Results & Lemmas (17)
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Theorem 2.1
Theorem 2.1. An analytic function f belongs in if and only if (2.1) for some analytic function q(z) subordinate to.
Theorem 2.1. An analytic function f belongs in $S_k^*$ if and only if
$$f(z) = zexp\left(\int_0^z \frac{q(\tau) - 1}{\tau} d\tau\right), \ (z \in \mathbb{U}),$$
(2.1)
for some analytic function q(z) subordinate to $\varphi_k$ .
Corollary 2.1
Corollary 2.1. An analytic function g belongs in if and only if for some analytic function q(z) subordinate to.
Corollary 2.1. An analytic function g belongs in $C_k$ if and only if
$$g(z) = \int_0^z exp\left(\int_0^w \frac{q(\tau) - 1}{\tau} d\tau\right) dw \ (z \in \mathbb{U}), \tag{2.3}$$
for some analytic function q(z) subordinate to $\varphi_k$ .
Theorem 2.2 · radius
Theorem 2.2. Let,, and |z| = r < 1. Then (i) (ii) (iii) Growth theorems give The equality holds if and only if f (or g) is a rotation of…
Theorem 2.2. Let $f \in \mathcal{S}_k^*$ , $g \in C_k$ , and |z| = r < 1. Then
(i)
$$\frac{zf'(z)}{f(z)} < \frac{zf'_k(z)}{f_k(z)} \text{ and } \frac{f(z)}{z} < \frac{f_k(z)}{z};$$
(ii)
$$\frac{zg''(z)}{g'(z)} < \frac{zg_k''(z)}{g_k'(z)} \text{ and } g'(z) < g_k'(z) .$$
(iii) Growth theorems give
$$-f_k(-r) \le |f(z)| \le f_k(r) \text{ and } -g_k(-r) \le |g(z)| \le g_k(r).$$
The equality holds if and only if f (or g) is a rotation of $f_k$ (or $g_k$ ).
(iv) Rotation theorems give
$$\left| arg\left( \frac{f\left( z \right)}{z} \right) \right| \le \max_{|z|=r} arg\left( \frac{f_k\left( z \right)}{z} \right) \quad and \quad \left| arg\left( g'\left( z \right) \right) \right| \le \max_{|z|=r} arg\left( g'_k\left( z \right) \right).$$
The equality holds if and only if f (or g) is a rotation of $f_k$ (or $g_k$ ).
The following result provides the convexity radius of $\varphi_k(z)$ :
Theorem 2.3
Theorem 2.3. is convex for whenever.
Theorem 2.3. $\varphi_k(z)$ is convex for $|z| < r_0$ whenever $r_0 \approx 0.7293401$ .
Theorem 3.1
Theorem 3.1. The class satisfies the following inclusion relations: - (a) whenever. - (b) whenever. - (c) for. - (d) whenever k > 2.624653.…
Theorem 3.1. The class $S_k^*$ satisfies the following inclusion relations:
- (a) $S_{\iota}^ \subset S^(\alpha) \subset S^*$ whenever $0 \le \alpha \le 0.46732$ .
- (b) $S_{\iota}^* \subset \mathcal{M}(\gamma)$ whenever $\gamma \geq 1.6155$ .
- (c) $S_{\nu}^ \subset SS^(\nu)$ for $0.5869227 \approx \nu_0 \leq \nu < 1$ .
- (d) $k ST \subset S_k^*$ whenever k > 2.624653.
- (e) $S_{ac}^ \subset S_k$ whenever $0 < c \le 0.70689$ .
- (f) $S_{\nu}^ \subset S_{\rho}$ .
- (g) $S_{\vartheta,I}^ \subset S_{\iota}$ whenever $\vartheta \geq 0.53268$ .
Theorem 3.2 · radius
Theorem 3.2. The subsequent results for the radius are valid: (2) (4) (5) (6) (7)
Theorem 3.2. The subsequent results for the radius are valid:
$$(1) \ R_{\mathcal{S}_k} \Big( \mathcal{S}_{\vartheta,e}^ \Big) = \rho_{\vartheta} := \log \left( (1.458603 - \vartheta) / (1 - \vartheta) \right), \ 0 \le \vartheta \le 0.733103.$$
(2)
$$R_{\mathcal{S}_k}(\mathcal{S}_{\vartheta,L}^) = \rho_{\vartheta} := (0.70639 - 0.917206\vartheta)/(1 - \vartheta)^2, \ 0 \le \vartheta \le 0.57728.$$
$$(3)\ R_{\mathcal{S}_k}(\mathcal{BS}^(\vartheta)) = \rho_\vartheta := \left(-1 + \sqrt{1 + 0.841266\vartheta}\right)/0.917206\vartheta, \ 0 \le \vartheta < 1.$$
(4)
$$R_{S_{\nu}}(S_C^) = \rho_C \approx 0.299193.$$
(5)
$$R_{\mathcal{S}_k}(\mathcal{S}_S^) = \rho_S \approx 0.443882.$$
(6)
$$R_{S_{k}^{}}(S_{R}^{}) = \rho_{R} \approx 0.642009.$$
(7)
$$R_{\mathcal{S}_{k}^{}}(\mathcal{S}_{\mathcal{O}}^{}) = \rho_{\mathcal{O}} \approx 0.382541.$$
Lemma 4.1
Lemma 4.1. [13, 33] Let, and then
Lemma 4.1. [13, 33] Let $p \in \mathbb{P}$ , and then
$$|\sigma_n| \le 2, \ n \ge 1,\tag{4.2}$$
$$|\sigma_{r+n} - \beta \sigma_r \sigma_n| < 2, \ 0 < \beta \le 1, \tag{4.3}$$
$$\left|\sigma_2 - \delta\sigma_1^2\right| < 2max\{1, |2\delta - 1|\}, \ \delta \in \mathbb{C}. \tag{4.4}$$
Lemma 4.2
Lemma 4.2. [34] Let with, and let be of the form (4.1). Then
Lemma 4.2. [34] Let $0 \le A \le 1$ with $A(2A - 1) \le B \le A$ , and let $p \in \mathbb{P}$ be of the form (4.1). Then
$$\left| B\sigma_1^3 - 2A\sigma_1\sigma_2 + \sigma_3 \right| \le 2. \tag{4.5}$$
Lemma 4.3
Lemma 4.3. [35, 36] Let be of the form (4.1). Then, for some with max.
Lemma 4.3. [35, 36] Let $p \in \mathbb{P}$ be of the form (4.1). Then,
$$2\sigma_2 = \sigma_1^2 + \gamma \left(4 - \sigma_1^2\right),\tag{4.6}$$
$$4\sigma_3 = \sigma_1^3 + 2\sigma_1 \left(4 - \sigma_1^2\right) \gamma - \sigma_1 \left(4 - \sigma_1^2\right) \gamma^2 + 2\left(4 - \sigma_1^2\right) \left(1 - |\gamma|^2\right) \eta,\tag{4.7}$$
for some $\gamma, \eta \in \mathbb{C}$ with max{ $|\gamma|, |\eta|$ } $\leq 1$ .
Theorem 4.1 · coeff
Theorem 4.1. Let. Then,,, and. These bounds are sharp.
Theorem 4.1. Let $f \in \mathcal{S}_k^*$ . Then,
$$|a_2| \le \frac{2}{3}$$
, $|a_3| \le \frac{1}{3}$ , and $|a_4| \le \frac{2}{9}$ .
These bounds are sharp.
Theorem 4.2 · coeff
Theorem 4.2. Let. Then for, we have The sharpness is given by (4.17) and (4.18).
Theorem 4.2. Let $f \in \mathcal{S}_k^*$ . Then for $\delta \in \mathbb{C}$ , we have
$$\left| a_3 - \delta a_2^2 \right| \le \frac{1}{3} \max \left\{ 1, \ \frac{|8\delta - 5|}{6} \right\}.$$
The sharpness is given by (4.17) and (4.18).
Theorem 4.3 · coeff
Theorem 4.3. Let. Then The sharpness is given by (4.19).
Theorem 4.3. Let $f \in \mathcal{S}_k^*$ . Then
$$|a_2a_3 - a_4| \le \frac{2}{9} \ .$$
The sharpness is given by (4.19).
Theorem 4.4 · coeff
Theorem 4.4. Let. Then The sharpness is given by (4.18).
Theorem 4.4. Let $f \in \mathcal{S}_k^*$ . Then
$$\left| a_2 a_4 - a_3^2 \right| \le \frac{1}{9} \ .$$
The sharpness is given by (4.18).
Theorem 4.5
Theorem 4.5. Let. Then, These bounds are sharp.
Theorem 4.5. Let $f \in \mathcal{S}_k^*$ . Then,
$$|\xi_n| \le \frac{1}{3n} \ (n=1,2,3).$$
These bounds are sharp.
Theorem 4.6
Theorem 4.6. Let. Then for, we have The sharpness is given by (4.17) and (4.18).
Theorem 4.6. Let $f \in \mathcal{S}_k^*$ . Then for $\delta \in \mathbb{C}$ , we have
$$\left|\xi_2 - \delta \xi_1^2\right| \le \frac{1}{6} \max\left\{1, \ \frac{|4\delta - 1|}{6}\right\}.$$
The sharpness is given by (4.17) and (4.18).
Theorem 4.7
Theorem 4.7. Let. Then The sharpness is given by (4.19) and (4.28).
Theorem 4.7. Let $f \in \mathcal{S}_k^*$ . Then
$$|\xi_1 \xi_2 - \xi_3| \le \frac{1}{9} \ .$$
The sharpness is given by (4.19) and (4.28).
Theorem 4.8
Theorem 4.8. Let. Then The sharpness is given by (4.18) and (4.27).
Theorem 4.8. Let $f \in \mathcal{S}_k^*$ . Then
$$\left|\xi_1\xi_3 - \xi_2^2\right| \le \frac{1}{36}.$$
The sharpness is given by (4.18) and (4.27).
Function classes studied:
Coefficient bounds & claims (14)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
|a_2| ≤ 2/3 for class S*_k (sharp) [Theorem 4.1]
coefficient_bound
|a_3| ≤ 1/3 for class S*_k (sharp) [Theorem 4.1]
coefficient_bound
|a_4| ≤ 2/9 for class S*_k (sharp) [Theorem 4.1]
coefficient_bound
|a_3 - delta*a_2^2| ≤ (1/3)*max(1, |8*delta-5|/6) for class S*_k (sharp) [Theorem 4.2]
coefficient_bound
|a_2*a_3 - a_4| ≤ 2/9 for class S*_k (sharp) [Theorem 4.3]
coefficient_bound
|a_2*a_4 - a_3^2| ≤ 1/9 for class S*_k (sharp) [Theorem 4.4]
coefficient_bound
|xi_1| (logarithmic coefficient) ≤ 1/3 for class S*_k (sharp) [Theorem 4.5]
coefficient_bound
|xi_2| (logarithmic coefficient) ≤ 1/6 for class S*_k (sharp) [Theorem 4.5]
coefficient_bound
|xi_3| (logarithmic coefficient) ≤ 1/9 for class S*_k (sharp) [Theorem 4.5]
coefficient_bound
|xi_2 - delta*xi_1^2| (logarithmic Fekete-Szego) ≤ (1/6)*max(1, |4*delta-1|/6) for class S*_k (sharp) [Theorem 4.6]
coefficient_bound
|xi_1*xi_2 - xi_3| ≤ 1/9 for class S*_k (sharp) [Theorem 4.7]
coefficient_bound
|xi_1*xi_3 - xi_2^2| (logarithmic H_2(1)) ≤ 1/36 for class S*_k (sharp) [Theorem 4.8]
function_family
Class S*_k: f in A: zf'(z)/f(z) subordinate to phi_k(z) = 2 - (4/(1+e^{2z^2}))^{1/3}
function_family
Class C_k: f in A: 1 + zf''(z)/f'(z) subordinate to phi_k(z)
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