On estimation of Hankel determinants for certain class of st

Ma-Minda φ-classes studied in this paper:

Abstract

In the present study, we consider two subclasses starlike and convex functions, denoted by $\mathcal{S}_{\mathcal{B}}^{*}$ and $\mathcal{C}_{\mathcal{B}}$ respectively, associated with a bean-shaped domain. Further, we estimate certain sharp initial coefficients, as well as second, third and fourth-order Hankel determinants for functions belonging to the class $\mathcal{S}_{\mathcal{B}}^{*}$. Additionally, we compute sharp second and third-order Hankel determinants for functions belonging to the

Results & Lemmas (8)

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. [13] Let be of the form. Then and

Lemma 2.1. [13] Let $p \in \mathcal{P}$ be of the form $1 + \sum_{n=1}^{\infty} p_n z^n$ . Then $$|p_1^4 - 3p_1^2p_2 + p_2^2 + 2p_1p_3 - p_4| \le 2 (2.6)$$ and $$|p_3 - 2p_1p_2 + p_1^3| \le 2. (2.7)$$

Lemma 2.2 Lemma 2.2. [14] Let be of the form. Then when or, the equality holds if and only if p(z) = (1+z)/(1-z) or one of its rotations. If, then…

Lemma 2.2. [14] Let $p \in \mathcal{P}$ be of the form $1 + \sum_{n=1}^{\infty} p_n z^n$ . Then $$|p_2 - \beta p_1^2| \le \begin{cases} 2 - 4\beta, & \beta \le 0; \\ 2, & 0 \le \beta \le 1; \\ 4\beta - 2, & \beta \ge 1 \end{cases}$$ when $\beta < 0$ or $\beta > 1$ , the equality holds if and only if p(z) = (1+z)/(1-z) or one of its rotations. If $0 < \beta < 1$ , then the inequality holds if and only if $p(z) = (1+z^2)/(1-z^2)$ or one of its rotations. If $\beta = 0$ , the equality holds if and only if $p(z) = (1+\eta)(1+z)/(2(1-z))+(1-\eta)(1-z)/(2(1+z))(0 \le \eta \le 1)$ or one of its rotations. If $\beta = 1$ , the equality holds if and only if p is the reciprocal of one of the functions such that the equality holds in case of $\beta = 0$ . Though the above upper bound is sharp for $0 < \beta < 1$ , still it can be improved as follows: $$|p_2 - \beta p_1^2| + \beta |p_1|^2 \le 2 \quad (0 < \beta \le 1/2) \tag{2.8}$$ and $$|p_2 - \beta p_1^2| + (1 - \beta)|p_1|^2 \le 2 \quad (1/2 < \beta \le 1).$$ Also, we recall that $$\max_{0 \le t \le 4} (At^2 + Bt + C) = \begin{cases} C, & B \le 0, A \le \frac{-B}{4}; \\ 16A + 4B + C, & B \ge 0, A \ge \frac{-B}{8} & \text{or} \quad B \le 0, A \ge \frac{-B}{4}; \\ \frac{4AC - B^2}{4A}, & B > 0, A \le \frac{-B}{8}. \end{cases}$$ (2.9)

Theorem 2.3 · coeff Theorem 2.3. If, then (i), (ii), (iii) and (iv). These bounds are sharp.

Theorem 2.3. If $f \in \mathcal{S}_{\mathcal{B}}^*$ , then (i) $|a_2| \le 1/2$ , (ii) $|a_3| \le 1/4$ , (iii) $|a_4| \le 1/6$ and (iv) $|a_5| \le 847/3216 \simeq 0.263371 \cdots$ . These bounds are sharp.

Lemma 2.4 Lemma 2.4. [12, 13] Let has the form. Then for some, and such that, and, we have and <span…

Lemma 2.4. [12, 13] Let $p \in \mathcal{P}$ has the form $1 + \sum_{n=1}^{\infty} p_n z^n$ . Then for some $\gamma$ , $\eta$ and $\rho$ such that $|\gamma| \leq 1$ , $|\eta| \leq 1$ and $|\rho| \leq 1$ , we have $$2p_2 = p_1^2 + \gamma(4 - p_1^2), (2.11)$$ $$4p_3 = p_1^3 + 2p_1(4 - p_1^2)\gamma - p_1(4 - p_1^2)\gamma^2 + 2(4 - p_1^2)(1 - |\gamma|^2)\eta, \tag{2.12}$$ and $$8p_4 = p_1^4 + (4 - p_1^2)\gamma(p_1^2(\gamma^2 - 3\gamma + 3) + 4\gamma) - 4(4 - p_1^2)(1 - |\gamma|^2)(p_1(\gamma - 1)\eta + \bar{\gamma}\eta^2 - (1 - |\eta|^2)\rho).$$ (2.13) 2.2. Sharp Hankel determinants for $\mathcal{S}_{\mathcal{B}}$ . The following theorem presents the sharp bound for $|H_3(1)|$ for functions belonging to the class $\mathcal{S}_{\mathcal{B}}$ . Theorem 2.5. Let $f \in \mathcal{S}_{\mathcal{B}}^*$ , then $$|H_3(1)| \le 1/36. \tag{2.14}$$ This result is sharp.

Lemma 2.7 · coeff Lemma 2.7. [8, 18] Let. Then and Now, we try to estimate the bound of sixth and seventh coefficient bounds of function as follows:

Lemma 2.7. [8, 18] Let $$p = 1 + \sum_{n=1}^{\infty} p_n z^n \in \mathcal{P}$$ . Then $$|p_n| \le 2, \quad n \ge 1,$$ $$|p_{n+k} - \nu p_n p_k| \le \begin{cases} 2, & 0 \le \nu \le 1; \\ 2|2\nu - 1|, & otherwise, \end{cases}$$ and $$|p_1^3 - \nu p_3| \le \begin{cases} 2|\nu - 4|, & \nu \le 4/3; \\ 2\nu\sqrt{\frac{\nu}{\nu - 1}}, & 4/3 < \nu. \end{cases}$$ Now, we try to estimate the bound of sixth and seventh coefficient bounds of function $f \in \mathcal{S}_{\mathcal{B}}^*$ as follows:

Lemma 2.8 · coeff Lemma 2.8. Let, then and.

Lemma 2.8. Let $f \in \mathcal{S}_{\mathcal{B}}^*$ , then $|a_6| \le 0.611233$ and $|a_7| \le 0.690994$ .

Theorem 2.9 Theorem 2.9. Let, then.

Theorem 2.9. Let $f \in \mathcal{S}_{\mathcal{B}}^*$ , then $|H_4(1)| \leq 0.169251$ .

Theorem 3.2 Theorem 3.2. Let, then This bound is sharp.

Theorem 3.2. Let $f \in C_B$ , then $$|H_2(3)| \le \frac{1}{576}. (3.15)$$ This bound is sharp.

Function classes studied: