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Abstract
In this paper we investigate Toeplitz and symmetric Toeplitz determinants of inverse functions for some classes of univalent functions and improve some previous results.
Results & Lemmas (6)
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Lemma 1
Lemma 1. Let denote the class of analytic functions of the form (7) If is given by (7), then, and where and are real and belongs to,, or,…
Lemma 1. Let $\Omega$ denote the class of analytic functions $\omega : \mathbb{D} \to \mathbb{D}$ of the form
(7)
$$\omega(z) = c_1 z + c_2 z^2 + \cdots.$$
If $\omega \in \Omega$ is given by (7), then
$$|c_1| \le 1$$
, $|c_2| \le 1 - |c_1|^2$
and
$$|c_3 + \mu c_1 c_2 + \nu c_1^3| \le |\nu|,$$
where $\mu$ and $\nu$ are real and $(\mu, \nu)$ belongs to $D_4$ , $D_6$ , or $D_7$ , where
$$D_4 = \left\{ (\mu, \nu) : |\mu| \ge \frac{1}{2}, \ \nu \le -\frac{2}{3}(|\mu| + 1) \right\},$$
$$D_6 = \left\{ (\mu, \nu) : |\mu| \ge 4, \ \nu \ge \frac{2}{3}(|\mu| - 1) \right\},$$
$$D_7 = \left\{ (\mu, \nu) : 2 \le |\mu| \le 4, \ \nu \ge \frac{1}{12}(\mu^2 + 8) \right\}.$$
Theorem 1
Theorem 1. If, then and.
Theorem 1. If $f \in \mathcal{S}$ , then
$$T_{2,1}(f^{-1}) = T_{2,1}(f)$$
and $T_{3,1}(f^{-1}) = T_{2,1}(f)$ .
Theorem 2
Theorem 2. If, then, and. All inequalities are sharp.
Theorem 2. If $f \in \mathcal{S}$ , then
$$|T_{2,2}^s(f^{-1})| \le 29$$
, $|T_{2,3}^s(f^{-1})| \le 221$ and $|T_{3,1}^s(f^{-1})| \le 24$ .
All inequalities are sharp.
Theorem 3
Theorem 3. If, then we have - All estimates are sharp for the function defined by i.e., for the function Its inverse function is
Theorem 3. If $f \in \mathcal{R}$ , then we have
- $\begin{array}{ll} (i) & |T_{2,2}^s(f^{-1})| \leq \frac{25}{9} = 2.77 \dots; \\ (ii) & |T_{2,3}^s(f^{-1})| \leq \frac{233}{36} = 6.472 \dots; \\ (iii) & |T_{3,2}^s(f^{-1})| \leq \frac{817}{108} = 7.5448 \dots. \end{array}$
All estimates are sharp for the function defined by
$$f'(z) = \frac{1+iz}{1-iz} = 1 + 2iz - 2z^2 - 2iz^3 + 2z^4 + \cdots,$$
i.e., for the function
$$f(z) = z + iz^2 - \frac{2}{3}z^3 - \frac{1}{2}iz^4 + \cdots$$
Its inverse function is
$$f^{-1}(w) = w - iw^2 - \frac{4}{3}w^3 + \frac{13}{6}iw^4 + \cdots$$
Theorem 4
Theorem 4. If, then we have - - All results are sharp.
Theorem 4. If $f \in \mathcal{S}^*$ , then we have
- $\begin{array}{ll} (i) & |T^s_{2,2}(f^{-1})| \leq 29; \\ (ii) & |T^s_{2,3}(f^{-1})| \leq 221; \end{array}$
- $(iii) |T_{3,2}^s(f^{-1})| \le 416.$
All results are sharp.
Theorem 5
Theorem 5. If, then we have - (i); - All results are sharp.
Theorem 5. If $f \in \mathcal{C}$ , then we have
- (i) $|T_{2,2}^s(f^{-1})| \le 2$ ;
- $|T_{2,3}^s(f^{-1})| \le 2;$ $|T_{3,2}^s(f^{-1})| \le 4.$
All results are sharp.
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