Hermitian-Toeplitz determinants for certain univalent functions

Lemma 1. [17, Lemma 3] If $p(z) = 1 + \sum_{n=1}^{\infty} p_n z^n \in \mathcal{P}$ , then $$2p_2 = p_1^2 + \left(4 - p_1^2\right)\zeta,$$ for some $\zeta \in \bar{\mathbb{D}}$ .

Theorem 2. Let $f \in S^*(\varphi)$ and $\varphi(z) = 1 + B_1 z + B_2 z^2 + B_3 z^3 + \cdots$ , then the following holds: 1. $$1 - B_1^2 \le \det T_{2,1}(f) \le 1$$ . 2. If $$3B_1^4 - 8B_1^2 + 2B_1^2B_2 - B_2^2 < 0$$ and $B_1 \le |B_2 + B_1^2|$ , then $$\det T_{3,1}(f) \le 1. \tag{4}$$ 3. If $3B_1^4 - 8B_1^2 + 2B_1^2B_2 - B_2^2 \ge 0$ and $B_1 \le |B_2 + B_1^2|$ , then $$\det T_{3,1}(f) \le B_1^2(B_1^2 + B_2) - \frac{1}{4} (B_1^2 + B_2)^2 - 2B_1^2 + 1.$$ (5) All these inequalities are sharp.

Corollary 2.1. 1. If $f \in S_{sin}^*$ , then $0 \le \det T_{2,1}(f) \le 1$ and $\det T_{3,1}(f) \le 1$ . 2. If $$f \in \mathcal{S}_{o}^{*}$$ , then $0 \leq \det T_{2,1}(f) \leq 1$ and $\det T_{3,1}(f) \leq 1$ . 3. If $f \in S_P$ , then $1 - (64/\pi^4) \le \det T_{2,1}(f) \le 1$ and $\det T_{3,1}(f) \le 1$ . Remark. Theorem 2 yields some already known results for different subclasses of $S^*$ , obtained by an appropriate choice of $\varphi$ . - 1. If $f \in \mathcal{S}_{SC}^*$ , then $3/4 \leq \det T_{2,1}(f) \leq 1$ and $\det T_{3,1}(f) \leq 1$ [3, Theorem 2.1]. - 2. If $f \in \mathcal{S}_{R}^{*}$ , then $0 \leq \det T_{2,1}(f) \leq 1$ and $\det T_{3,1}(f) \leq 1$ [3, Theorem 2.2]. - 3. If $f \in \Delta^*$ , then $\det T_{3,1}(f) \le 1$ [15, Theorem 2]. - 4. If $f \in S^*$ , then $\det T_{3,1}(f) \le 8$ [4, Corollary 3], [11, Corollary 2]. - 5. If $f \in S^*(1/2)$ , then $\det T_{3,1}(f) \le 1$ [4, Corollary 4]. - 6. If $f \in \mathcal{S}_{N_e}^*$ , then $\det T_{3,1}(f) \leq 1$ [13, Theorem 4.2]. - 7. If we take $\varphi(z) = (1 + (1 2\alpha)z)/(1 z)$ , $\alpha \in (0, 1]$ , then we obtain the upper bound of $\det T_{3,1}(f)$ for $f \in \mathcal{S}^*(\alpha)$ [4, Theorem 3]. - 8. For $\varphi(z) = ((1+z)/(1-z))^{\alpha}$ , $\alpha \in [1/3,1]$ , we get the bound of $T_{3,1}(f)$ for function f belonging to the class of strongly starlike function $SS^*(\alpha)$ [15, Theorem 1], [11, Theorem 3]. - 9. For $\varphi(z) = (1 + Az)/(1 + Bz)$ , where $-1 \le B < A \le 1$ with $A B \le |A^2 3AB + 2B^2|$ , we get the bound of $T_{3,1}(f)$ for the class $S^*[A, B]$ [16, Theorem 2]. In a similar fashion, the bounds of $\det T_{2,1}(f)$ can also be found for all the above mentioned classes.

Theorem 3. If $f \in \mathcal{C}(\varphi)$ and $\varphi(z) = 1 + B_1 z + B_2 z^2 + B_3 z^3 + \cdots$ , then the following hold: - 1. $1 \frac{B_1^2}{4} \le \det T_{2,1}(f) \le 1$ . - 2. If $B_1 \leq |B_2 + B_1^2|$ , then $$\det T_{3,1}(f) \le 1. \tag{7}$$ All these estimates are sharp.

Theorem 4. Let $f \in S^*(\varphi)$ and $B_1^2 \geq B_2$ , then the following inequalities hold: 1. If $\mu \notin [0, 4]$ , then $$\det T_{3,1}(f) \ge \min \left\{ 1 - \frac{B_1^2}{4}, 1 - 2B_1^2 + \frac{3B_1^4}{4} + \frac{B_1^2 B_2}{2} - \frac{B_2^2}{4} \right\}. \tag{8}$$ 2. If $\mu = 4$ , then $$\det T_{3,1}(f) \ge 1 - 2B_1^2 + \frac{3B_1^4}{4} + \frac{B_1^2 B_2}{2} - \frac{B_2^2}{4}. \tag{9}$$ 3. If $\mu \in (0,4)$ , then $$\det T_{3,1}(f) \ge 1 - \frac{B_1^2}{4} - \frac{B_1^2(B_1^2 + 3B_1 - B_2)^2}{4(B_1(2B_1^2 - B_1 - 2B_2) + (3B_1^2 - B_2)(B_1^2 + B_2))},\tag{10}$$ where $$\mu = \frac{4B_1(B_1^2 + 3B_1 - B_2)}{(3B_1^2 - B_2)(B_1^2 + B_2) + B_1(2B_1^2 - 2B_2 - B_1)}.$$ The first two inequalities are sharp.

Corollary 4.1. Let $f \in \mathcal{S}^*[A, B]$ , then the following hold: 1. If $\mu \notin [0, 4]$ , then $$\det T_{3,1}(f) \ge \min \left\{ 1 - \frac{(A-B)^2}{4}, 1 + \frac{(A-B)^2(3A^2 - 8AB + 4B^2 - 8)}{4} \right\}.$$ 2. If $\mu = 4$ , then $$\det T_{3,1}(f) \ge 1 + \frac{(A-B)^2(3A^2 - 8AB + 4B^2 - 8)}{4}.$$ 3. If $\mu \in (0,4)$ , then $$\det T_{3,1}(f) \ge 1 - \frac{(A-B)^2(A^2 - 2AB + B^2 + 2A + 2)}{(+3A^2 - 8AB + 4B^2 + 2A - 1)},$$ where $\mu = 4(3+A)/(+3A^2+4B^2-8AB+2A-1)$ . The first two inequalities are sharp. In the following corollary, bounds are given for certain classes when coefficients $B_1$ and $B_2$ of $\varphi(z)$ satisfy the condition (8) or (9).

Corollary 4.2. 1. If $f \in \mathcal{S}_{o}^{*}$ , then $\det T_{3,1}(f) \geq 0$ . - 2. If $f \in \mathcal{S}_{\sin}^*$ , then $\det T_{3,1}(f) \geq -1/4$ . - 3. If $f \in S_P$ , then $\det T_{3,1}(f) \ge 1 64 \left(19\pi^4 24\pi^2 432\right)/(9\pi^8)$ . - 4. If $f \in \mathcal{S}_{RL}^*$ , then $\det T_{3,1}(f) \ge -9(4130\sqrt{2} 5861)/256$ . Remark. Some of the previously known results are established as special cases of the result, which are given below. - 1. If $f \in \mathcal{S}_{SG}^*$ , then $\det T_{3,1}(f) \geq 35/64$ [3, Theorem 2.1]. - 2. If $f \in \mathcal{S}_B^*$ , then $\det T_{3,1}(f) \geq 0$ [3, Theorem 2.2]. - 3. If $f \in \mathcal{S}_{N_e}^*$ , then $\det T_{3,1}(f) \geq -1/4$ [13, Theorem 4.2]. - 4. If $f \in \mathcal{S}_L^*$ , then $\det T_{3,1}(f) \geq 135/256$ [15, Theorem 3]. - 5. If $f \in S^*(1/2)$ , then $\det T_{3,1}(f) \geq 0$ [4, Corollary 4]. Note that, the choice of $\varphi(z)$ satisfy the condition in (10) for the classes $\mathcal{S}$ and $\Delta$ and by Theorem 4, we have the following results. Remark. 1. If $f \in \mathcal{S}^*$ , then $\det T_{3,1}(f) \geq -1$ [4, Corollary 3],[11, Corollary 2]. 2. If $f \in \Delta^*$ , then $\det T_{3,1}(f) \ge -1/15$ [15, Theorem 2].

Theorem 5. If $f \in \mathcal{C}(\varphi)$ such that $B_1^2 \geq 2B_2$ , then $$T_{3,1}(f) \geq \begin{cases} \min\left\{1 - \frac{B_1^2}{36}, 1 - \frac{B_1^2}{2} + \frac{B_1^4}{18} + \frac{B_1^2 B_2}{36} - \frac{B_2^2}{36}\right\}, & \sigma \notin [0, 4], \end{cases}$$ $$T_{3,1}(f) \geq \begin{cases} 1 - \frac{B_1^2}{2} + \frac{B_1^4}{18} + \frac{B_1^2 B_2}{36} - \frac{B_2^2}{36}, & \sigma = 4, \end{cases}$$ $$1 - \frac{B_1^3 (B_1^3 + 4B_1^2 + 28B_1 - 8B_2)}{16(2B_1^4 + B_1^3 - B_1^2 - 2B_1B_2 + B_1^2B_2 - B_2^2)}, & \sigma \in (0, 4), \end{cases}$$ where $$\sigma = \frac{2B_1(B_1^2 + 16B_1 - 2B_2)}{2B_1^4 + B_1^3 - B_1^2 - 2B_1B_2 + B_1^2B_2 - B_2^2}.$$ The first two inequalities are sharp.

Corollary 5.1. Let $f \in C[A, B]$ and $(A - B)^2 \ge 2(B^2 - AB)$ , then the following bounds hold: 1. If $\sigma \notin [0,4]$ , then $$T_{3,1}(f) \ge \min\left\{1 - \frac{(A-B)^2}{36}, 1 + \frac{(A-B)^2(2A^2 + 2B^2 - 5AB - 18)}{36}\right\}.$$ 2. If $\sigma = 4$ , then $$T_{3,1}(f) \ge 1 + \frac{(A-B)^2(2A^2 + 2B^2 - 5AB - 18)}{36}.$$ 3. If $\sigma \in (0,4)$ , then $$T_{3,1}(f) \ge 1 - \frac{(A-B)^2(A^2+B^2+4A+4B-2AB+28)}{16(2A^2+2B^2+A+B-5AB-1)},$$ where $\sigma = 2(A+B+16)/(2A^2+2B^2+A+B-5AB-1)$ . The first two inequalities are sharp.

Corollary 5.2. If $f \in \mathcal{C}$ , then $\det T_{3,1}(f) \geq 0$ . The bound is sharp. Recently, the lower and upper bounds of $\det T_{2,1}(f)$ and $\det T_{3,1}(f)$ for functions f in $\mathcal{F}_1$ , $\mathcal{F}_2$ , $\mathcal{F}_3$ and $\mathcal{F}_4$ are obtained [11, 12, 19]. By generalizing these works, when $f \in \mathcal{K}$ , we obtain the bounds of $\det T_{2,1}(f)$ and $\det T_{3,1}(f)$ for different choices of $g \in \mathcal{S}^*$ .

Theorem 6. Let $f \in \mathcal{K}$ for some function $g(z) = z + b_2 z^2 + b_3 z^3 + \cdots \in \mathcal{S}^*$ , such that $$6|b_2|^3 - 4|b_2|(|b_3| - 1) - 4(|b_3| - 1)^2 + |b_2|^2(3|b_3| + 5) \ge 0, (13)$$ then 1. $$1 - \left(1 + \frac{|b_2|}{2}\right)^2 \le \det T_{2,1}(f) \le 1$$ . 2. If $$6|b_2|^3 + |b_2|^2(3|b_3| + 13) + 4|b_2|(|b_3| - 1) - 2(1 - |b_3|)^2 - 18 \le 0$$ , then $$\det T_{3,1}(f) \le 1. \tag{14}$$ 3. If $6|b_2|^3 + |b_2|^2(3|b_3| + 13) + 4|b_2|(|b_3| - 1) - 2(1 - |b_3|)^2 - 18 > 0$ , then $$\det T_{3,1}(f) \ge \frac{1}{18} \left( 6|b_2|^3 + |b_2|^2 (3|b_3| + 13) + 4|b_2|(|b_3| - 1) - 2(1 - |b_3|)^2 \right). \tag{15}$$ All these bounds are sharp.

Theorem 7. Let $f \in \mathcal{K}$ and $g(z) = z + \sum_{n=2}^{\infty} b_n z^n \in \mathcal{S}$ . If $\tilde{g}(z) = z + \sum_{n=2}^{\infty} i^{n-1} b_n z^n \in \mathcal{S}$ , then the following sharp bound hold: $$|\det T_{2,2}(f)| \le \frac{1}{4} (2 + |b_2|)^2 + \frac{1}{9} (|b_3| + 2|b_2| + 2)^2.$$

Corollary 7.1. 1. If f ∈ F1, then |det T2,2(f)| ≤ 181/36 and the bound is sharp for $$f(z) = \int_0^z \left(\frac{1+it}{1-it}\right) \frac{\tilde{g}(t)}{t} dt,$$ where ^g˜(z) = z/(1 ⁻ iz) ∈ S^∗ . 2. If f ∈ F2, then |det T2,2(f)| ≤ 2 and the bound is sharp for $$f(z) = \int_0^z \left(\frac{1+it}{1-it}\right) \frac{\tilde{g}(t)}{t} dt,$$ where g˜(z) = z/(1 + z 2 ) ∈ S^∗ . 3. If f ∈ F³, then |det T²,²(f)| ≤ 13 and the bound is sharp for $$f(z) = \int_0^z \left(\frac{1+it}{1-it}\right) \frac{\tilde{g}(t)}{t} dt,$$ where g˜(z) = z/(1 − iz) ² ∈ S^∗ . 4. If f ∈ F⁴, then |det T²,²(f)| ≤ 145/36 and the bound is sharp for $$f(z) = \int_0^z \left(\frac{1+it}{1-it}\right) \frac{\tilde{g}(t)}{t} dt,$$ where g˜(z) = z/(1 − iz + z 2 ) ∈ S^∗ . 5. If f ∈ R, then |det T²,²(f)| ≤ 13/9 and the bound is sharp for $$f(z) = \int_0^z \left(\frac{1+it}{1-it}\right) dt.$$