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Abstract
The classes of analytic univalent functions on the unit disk defined by
$$ \mathcal{S}^*(\varphi)= \bigg\{ f \in \mathcal{A}: \frac{z f'(z)}{f(z)} \prec \varphi(z)\bigg\}$$
and
$$ \mathcal{C}(\varphi)=\bigg\{ f \in \mathcal{A}: 1 + \frac{z f''(z)}{f'(z)} \prec \varphi(z)\bigg\} $$
generalize various subclasses of starlike and convex functions, respectively. In this paper, sharp bounds are established for certain Toeplitz determinants constructed over the coefficients and logarithmic coef
Results & Lemmas (22)
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. [24] If and, then where and
Lemma 1. [24] If
$$\omega(z) = \sum_{n=1}^{\infty} c_n z^n \in \mathcal{B}_0$$
and $(\sigma, \mu) \in \bigcup_{i=1}^{3} \Omega_i$ , then
$$|c_3 + \sigma c_1 c_2 + \mu c_1^3| \le |\mu|,$$
where
$$\Omega_1 = \left\{ (\sigma, \mu) : |\sigma| \le 2, \ \mu \ge 1 \right\}, \ \Omega_2 = \left\{ (\sigma, \mu) : 2 \le |\sigma| \le 4, \ \mu \ge \frac{1}{12} (\sigma^2 + 8) \right\},$$
and
$$\Omega_3 = \left\{ (\sigma, \mu) : |\sigma| \ge 4, \ \mu \ge \frac{2}{3} (|\sigma| - 1) \right\}.$$
Theorem 2
Theorem 2. Let and. If holds, The inequality is sharp.
Theorem 2. Let $f \in S^*(\varphi)$ and $\varphi(z) = 1 + B_1 z + B_2 z^2 + B_3 z^3 + \cdots$ . If $|B_2 - 2B_1^2| \ge B_1$ holds,
$$|T_{2,1}(F_{f^{-1}})| \le \frac{B_1^2}{4} + \frac{1}{16}(2B_1^2 - B_2)^2.$$
The inequality is sharp.
Theorem 3
Theorem 3. Let and. If holds, then The inequality is sharp.
Theorem 3. Let $f \in C(\varphi)$ and $\varphi(z) = 1 + B_1 z + B_2 z^2 + B_3 z^3 + \cdots$ . If $|B_2 - \frac{5}{4}B_1^2| \ge B_1$ holds, then
$$|T_{2,1}(F_{f^{-1}}/2)| \le \frac{B_1^2}{16} + \frac{1}{144} \left(B_2 - \frac{5}{4}B_1^2\right)^2.$$
The inequality is sharp.
Theorem 4
Theorem 4. Let and. If and hold, then where and The bound is sharp.
Theorem 4. Let $f \in S^*(\varphi)$ and $\varphi(z) = 1 + B_1 z + B_2 z^2 + B_3 z^3 + \cdots$ . If $|B_2 - 2B_1^2| \ge B_1$ and $(\sigma_1, \mu_1) \in \bigcup_{i=1}^3 \Omega_i$ hold, then
$$|T_{2,2}(F_{f^{-1}})| \le \frac{(B_2 - 2B_1^2)^2}{16} + \frac{(9B_1^3 - 9B_1B_2 + 2B_3)^2}{144},$$
where
$$\sigma_1 = -\frac{(9B_1^2 - 4B_2)}{2B_1}$$
and $\mu_1 = \frac{(9B_1^3 - 9B_1B_2 + 2B_3)}{2B_1}$
The bound is sharp.
Theorem 5
Theorem 5. Let and. If and hold, then (21) where and. The estimate is sharp.
Theorem 5. Let $f \in \mathcal{K}(\varphi)$ and $\varphi(z) = 1 + B_1 z + B_2 z^2 + B_3 z^3 \cdots$ . If $|B_2 - \frac{5}{4} B_1^2| \ge B_1$ and $(\sigma_2, \mu_2) \in \bigcup_{i=2}^3 \Omega_i$ hold, then
$$|T_{2,2}(F_{f^{-1}}/2)| \le \frac{1}{144} \left(B_2 - \frac{5}{4}B_1^2\right)^2 + \frac{(3B_1^3 - 5B_1B_2 + 2B_3)^2}{2304},$$
(21)
where
$$\sigma_2 = -\frac{(5B_1^2 - 4B_2)}{2B_1}$$
and $\mu_2 = \frac{(3B_1^3 - 5B_1B_2 + 2B_3)}{2B_1}$ .
The estimate is sharp.
Theorem 6
Theorem 6. Let and. If holds, then The estimate is sharp.
Theorem 6. Let $f \in S^*(\varphi)$ and $\varphi(z) = 1 + B_1 z + B_2 z^2 + \cdots$ . If $B_1 \le |3B_1^2 - B_2|$ holds, then
$$|T_{2,1}(f^{-1})| \le B_1^2 + \frac{(3B_1^2 - B_2)^2}{4}.$$
The estimate is sharp.
Theorem 7
Theorem 7. Let and. If holds, then The estimate is sharp.
Theorem 7. Let $f \in C(\varphi)$ and $\varphi(z) = 1 + B_1 z + B_2 z^2 + \cdots$ . If $B_1 \le |2B_1^2 - B_2|$ holds, then
$$|T_{2,1}(f^{-1})| \le \frac{B_1^2}{4} + \frac{(2B_1^2 - B_2)^2}{36}$$
The estimate is sharp.
Theorem 8
Theorem 8. Let and. If and hold, then where and. The estimate is sharp.
Theorem 8. Let $f \in S^*(\varphi)$ and $\varphi(z) = 1 + B_1 z + B_2 z^2 + \cdots$ . If $(\sigma_3, \mu_3) \in \bigcup_{i=2}^3 \Omega_i$ and $B_1 \leq |3B_1^2 - B_2|$ hold, then
$$|T_{2,2}(f^{-1})| \le \frac{(B_2 - 3B_1^2)^2}{4} + \frac{(8B_1^3 - 6B_1B_2 + B_3)^2}{9},$$
where
$$\sigma_3 = \frac{2(B_2 - 3B_1^2)}{B_1}$$
and $\mu_3 = \frac{(8B1^3 - 6B1B2 + B3)}{B_1}$ .
The estimate is sharp.
Theorem 9
Theorem 9. Let and. If and hold, then where The estimate is sharp.
Theorem 9. Let $f \in C(\varphi)$ and $\varphi(z) = 1 + B_1 z + B_2 z^2 + \cdots$ . If $(\sigma_4, \mu_4) \in \bigcup_{i=1}^3 \Omega_i$ and $B_1 \leq |2B_1^2 - B_2|$ hold, then
$$|b_3^2 - b_4^2| \le \frac{(B_2 - 2B_1^2)^2}{36} + \frac{(6B_1^3 - 7B_1B_2 + 2B_3)^2}{576}$$
where
$$\sigma_4 = \frac{4B_2 - 7B_1^2}{2B_1}, \ \mu_4 = \frac{6B_1^3 - 7B_1B_2 + 2B_3}{2B_1}.$$
The estimate is sharp.
Corollary 9.1
Corollary 9.1. Let. Then the following sharp bounds hold: 1. If |2A - B| > 1, then 2. If and, then where and. 3. If, then 4. If and, then…
Corollary 9.1. Let $f \in S^*[A, B]$ . Then the following sharp bounds hold:
1. If |2A - B| > 1, then
$$|T_{2,1}(F_{f^{-1}})| \le \frac{1}{16}(A-B)^2(4A^2-4AB+B^2+4).$$
2. If $|2A - B| \ge 1$ and $(\sigma_1, \mu_1) = \bigcup_{i=1}^{3} \Omega_i$ , then
$$|T_{2,2}(F_{f^{-1}})| \le \frac{(A-B)^2((9A^2-9AB+2B^2)^2+9(B-2A)^2)}{144},$$
where $\sigma_1 = (5B - 9A)/2$ and $\mu_1 = (3A - 2B)(3A - B)/2$ .
3. If $|3A - 2B| \ge 1$ , then
$$|T_{2,1}(f^{-1})| \le \frac{(A-B)^2((3A-2B)^2+4)}{4}.$$
4. If $|3A - 2B| \ge 1$ and $(\sigma_3, \mu_3) = \bigcup_{i=2}^3 \Omega_i$ , then
$$|T_{2,2}(f^{-1})| \le \frac{(A-B)^2(9(3A-2B)^2+4(B-2A)^2(4A-3B)^2)}{36},$$
where $\sigma_3 = 4B - 6A$ and $\mu_3 = 8A^2 - 10AB + 3B^2$ .
Corollary 9.2
Corollary 9.2. Let. Then the following sharp bounds hold: 1. If, then 2. If and, then where and. 3. If, then 4. If and, then where and.
Corollary 9.2. Let $f \in C[A, B]$ . Then the following sharp bounds hold:
1. If $|5A - B| \ge 4$ , then
$$|T_{2,1}(F_{f^{-1}})| \le \frac{(A-B)^2(25A^2 - 10AB + B^2 + 144)}{2304},$$
2. If $|5A - B| \ge 4$ and $(\sigma_2, \mu_2) \in \bigcup_{i=2}^{3} \Omega_i$ , then
$$|T_{2,2}(F_{f^{-1}})| \le \frac{(A-B)^2(A^2(B-3A)^2 + (B-5A)^2)}{2304},$$
where $\sigma_2 = (B - 5A)/2$ and $\mu_2 = A(3A - B)/2$ .
3. If $|2A - B| \ge A - B$ , then
$$|T_{2,1}(f^{-1})| \le \frac{(A-B)^2((B-2A)^2+9)}{36}.$$
4. If $|2A - B| \ge A - B$ and $(\sigma_4, \mu_4) \in \bigcup_{i=1}^{3} \Omega_i$ , then
$$|T_{2,2}(f^{-1})| \le \frac{(2A^2 - 3AB + B^2)^2((B - 3A)^2 + 16)}{576},$$
where $\sigma_4 = (7A - 3B)/2$ and $\mu_4 = (6A^2 - 5AB + B^2)/2$ .
Corollary 9.3
Corollary 9.3. Let. Then the following hold: 1. If, then 2. If, then 3. If, then 4. If, then
Corollary 9.3. Let $f \in S^*(\alpha)$ . Then the following hold:
1. If $\alpha \in [0, 1/2]$ , then
$$|T_{2,1}(F_{f^{-1}})| \le \frac{(1-\alpha)^2((3-4\alpha)^2+4)}{4}.$$
2. If $\alpha \in [0, 7/15]$ , then
$$|T_{2,2}(F_{f^{-1}})| \le \frac{(1-\alpha)^2(9(3-4\alpha)^2+4(2-3\alpha)^2(5-6\alpha)^2)}{36}.$$
3. If $\alpha \in [0, 2/3]$ , then
$$|T_{2,1}(f^{-1})| \le (1-\alpha)^2 (36\alpha^2 - 60\alpha + 29).$$
4. If $\alpha \in [0, 3/5]$ , then
$$|T_{2,2}(f^{-1})| \le \frac{(1-\alpha)^2(9(5-6\alpha)^2+4(3-4\alpha)^2(7-8\alpha)^2)}{9}.$$
Corollary 9.4
Corollary 9.4. If, Then the following hold: 1. If, then 2. If, then 3. If, then 4. If, then
Corollary 9.4. If $f \in C(\alpha)$ , Then the following hold:
1. If $\alpha \in [0, 1/5]$ , then
$$|T_{2,1}(F_{f^{-1}})| \le \frac{5}{144}(1-\alpha)^2(5\alpha^2-6\alpha+9).$$
2. If $\alpha \in [0, 7/47]$ , then
$$|T_{2,2}(F_{f^{-1}})| \le \frac{(\alpha-1)^2((6\alpha^2-7\alpha+2)^2+(3-5\alpha)^2)}{144}.$$
3. If $\alpha \in [0, 1/2]$ , then
$$|T_{2,1}(f^{-1})| \le \frac{((1-\alpha)^2(3-4\alpha)^2+9)}{9}.$$
4. If $\alpha \in [0, 39/95]$ , then
$$|T_{2,2}(f^{-1})| \le \frac{(1-\alpha)^2((2-3\alpha)^2+4)(3-4\alpha)^2}{36}.$$
Corollary 9.5
Corollary 9.5. If, Then the following hold: 1. If, then 2. If, then and
Corollary 9.5. If $f \in \mathcal{SS}^*(\beta)$ , Then the following hold:
1. If $\beta \in [1/3, 1]$ , then
$$|T_{2,1}(F_{f^{-1}})| \le \frac{\beta^2(9\beta^2+4)}{4} \text{ and } |T_{2,2}(F_{f^{-1}})| \le \frac{\beta^2(3364\beta^4+961\beta^2+4)}{324}.$$
2. If $\beta \in [1/5, 1]$ , then
$$|T_{2,1}(f^{-1})| \le \beta^2 (25\beta^2 + 4)$$
and $|T_{2,2}(f^{-1})| \le \frac{1}{81}\beta^2 (15376\beta^4 + 2521\beta^2 + 4).$
Corollary 9.6
Corollary 9.6. Let. Then the following hold: 1. If, then and. 2. If, then 3. If, then
Corollary 9.6. Let $f \in \mathcal{CC}(\beta)$ . Then the following hold:
1. If $\beta \in [2/3, 1]$ , then
$$|T_{2,1}(F_{f^{-1}})| \le \frac{\beta^2(\beta^2+4)}{16}$$
and $|T_{2,2}(F_{f^{-1}})| \le \frac{\beta^2(25\beta^4+91\beta^2+1)}{1296}$ .
2. If
$$\beta \in [1/3, 1]$$
, then
$$|T_{2,1}(f^{-1})| \le \beta^2(\beta^2 + 1).$$
3. If
$$\beta \in [\sqrt{2/17}, 1]$$
, then
$$|T_{2,2}(f^{-1})| \le \frac{\beta^2(289\beta^4 + 358\beta^2 + 1)}{324}.$$
Corollary 9.7
Corollary 9.7. Let. Then the following sharp bounds hold:
Corollary 9.7. Let $f \in S^*$ . Then the following sharp bounds hold:
$$|T_{2,1}(F_{f^{-1}})| \le \frac{13}{4}, |T_{2,2}(F_{f^{-1}})| \le \frac{481}{36}, |T_{2,1}(f^{-1})| \le 29, |T_{2,2}(f^{-1})| \le 221.$$
Corollary 9.8
Corollary 9.8. Let. Then the following sharp bounds hold:
Corollary 9.8. Let $f \in C$ . Then the following sharp bounds hold:
$$|T_{2,1}(F_{f^{-1}})| \le \frac{5}{16}, |T_{2,2}(F_{f^{-1}})| \le \frac{13}{144}, |T_{2,1}(f^{-1})| \le 2, |T_{2,2}(f^{-1})| \le 2.$$
Corollary 9.9
Corollary 9.9. If, then the following sharp estimates hold:
Corollary 9.9. If $f \in \mathcal{S}_{\varrho}^*$ , then the following sharp estimates hold:
$$|T_{2,1}(F_{f^{-1}})| \le \frac{5}{16}, \quad |T_{2,1}(f^{-1})| \le 2 \quad and \quad |T_{2,2}(f^{-1})| \le \frac{61}{36}$$
Corollary 9.10
Corollary 9.10. If, then the following sharp estimates hold: and
Corollary 9.10. If $f \in S_P$ , then the following sharp estimates hold:
$$|T_{2,1}(f^{-1})| \le \frac{128(648 - 36\pi^2 + 5\pi^4)}{9\pi^8}$$
and
$$|T_{2,2}(f^{-1}) \le \frac{64(\pi^2 - 36)^2}{9\pi^8} + \frac{64(23040 - 1440\pi^2 + 23\pi^4)^2}{18225\pi^{12}}.$$
Corollary 9.11
Corollary 9.11. If, then the following sharp estimates hold:
Corollary 9.11. If $f \in \mathcal{S}_e^*$ , then the following sharp estimates hold:
$$|T_{2,1}(F_{f^{-1}})| \le \frac{25}{64}, |T_{2,2}(F_{f^{-1}})| \le \frac{785}{2592}, |T_{2,1}(f^{-1})| \le \frac{41}{16}, |T_{2,2}(f^{-1})| \le \frac{5869}{1296}.$$
Corollary 9.12
Corollary 9.12. If, then the following sharp estimates hold:
Corollary 9.12. If $f \in \Delta^*$ , then the following sharp estimates hold:
$$|T_{2,1}(F_{f^{-1}})| \le \frac{25}{64}, |T_{2,2}(F_{f^{-1}})| \le \frac{9}{32}, |T_{2,1}(f^{-1})| \le \frac{41}{16}, |T_{2,2}(f^{-1})| \le \frac{625}{144}.$$
Corollary 9.13
Corollary 9.13. If, then the following sharp estimates hold:
Corollary 9.13. If $f \in \mathcal{S}_L^*$ , then the following sharp estimates hold:
$$|T_{2,1}(F_{f^{-1}})| \le \frac{1}{64} \text{ and } |T_{2,1}(f^{-1})| \le \frac{1}{16}.$$
Function classes studied:
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