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Abstract
In this paper, we consider estimates of symmetric Toeplitz determinants $T_{q,n}(f)$ for the class ${\mathcal U}$ and for the general class ${\mathcal S}$ for certain values of $q$ and $n$ ($q,n=1,2,3\ldots$).
Results & Lemmas (4)
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. Let. Then (4) where, and, for all. From (4), for, follows (5),,, where (6),,.
Lemma 1. Let $f(z) = z + a_2 z^2 + \cdots$ . Then
(4)
$$f \in \mathcal{U} \quad \Leftrightarrow \quad \frac{z}{f(z)} = 1 - a_2 z - z\omega(z),$$
where $\omega(0) = 0$ , and $|\omega(z)| < 1$ , $|\omega'(z)| \le 1$ for all $z \in \mathbb{D}$ . From (4), for $\omega(z) = c_1 z + c_2 z^2 + \cdots$ , follows
(5)
$$a_3 = a_2^2 + c_1$$
, $a_4 = c_2 + 2a_2c_1 + a_2^3$ , $a_5 = c_3 + 2a_2c_2 + c_1^2 + 3a_2^2c_1 + a_2^4$ , where
(6)
$$|c_1| \le 1$$
, $|c_2| \le \frac{1}{2} (1 - |c_1|^2)$ , $|c_3| \le \frac{1}{3} \left( 1 - |c_1|^2 - \frac{4|c_2|^2}{1 + |c_1|} \right)$ .
Theorem 2 · coeff
Theorem 2. Let be of the form (1) with. Then - (i); - (ii); - (iii); - -. The inequalities (i)-(iv) are sharp.
Theorem 2. Let $f \in \mathcal{U}$ be of the form (1) with $a_2 = 0$ . Then
- (i) $|T_{2,2}(f)| \leq 1$ ;
- (ii) $|T_{2,3}(f)| \leq 1$ ;
- (iii) $|T_{3,1}(f)| \le 2$ ;
- $(iv) |T_{3,2}(f)| \leq \frac{3}{16};$
- $(v) |T_{3,3}(f)| \leq \frac{9}{2}$ .
The inequalities (i)-(iv) are sharp.
Theorem 3 · coeff
Theorem 3. If has the form (1), then - (i); - Proof. (i) Similarly as in the proof of Theorem 1 we have where we used Lemma 3(a). (ii)…
Theorem 3. If $f \in \mathcal{S}$ has the form (1), then
- (i) $|T_{3,2}(f)| \leq 86.1684...$ ;
- $(ii) |T_{2,3}(f)| \le 239.1895...$
Proof.
(i) Similarly as in the proof of Theorem 1 we have
$$|T_{3,2}(f)| \le (|a_2| + |a_4|) (|a_2|^2 + |a_3|^2 + |H_{2,2}(f)|)$$
$\le 6 \cdot (13 + 1.3614...) = 86.1684...,$
where we used Lemma 3(a).
(ii) Also,
$$|T_{3,3}(f)| \le (|a_3| + |a_5|) (|a_3|^2 + |a_4|^2 + |H_{2,3}(f)|)$$
$\le 8 \cdot (25 + 4.89869...) = 239.1895...,$
where we used Lemma 3(b).
Theorem 4 · coeff
Theorem 4. If has the form (1) with, then - (i); -
Theorem 4. If $f \in \mathcal{S}$ has the form (1) with $a_2 = 0$ , then
- (i) $|T_{3,2}(f)| \le \frac{4}{3}$ ;
- $(ii) |T_{2,3}(f)| \le 7.3883...$
Function classes studied:
Coefficient bounds & claims (16)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
T_{2,2}(f) ≤ 13 for class U (sharp) [Theorem 1(i)]
coefficient_bound
T_{2,3}(f) ≤ 25 for class U (sharp) [Theorem 1(ii)]
coefficient_bound
T_{3,1}(f) ≤ 24 for class U (sharp) [Theorem 1(iii)]
coefficient_bound
T_{3,2}(f) ≤ 84 for class U (sharp) [Theorem 1(iv)]
coefficient_bound
T_{3,3}(f) ≤ 211.8772 for class U [Theorem 1(v)]
coefficient_bound
T_{2,2}(f) ≤ 1 for class U (a2=0) (sharp) [Theorem 2(i)]
coefficient_bound
T_{2,3}(f) ≤ 1 for class U (a2=0) (sharp) [Theorem 2(ii)]
coefficient_bound
T_{3,1}(f) ≤ 2 for class U (a2=0) (sharp) [Theorem 2(iii)]
coefficient_bound
T_{3,2}(f) ≤ 3/16 for class U (a2=0) (sharp) [Theorem 2(iv)]
coefficient_bound
T_{3,3}(f) ≤ 9/2 for class U (a2=0) [Theorem 2(v)]
coefficient_bound
T_{3,2}(f) ≤ 86.1684 for class S [Theorem 3(i)]
coefficient_bound
T_{3,3}(f) ≤ 239.1895 for class S [Theorem 3(ii)]
coefficient_bound
T_{3,2}(f) ≤ 4/3 for class S (a2=0) [Theorem 4(i)]
coefficient_bound
T_{3,3}(f) ≤ 7.3883 for class S (a2=0) [Theorem 4(ii)]
function_family
Class U: |(z/f(z))^2 * f'(z) - 1| < 1, z in D
function_family
Class S: Class of all univalent functions in the unit disk with f(0)=f'(0)-1=0
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