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Abstract
Let $\mathcal{S}_u^*$ denote the class of all analytic functions $f$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$, normalized by $f(0)=f'(0)-1=0$ that satisfies the inequality $\left|zf'(z)/f(z)-1\right|<1$ in $\mathbb{D}$. In the present article, we obtain the sharp estimate of Hankel determinants whose entries are coefficients of $f\in\mathcal{S}_u^*$, logarithmic coefficients of $f\in\mathcal{S}_u^*$ and coefficients of inverse of $f\in\mathcal{S}_u^*$, respectively. We also obtain
Results & Lemmas (6)
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 · coeff
Lemma 2.1. If is of the form (1.3), then there exist such that If is of the form then called the rotation If is of the form (1.1) then,,…
Lemma 2.1. If $p \in \mathcal{P}$ is of the form (1.3), then there exist $z_1, z_2, z_3, z_4 \in \overline{\mathbb{D}}$ such that
$$(2.1) \begin{cases} p_1 &= 2z_1, \\ p_2 &= 2z_1^2 + 2(1 - |z_1|^2)z_2, \\ p_3 &= 2z_1^3 + 4(1 - |z_1|^2)z_1z_2 - 2(1 - |z_1|^2)z_1z_2^2 + 2(1 - |z_1|^2)(1 - |z_2|^2)z_3, \\ p_4 &= 2z_1^4 + 2(1 - |z_1|^2)(z_1^2z_2^2 - 3z_1^2z_2 + 3z_1^2 + z_2)z_2 \\ &+ 2(1 - |z_1|^2)(1 - |z_2|^2)(2z_1 - 2z_1z_2 - \bar{z}_2z_3)z_3 \\ &+ 2(1 - |z_1|^2)(1 - |z_2|^2)(1 - |z_3|^2)z_4. \end{cases}$$
If $f \in \mathcal{S}^*$ is of the form $(1, 1)$ then $f_2(z) = e^{-i\theta} f(e^{i\theta}z), \ \theta \in \mathbb{R}$ called the rotation
If $f \in \mathcal{S}_u$ is of the form (1.1) then $f_{\theta}(z) = e^{-i\theta} f(e^{i\theta}z)$ , $\theta \in \mathbb{R}$ , called the rotation of f, is also belongs to $\mathcal{S}_u$ . Thus the class $\mathcal{S}_u^*$ is rotationally invariant. Further, it is an easy exercise to verify that
$$|H_{2,2}(f)| = |H_{2,2}(f_{\theta})|, |H_{3,1}(f)| = |H_{3,1}(f_{\theta})|,$$
$$\left|H_{2,2}(f^{-1})\right| = \left|H_{2,2}(f_{\theta}^{-1})\right|, \left|H_{2,1}\left(\frac{\mathcal{F}_f}{2}\right)\right| = \left|H_{2,1}\left(\frac{\mathcal{F}_{f_{\theta}}}{2}\right)\right|.$$
Therefore, the functionals $|H_{2,2}(f)|, |H_{2,2}(f)|, |H_{2,2}(f^{-1})|$ and $|H_{2,1}(\frac{\mathbb{F}_f}{2})|$ are also rotationally invariant.
If $f \in \mathcal{S}_u^*$ then there exists a function $p \in \mathcal{P}$ such that
$$\frac{zf'(z)}{f(z)} - 1 = \frac{p(z) - 1}{p(z) + 1}.$$
Comparing the coefficients of $z^n$ , n=2,3,4 and 5, we get the following relations
$$(2.2) \quad a_2 = \frac{p_1}{2}, \quad a_3 = \frac{p_2}{4}, \quad a_4 = \frac{p_3}{6} - \frac{p_1 p_2}{24}, \quad a_5 = \frac{1}{8} \left( p_4 - \frac{p_2^2}{4} - \frac{p_1 p_3}{3} + \frac{p_1^2 p_2}{12} \right).$$
If $f \in \mathcal{S}_u^*$ is of the form (1.1), then from (2.2) and using [22, Lemma 1], we have
$$|H_{2,1}(f)| = |a_3 - a_2| = \frac{1}{4} |p_2 - p_1| \le \frac{1}{2}.$$
The equality occurs in the above estimate for the function $f_2(z) = ze^z$ . In the following theorem we obtain the sharp bound for the Hankel determinant $|H_{2,2}(f)|$ for functions in the class $\mathcal{S}_{\nu}^*$ .
Theorem 2.1
Theorem 2.1. Let be of the form (1.1). Then and the estimate is sharp.
Theorem 2.1. Let $f \in \mathcal{S}_u^*$ be of the form (1.1). Then
$$|H_{2,2}(f)| \le \frac{1}{4}$$
and the estimate is sharp.
Theorem 2.2
Theorem 2.2. Let be of the form (1.1). Then and the estimate is sharp.
Theorem 2.2. Let $f \in \mathcal{S}_u^*$ be of the form (1.1). Then
$$|H_{3,1}(f)| \le \frac{1}{9}$$
and the estimate is sharp.
Theorem 2.3
Theorem 2.3. Let be of the form (1.1) and is given by (1.6). Then and the estimate is sharp.
Theorem 2.3. Let $f \in \mathcal{S}_u^*$ be of the form (1.1) and $F_f$ is given by (1.6). Then
$$\left| H_{2,1}\left(\frac{\mathbf{F}_f}{2}\right) \right| \le \frac{1}{16},$$
and the estimate is sharp.
Theorem 2.4
Theorem 2.4. Let be of the form (1.1) and be the inverse of f of the form (1.10). Then and the estimate is sharp.
Theorem 2.4. Let $f \in \mathcal{S}_u^*$ be of the form (1.1) and $f^{-1}$ be the inverse of f of the form (1.10). Then
$$\left| H_{2,2}(f^{-1}) \right| \le \frac{5}{12},$$
and the estimate is sharp.
Theorem 2.5 · coeff
Theorem 2.5. Let be of the form (1.1). Then Both the inequalities are sharp.
Theorem 2.5. Let $f \in \mathcal{S}_u^*$ be of the form (1.1). Then
$$-\frac{1}{n-1} \le |a_{n+1}| - |a_n| \le \frac{1}{n} \quad for \ n \ge 2.$$
Both the inequalities are sharp.
Function classes studied:
Coefficient bounds & claims (6)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
H_{2,1}(f) = a_3 - a_2^2 ≤ 1/2 for class S*_u (sharp) [before Theorem 2.1]
coefficient_bound
H_{2,2}(f) = a_2*a_4 - a_3^2 ≤ 1/4 for class S*_u (sharp) [Theorem 2.1]
coefficient_bound
H_{3,1}(f) ≤ 1/9 for class S*_u (sharp) [Theorem 2.2]
coefficient_bound
H_{2,1}(F_f/2) = gamma_1*gamma_3 - gamma_2^2 ≤ 1/16 for class S*_u (sharp) [Theorem 2.3]
coefficient_bound
H_{2,2}(f^{-1}) = A_2*A_4 - A_3^2 ≤ 5/12 for class S*_u (sharp) [Theorem 2.4]
function_family
Class S*_u: f in A: |zf'(z)/f(z) - 1| < 1, i.e., zf'(z)/f(z) subordinate to 1+z
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