Ψ ∈ C(Ψ) and for this function, we have $$a_3 = -\frac{1}{6} \left( (\Psi'(0))^2 + \frac{\Psi''(0)}{2} \right) \text{ and } a_4 = -\frac{i}{24} \left( (\Psi'(0))^3 + \frac{3\Psi'(0)\Psi''(0)}{2} + \frac{\Psi'''(0)}{3} \right),$$ which together with [\(2\)](#page-1-0) shows that equality case holds in [\(8\)](#page-5-4) for the function fΨ. For Ψ(z) = (1 + z)/(1 − z), Ψ(z) = (1 + (1 − 2α)z)/(1 − z) and Ψ(z) = ((1 + z)/(1 − z))^β , the class C(Ψ) corresponds to the classes C, C(α) and CC(β), respectively. Consequenlty, the following results are obtained as immediate consequences of Theorem [3](#page-4-2) for these subclasses.

Corollary 4.1. If f ∈ C, then |T2,3(f)| ≤ 2.

Corollary 4.2. If f ∈ C(α), then $$|T_{2,3}(f)| \le \frac{(1-\alpha)^2(3-2\alpha)^2}{9} + \frac{(1-\alpha)^2(2-\alpha)^2(3-2\alpha)^2}{36}, \quad \alpha \in [0,1].$$

Corollary 4.3. If f ∈ CC(β), then $$|T_{2,3}(f)| \le \beta^4 + \frac{\beta^2 (1 + 17\beta^2)^2}{324}, \quad \beta \in [2/3, 1].$$

Theorem 5. Let $f \in \mathcal{H}(\mathbb{B},\mathbb{C})$ with f(0) = 1, $f(z) \neq 0$ , $z \in \mathbb{B}$ and suppose F(z) = zf(z). If $(DF(z))^{-1}(D^2F(z)(z^2) + DF(z)(z)) \in \mathcal{M}_{\Psi}$ such that $$|\Psi''(0) + 2(\Psi'(0))^2| \ge 2\Psi'(0)$$ and $(r_1, r_2) \in \bigcup_{i=1}^7 \Theta_i$ , then $$\left| \left( \frac{l_z(D^3 F(0)(z^3))}{3! \|z\|^3} \right)^2 - \left( \frac{l_z(D^4 F(0)(z^4))}{4! \|z\|^4} \right)^2 \right| \le \frac{1}{144} \left( 2(\Psi'(0))^2 + \Psi''(0) \right)^2 + \frac{1}{576} \left( (\Psi'(0))^3 + \frac{3\Psi'(0)\Psi''(0)}{2} + \frac{\Psi'''(0)}{3} \right)^2, \ z \in \mathbb{B} \setminus \{0\}.$$ The bound is sharp.

Theorem 6. Let $f \in \mathcal{H}(\mathbb{U}^n,\mathbb{C})$ with f(0) = 1, $f(z) \neq 0$ , $z \in \mathbb{U}^n$ and suppose F(z) = zf(z). If $(DF(z))^{-1}(D^2F(z)(z^2) + DF(z)(z)) \in \mathcal{M}_{\Psi}$ such that $$|\Psi''(0) + 2(\Psi'(0))^2| \ge 2\Psi'(0)$$ and $(r_1, r_2) \in \bigcup_{i=1}^7 \Theta_i$ , then $$\left\| \frac{1}{4!} D^{4} F(0) \left( z^{3}, \frac{D^{4} F(0)(z^{4})}{4!} \right) - \frac{1}{3!} D^{3} F(0) \left( z^{2}, \frac{D^{3} F(0)(z^{3})}{3!} \right) \right\| \\ \leq \frac{\|z\|^{7}}{576} \left| (\Psi'(0))^{3} + \frac{3\Psi'(0)\Psi''(0)}{2} + \frac{\Psi'''(0)}{3} \right|^{2} \\ + \frac{(\Psi'(0))^{2} \|z\|^{5}}{36} \left( \frac{1}{2} \frac{\Psi''(0)}{\Psi'(0)} + \Psi'(0) \right)^{2}, \ z \in \mathbb{U}^{n}.$$ (15) The estimate is sharp.

Corollary 6.1. Let $f \in \mathcal{H}[\mathbb{B},\mathbb{C}]$ and $F(z) = zf(z) \in \mathcal{C}(\mathbb{B})$ . Then $$\left| \left( \frac{l_z(D^3 F(0)(z^3))}{3! \|z\|^3} \right)^2 - \left( \frac{l_z(D^4 F(0)(z^4))}{4! \|z\|^4} \right)^2 \right| \le 2, \quad z \in \mathbb{B} \setminus \{0\}.$$ If $\mathbb{B} = \mathbb{U}^n$ and $X = \mathbb{C}^n$ , then $$\left\| \frac{1}{4!} D^4 F(0) \left( z^3, \frac{D^4 F(0)(z^4)}{4!} \right) - \frac{1}{3!} D^3 F(0) \left( z^2, \frac{D^3 F(0)(z^3)}{3!} \right) \right\| \le \|z\|^5 + \|z\|^7, \ z \in \mathbb{U}^n.$$ All these estimates are sharp.

Corollary 6.2. Let $f \in \mathcal{H}[\mathbb{B}, \mathbb{C}]$ and $F(z) = zf(z) \in \mathcal{C}_{\alpha}(\mathbb{B})$ . Then $$\begin{split} \left| \left( \frac{l_z(D^3 F(0)(z^3))}{3! \|z\|^3} \right)^2 - \left( \frac{l_z(D^4 F(0)(z^4))}{4! \|z\|^4} \right)^2 \right| &\leq \frac{(1-\alpha)^2 (3-2\alpha)^2}{9} \\ &\qquad \qquad + \frac{(1-\alpha)^2 (2-\alpha)^2 (3-2\alpha)^2}{36}, \ \ z \in \mathbb{B} \setminus \{0\}. \end{split}$$ If $\mathbb{B} = \mathbb{U}^n$ and $X = \mathbb{C}^n$ , then $$\begin{split} \left\| \frac{1}{4!} D^4 F(0) \bigg( z^3, \frac{D^4 F(0)(z^4)}{4!} \bigg) - \frac{1}{3!} D^3 F(0) \bigg( z^2, \frac{D^3 F(0)(z^3)}{3!} \bigg) \right\| \\ & \leq \frac{\|z\|^5 (1-\alpha)^2 (3-2\alpha)^2}{9} + \frac{\|z\|^7 (1-\alpha)^2 (2-\alpha)^2 (3-2\alpha)^2}{36}, \ z \in \mathbb{U}^n. \end{split}$$ All these estimates are sharp.

Definition 1. [12] Suppose $\alpha \in [0,1)$ and $f: \mathbb{B} \to X$ be a normalized locally biholomorphic mapping. If Re $$\{l_z[(Df(z))^{-1}(D^2f(z)(z^2) + Df(z)(z))]\} \ge \alpha ||z||, l_z \in T_z, z \in \mathbb{B} \setminus \{0\},$$ then f is called a quasi convex mapping of type B and order $\alpha$ on $\mathbb{B}$ . If $\mathbb{B} = \mathbb{U}^n$ and $X = \mathbb{C}^n$ , then the above condition reduces to $$\left| \frac{q_k(z)}{z_k} - \frac{1}{2\alpha} \right| < \frac{1}{2\alpha}, \quad \forall z \in \mathbb{U}^n \setminus \{0\},$$ where $$q(z) = (q_1(z), q_2(z), \dots, q_n(z))' = (Df(z))^{-1}(D^2f(z)(z^2) + Df(z)(z))$$ is a column vector in $\mathbb{C}^n$ and k satisfies $$|z_k| = ||z|| = \max_{1 \le j \le n} \{|z_j|\}.$$ In the case $\mathbb{B} = \mathbb{U}$ and $X = \mathbb{C}$ , the condition is equivalent to $$\operatorname{Re}\left(1+\frac{zf''(z)}{f'(z)}\right) > \alpha, \quad z \in \mathbb{U}.$$ We denote by $\mathcal{C}_{\alpha}(\mathbb{B})$ the class of quasi convex mappings of type B and order $\alpha$ . When $\alpha = 0$ , Definition 1 coincides with the definition of quasi convex mapping of type B, denoted by $\mathcal{C}(\mathbb{B})$ , introduced by Roper and Suffridge [15]. Definition. For a biholomorphic function $\Psi : \mathbb{U} \to \mathbb{C}$ satisfying $\operatorname{Re} \Psi(z) > 0$ , $\Psi(0) = 1$ , $\Psi'(0) > 0$ and $\Psi''(0) \in \mathbb{R}$ . Let $\mathcal{M}_{\Psi}$ , Graham et al. [11] introduced the class $$\mathcal{M}_{\Psi} = \left\{ p \in \mathcal{H}(\mathbb{B}) : p(0) = Dp(0) = I, \ \frac{l_z(p(z))}{\|z\|} \in \Psi(\mathbb{U}), \ z \in \mathbb{B} \setminus \{0\}, l_z \in T_z \right\}.$$ In case of $\mathbb{B} = \mathbb{U}^n$ and $X = \mathbb{C}^n$ , we have $$\mathcal{M}_{\Psi} = \left\{ p \in \mathcal{H}(\mathbb{B}) : p(0) = Dp(0) = I, \ \frac{p_j(z)}{z_j} \in \Psi(\mathbb{U}), \ z \in \mathbb{U}^n \setminus \{0\} \right\},$$ where $p(z) = (p_1(z), p_2(z), \dots, p_n(z))'$ is a column vector in $\mathbb{C}^n$ and j satisfies $$|z_j| = ||z|| = \max_{1 \le k \le n} \{|z_k|\}.$$ For $\mathbb{B} = \mathbb{U}$ and $X = \mathbb{C}$ , the relation is equivalent to $$\mathcal{M}_{\Psi} = \left\{ p \in \mathcal{H}(\mathbb{U}) : p(0) = 0, p'(0) = 1, \frac{p(z)}{z} \in \Psi(\mathbb{U}), z \in \mathbb{U} \right\}.$$ In order to establish the main results, we impose additional geometric constraints on the function $\Psi$ . Assumption. In addition to the existing properties, we assume that the biholomorphic function $\Psi$ satisfies $\Psi'(0) > 0$ , $\Psi(\mathbb{U})$ is starlike with respect to 1 and symmetric about the real axis in the right half plane. Extending the Fekete-Szegö inequality from one dimension to higher dimensions, Xu et al. [18] established sharp estimate for the Fekete-Szegö functional for the classes $\mathcal{C}_{\alpha}(\mathbb{B})$ and $\mathcal{C}(\mathbb{B})$ , defined on the unit ball in X and on the unit polydisk in $\mathbb{C}^n$ . This work was further generalized for the mappings having a zero of order k at z = 0, where $k \in \mathbb{N}$ [17]. Considering the Toeplitz determinants, Giri and Kumar [8] derived sharp bounds of $|T_{2,2}(f)|$ and $|T_{3,1}(f)|$ for the class $\mathcal{C}_{\alpha}(\mathbb{B})$ ; however, the case $|T_{2,3}(f)|$ remained unaddressed. Focusing on this problem, in section 4, we obtain sharp estimates of $|T_{2,3}(f)|$ for a class of holomorphic mappings on the unit ball in X and on the unit polydisk in $\mathbb{C}^n$ . As a consequence, these results yield bounds for the classes $\mathcal{C}_{\alpha}(\mathbb{B})$ and $\mathcal{C}(\mathbb{B})$ , thereby extending the study of Toeplitz determinants for convex mappings from one dimension to higher dimensions.