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Abstract
We determine sharp bounds on some Hankel determinants involving initial coefficients, inverse coefficients, and logarithmic inverse coefficients for two subclasses of Sakaguchi functions which are associated with the right half of the lemniscate of Bernoulli and the exponential function. Further, we compute sharp bounds on the second Hermitian-Toeplitz determinants involving logarithmic coefficients and logarithmic inverse coefficients. We also discuss invariant property for the obtained estimat
Results & Lemmas (15)
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 2.1
Lemma 2.1. [5] Let be a Schwarz function. Then,,,.
Lemma 2.1. [5] Let $w(z) = c_1 z + c_2 z^2 + c_3 z^3 + c_4 z^4 + ...$ be a Schwarz function. Then
$$|c_1| \le 1$$
, $|c_2| \le 1 - |c_1|^2$ , $|c_3| \le 1 - |c_1|^2 - \frac{|c_2|^2}{1 + |c_1|}$ , $|c_4| \le 1 - |c_1|^2 - |c_2|^2$ .
Lemma 2.2
Lemma 2.2. [18] Let be the class of analytic functions having the Taylor series of the form (2.1) satisfying the condition. Then for some…
Lemma 2.2. [18] Let $\mathcal{P}$ be the class of analytic functions having the Taylor series of the form
$$p(z) = 1 + p_1 z + p_2 z^2 + p_3 z^3 + \cdots$$
(2.1)
satisfying the condition $\operatorname{Re} p(z) > 0 \ (z \in \mathbb{D})$ . Then
$$2p_{2} = p_{1}^{2} + t\xi,$$
$$4p_{3} = p_{1}^{3} + 2p_{1}t\xi - p_{1}t\xi^{2} + 2t(1 - |\xi|^{2})\eta,$$
$$8p_{4} = p_{1}^{4} + 3p_{1}^{2}t\xi + (4 - 3p_{1}^{2})t\xi^{2} + p_{1}^{2}t\xi^{3} + 4t(1 - |\xi|^{2})(1 - |\eta|^{2})\gamma$$
$$+ 4t(1 - |\xi|^{2})(p_{1}\eta - p_{1}\xi\eta - \bar{\xi}\eta^{2}),$$
for some $\xi, \eta, \gamma \in \overline{\mathbb{D}}$ and $t = (4 - p_1^2)$ .
Theorem 2.3 · coeff
Theorem 2.3. If the function, then The inequality is sharp.
Theorem 2.3. If the function $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{S}_{S.e.}^*$ , then
$$|H_{2,1}(F_{f^{-1}}/2)| \le \frac{1}{16}.$$
The inequality is sharp.
Theorem 2.5 · coeff
Theorem 2.5. If the function, then The inequality is sharp.
Theorem 2.5. If the function $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{S}_{S,e}^*$ , then
$$|H_{2,2}(f^{-1})| \le \frac{1}{4}.$$
The inequality is sharp.
Corollary 2.7 · coeff
Corollary 2.7. If, then.
Corollary 2.7. If
$$f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{S}_{S,e}^*$$
, then $|H_{2,2}(f^{-1}) - H_{2,2}(f)| \leq \frac{1}{32}$ .
Theorem 2.8 · coeff
Theorem 2.8. If the function, then. The inequality is sharp.
Theorem 2.8. If the function $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{S}_{S,e}^*$ , then $|H_{2,3}(f^{-1})| \leq \frac{1}{4}$ . The inequality is sharp.
Corollary 2.9 · coeff
Corollary 2.9. If the function, then we have The inequality is sharp for the function defined by (1.1).
Corollary 2.9. If the function $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{S}_{S,e}^*$ , then we have
$$|H_{2,3}(f^{-1}) - H_{2,3}(f)| \le \frac{3}{8}.$$
The inequality is sharp for the function $f_1$ defined by (1.1).
Theorem 2.10 · coeff
Theorem 2.10. If the function, then The inequality is sharp.
Theorem 2.10. If the function $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{S}_{S,L}^*$ , then
$$|H_{2,1}(F_{f^{-1}}/2)| \le \frac{1}{64}.$$
The inequality is sharp.
Theorem 2.12 · coeff
Theorem 2.12. If the function, then The inequality is sharp.
Theorem 2.12. If the function $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{S}_{S,L}^*$ , then
$$|H_{2,2}(f)|, |H_{2,2}(f^{-1})| \le \frac{1}{16}.$$
The inequality is sharp.
Corollary 2.13 · coeff
Corollary 2.13. If the function, then The inequality is sharp.
Corollary 2.13. If the function $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{S}_{S,L}^*$ , then
$$|H_{2,2}(f^{-1}) - H_{2,2}(f)| \le \frac{1}{128}.$$
The inequality is sharp.
Theorem 2.14 · coeff
Theorem 2.14. If the function, then. The inequality is sharp.
Theorem 2.14. If the function $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{S}_{S,L}^*$ , then $|H_{2,3}(f^{-1})| \leq \frac{3}{64}$ . The inequality is sharp.
Corollary 2.15 · coeff
Corollary 2.15. If the function, then The inequality is sharp for the function defined by (1.3).
Corollary 2.15. If the function $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{S}_{S,L}^*$ , then
$$|H_{2,3}(f^{-1}) - H_{2,3}(f)| \le \frac{3}{64}.$$
The inequality is sharp for the function $g_1$ defined by (1.3).
Theorem 3.1
Theorem 3.1. If the function, then The lower and upper bounds are sharp.
Theorem 3.1. If the function $f \in \mathcal{S}_{S,e}^*$ , then
$$-\frac{1}{16} \le T_{2,1}(F_f/\gamma), T_{2,1}(F_{f^{-1}}/\Gamma) \le \frac{15}{256}.$$
The lower and upper bounds are sharp.
Theorem 3.2
Theorem 3.2. If the function, then The bounds are sharp.
Theorem 3.2. If the function $f \in \mathcal{S}_{S,L}^*$ , then
$$-\frac{1}{64} \le T_{2,1}(F_f/\gamma) \le \frac{55}{4096}.$$
The bounds are sharp.
Theorem 3.3
Theorem 3.3. Let the function. Then The inequality is sharp.
Theorem 3.3. Let the function $f \in \mathcal{S}_{S,L}^*$ . Then
$$-\frac{1}{64} \le T_{2,1}(F_{f^{-1}}/\Gamma) \le \frac{39}{4096}.$$
The inequality is sharp.
Function classes studied:
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