Starlike And Convex Functions Associated with A Nephroid domain having Cusps On The Real Axis

Theorem 1.1. The function $\varphi_{Ne}(z) = 1 + z - z^3/3$ maps $\mathbb{D}$ onto the region bounded by the nephroid $$\left( (u-1)^2 + v^2 - \frac{4}{9} \right)^3 - \frac{4v^2}{3} = 0,$$ (1.2) which is symmetric about the real axis and lies completely inside the right-half plane u > 0 (see Figure 1).

Lemma 2.1. For 0 < r < 1, the function $\varphi_{Ne}(z)$ satisfies $$\min_{|z|=r} \operatorname{Re}(\varphi_{Ne}(z)) = \begin{cases} 1 - r + \frac{1}{3}r^3, & \text{if } r \leq \frac{1}{\sqrt{3}} \\ 1 - \frac{1}{3}(1 + r^2)^{\frac{3}{2}}, & \text{if } r \geq \frac{1}{\sqrt{3}} \end{cases}$$ and $$\max_{|z|=r} \operatorname{Re}(\varphi_{Ne}(z)) = \begin{cases} 1 + r - \frac{1}{3}r^3, & \text{if } r \leq \frac{1}{\sqrt{3}} \\ 1 + \frac{1}{3}(1 + r^2)^{\frac{3}{2}}, & \text{if } r \geq \frac{1}{\sqrt{3}}. \end{cases}$$

Lemma 2.2. Let 1/3 < a < 5/3. Let $r_a$ and $R_a$ be given by $$r_a = \begin{cases} a - \frac{1}{3}, & \frac{1}{3} < a \le 1\\ \frac{5}{3} - a, & 1 \le a < \frac{5}{3}. \end{cases}$$ and $$R_a = \frac{1}{9} \sqrt{\frac{2(1+18(a-1)^2)^{3/2} + 9a(a-2)(28+9a(a-2)) + 169}{(a-1)^2}}.$$ Then $$\{w \in \mathbb{C} : |w - a| < r_a\} \subseteq \Omega_{Ne} \subseteq \{w \in \mathbb{C} : |w - a| < R_a\}.$$

Theorem 2.1. A function f belongs to the class $\mathcal{S}_{Ne}^*$ if, and only if, there exists an analytic function q(z), satisfying $q(z) \prec \varphi_{Ne}(z) = 1 + z - z^3/3$ such that $$f(z) = z \exp\left(\int_0^z \frac{q(\zeta) - 1}{\zeta} d\zeta\right), \quad z \in \mathbb{D}.$$ (2.1)

Corollary 2.1. A function f belongs to the class $C_{Ne}$ if, and only if, there exists an analytic functions q(z), satisfying $q(z) \prec \varphi_{Ne}(z)$ such that $$f(z) = \int_0^z \left( \exp \int_0^\zeta \frac{q(\eta) - 1}{\eta} d\eta \right) d\zeta, \quad z \in \mathbb{D}.$$ (2.3) Next, consider the analytic functions $q_i: \mathbb{D} \to \mathbb{C}, i = 1, 2, 3, 4, 5$ , defined as follows. $$q_1(z) := \sqrt{1+cz}$$ with $0 < c \le 8/9$ , $q_2(z) := 1 + \frac{z}{2}\sin z$ , $q_3(z) := 1 + \frac{2}{9}ze^z$ , $q_4(z) := \frac{2}{1+e^{-z}}$ and $q_5(z) := \frac{5+3z^2}{5+z^2}$ , for $z \in \mathbb{D}$ . For each i (i = 1, 2, 3, 4, 5), it is easy to verify that $q_i(0) = 1 = \varphi_{Ne}(0)$ and $q_i(\mathbb{D}) \subset \varphi_{Ne}(\mathbb{D})$ . Since the function $\varphi_{Ne}(z) = 1 + z - z^3/3$ is univalent, we conclude that $q_i(z) \prec \varphi_{Ne}(z)$ for every i (i = 1, 2, 3, 4, 5). Consequently, it follows from Theorem 2.1 that the functions $f_i(z)$ (i = 1, 2, 3, 4, 5) given below are members of the class $\mathcal{S}_{Ne}^*$ . $$f_1(z) = \frac{4z \exp\left(2\sqrt{1+cz}-2\right)}{\left(\sqrt{1+cz}+1\right)^2}, \quad 0 < c \le 8/9,$$ $$f_2(z) = z \exp\left(\frac{1-\cos z}{2}\right),$$ $$f_3(z) = z \exp\left(\frac{2e^z-2}{9}\right),$$ $$f_4(z) = z + \frac{z^2}{2} + \frac{z^3}{8} + \frac{z^4}{144} - \frac{5z^5}{1152} - \frac{37z^6}{57600} + \frac{509z^7}{2073600} + \cdots,$$ $$f_5(z) = \frac{z}{5} \left(z^2 + 5\right).$$ In particular, if we choose $q(z) = \varphi_{Ne}(z) = 1 + z - z^3/3$ , then the function $$f_{Ne}(z) := ze^{z-\frac{z^3}{9}} = z + z^2 + \frac{z^3}{2} + \frac{z^4}{18} - \frac{5z^5}{72} - \frac{17z^6}{360} - \frac{71z^7}{6480} + \cdots$$ is a member of the class $\mathcal{S}_{Ne}$ . In the sequel, we will see that the function $f_{Ne}(z) = ze^{z-\frac{z^3}{9}}$ plays the role of an extremal function for several extremal problems in the class $\mathcal{S}_{Ne}$ . Furthermore, the corresponding functions belonging to the class $\mathcal{C}_{Ne}$ can be easily constructed using (2.3). In particular, the function $$\widehat{f}_{Ne}(z) := \int_0^z \frac{f_{Ne}(\zeta)}{\zeta} d\zeta = \int_0^z e^{\zeta - \frac{\zeta^3}{9}} d\zeta = z + \frac{z^2}{2} + \frac{z^3}{6} + \frac{z^4}{72} - \frac{z^5}{72} - \frac{17z^6}{2160} + \cdots$$ is a member of the class $\mathcal{C}_{Ne}$ . The function $\widehat{f}_{Ne}(z)$ plays central role in $\mathcal{C}_{Ne}$ for the extremal problems. Following Ma and Minda [18, Section 3], the following results are easy to establish and hence we omit the

Theorem 2.2. Let $f \in \mathcal{S}_{Ne}^*$ and let |z| = r < 1. Then - (i) Subordination results: $\frac{zf'(z)}{f(z)} \prec \frac{zf'_{Ne}(z)}{f_{Ne}(z)}$ and $\frac{f(z)}{z} \prec \frac{f_{Ne}(z)}{z}$ . (ii) Growth Theorem: $-f_{Ne}(-r) \leq |f(z)| \leq f_{Ne}(r)$ . Equality holds for some non-zero zif, and only if, f is a rotation of $f_{Ne}(z)$ . - (iii) Covering Theorem: Either f is a rotation of $f_{Ne}$ or $f(\mathbb{D})$ contains the disk $\Delta^* = \{w : |w| < -f_{Ne}(-1) = e^{\frac{-8}{9}} \approx 0.411112\}.$ - (iv) Rotation Theorem: $\left|\arg\frac{f(z)}{z}\right| \leq \max_{|z|=r} \arg\left(\frac{f_{Ne}(z)}{z}\right)$ . Equality holds for some non-zero z if, and only if, f is a rotation of $f_{\rm Ne}(z)$  FIGURE 2. Boundary curve of $f_{Ne}(\mathbb{D})$ , where $f_{Ne}(z) = ze^{z-\frac{z^3}{9}}$ .

Theorem 2.3. Let $f \in \mathcal{C}_{Ne}$ and let |z| = r < 1. Then - (i) Subordination results: $\frac{zf'(z)}{f(z)} \prec \frac{z\widehat{f}'_{Ne}(z)}{\widehat{f}_{Ne}(z)}$ and $\frac{f(z)}{z} \prec \frac{\widehat{f}_{Ne}(z)}{z}$ . - (ii) Distortion Theorem: $\hat{f}'_{Ne}(-r) \leq |f'(z)| \leq \hat{f}'_{Ne}(r)$ . - (iii) Growth Theorem: $-\hat{f}_{Ne}(-r) \le |f(z)| \le \hat{f}_{Ne}(r)$ . - (iv) Covering Theorem: Either f is a rotation of $\widehat{f}_{Ne}$ or $f(\mathbb{D})$ contains the disk $\Delta^c=$ $\{w: |w| < -\hat{f}_{Ne}(-1) \approx 0.645159\}.$ (v) Rotation Theorem: $$\left|\arg\frac{f(z)}{z}\right| \leq \max_{|z|=r} \arg\left(\frac{\widehat{f}_{Ne}(z)}{z}\right)$$ . Equality in (ii), (iii) and (iv) holds for some non-zero z if, and only if, f is a rotation of $f_{Ne}(z)$ . Remark 2. In the class $S_{Ne}^*$ , the function $f_{Ne}(z)$ is not an extremal for the distortion theorem. It is evident from the function $f_{Ne}(z)$ itself, as the inequality $$f'_{Ne}(-r) \le |f'_{Ne}(z)| \le f'_{Ne}(r)$$ is not true for $|z| = r > 1/\sqrt{3}$ , whereas for $|z| \le 1/\sqrt{3}$ , the result is true due to Lemma 2.1 and [18, Theorem 2].

Theorem 3.1. The function class $S_{Ne}^*$ satisfies the following inclusion properties: - (a) $\mathcal{S}_{Ne}^ \subset \mathcal{S}^(\alpha)$ whenever $0 \le \alpha \le 1 \frac{2\sqrt{2}}{3} \approx 0.057191$ . (b) $\mathcal{S}_{Ne}^ \subset \mathcal{SS}^(\beta)$ whenever $\beta_0 \le \beta \le 1$ , where $\beta_0 \approx 0.929121$ . - (c) $\mathcal{S}_{q_c}^ \subset \mathcal{S}_{Ne}$ whenever $0 < c \leq \frac{8}{9}$ - (d) $k \mathcal{ST} \subset \mathcal{S}_{Ne}^*$ whenever $k \geq \frac{5}{2}$ - (e) $\mathcal{S}_{\alpha,e}^ \subset \mathcal{S}_{Ne}$ whenever $\alpha \geq 1 \frac{2/3}{e-1} \approx 0.612016$ . (f) $\mathcal{S}_L^(\alpha) \subset \mathcal{S}_{Ne}$ whenever $\alpha \geq \frac{1}{3}$ . The constant in each part is best possible. (a): Let $f \in \mathcal{S}_{Ne}^*$ . Then $zf'(z)/f(z) \prec \varphi_{Ne}(z)$ . In view of Lemma 2.1, it can be easily deduced that $$1 - \frac{2\sqrt{2}}{3} = \min_{|z|=1} \operatorname{Re}\left(\varphi_{Ne}(z)\right) < \operatorname{Re}\left(\frac{zf'(z)}{f(z)}\right).$$ This implies that $f \in \mathcal{S}^(1-2\sqrt{2}/3)$ . Therefore, $\mathcal{S}_{Ne}^ \subset \mathcal{S}^*(1-2\sqrt{2}/3)$ and proves (a) (b): It is clear that $f \in \mathcal{S}_{Ne}^*$ implies $$\left| \arg \left( \frac{zf'(z)}{f(z)} \right) \right| < \max_{|z|=1} \arg \left( \varphi_{Ne}(z) \right)$$ $$= \max_{t \in (-\pi,\pi]} \arg \left( \varphi_{Ne}(e^{it}) \right)$$ $$= \max_{t \in (-\pi,\pi]} \tan^{-1} \left( \frac{\sin t - \frac{1}{3}\sin 3t}{1 + \cos t - \frac{1}{3}\cos 3t} \right).$$ Since $\tan^{-1}\theta$ ( $\theta \in \mathbb{R}$ ) is an increasing function, an easy computation leads us to the problem of finding $\tan^{-1}(\max Q(t))$ , where $Q(t) = 4\sin^3 t/(3+6\cos t-4\cos^3 t)$ . Simple verification shows that Q(t) attains its maximum at $t = \cos^{-1}(-2/3)$ and hence, we conclude that $$\left| \arg \left( \frac{zf'(z)}{f(z)} \right) \right| < \tan^{-1} \left( Q \left( \cos^{-1}(-2/3) \right) \right) \approx 1.45946 \approx (0.929121) \frac{\pi}{2}.$$ This implies that $f \in \mathcal{SS}^*(\beta_0)$ , where $\beta_0 \approx 0.929121$ , and establishes (b). - (c): It has been proved [32] that the function $q_c(z) = \sqrt{1+cz}$ (0 < $c \le 1$ ) maps $\mathbb{D}$ onto the interior of the right loop of the Cassini's curve $|w^2 1| = c$ . A simple analysis shows that $q_c(\mathbb{D})$ is contained in $\Omega_{Ne}$ if, and only if, $\sqrt{1-c} \ge 1/3$ , i.e., if, and only if, $c \le 8/9$ , which is nothing but (c). - (d): If $f \in k \mathcal{ST}$ , then $zf'(z)/f(z) \prec p_k(z)$ , where $$p_k(z) = \frac{1}{1 - k^2} \cosh\left(A \log \frac{1 + \sqrt{z}}{1 - \sqrt{z}}\right) - \frac{k^2}{1 - k^2},$$ with $A = \frac{2}{\pi} \cos^{-1} k$ , $0 \le k < \infty$ . For k > 1, the function $p_k(z)$ maps $\mathbb{D}$ onto the region $\Omega_k$ bounded by the ellipse (cf. [12, p. 331]) $$\frac{\left(u - \frac{k^2}{k^2 - 1}\right)^2}{\left(\frac{k}{k^2 - 1}\right)^2} + \frac{v^2}{\frac{1}{k^2 - 1}} = 1.$$ For the region $\Omega_k$ to lie inside $\Omega_{Ne}$ , we should have $$\frac{k^2}{k^2 - 1} + \frac{k}{k^2 - 1} \le \frac{5}{3}.$$ This gives $k \geq 5/2$ , as in (d) (e): Since the function $\varphi_{\alpha,e}(z) = \alpha + (1-\alpha)e^z$ maps $\mathbb{D}$ onto the region $$\left\{w: \left|\log\left(\frac{w-\alpha}{1-\alpha}\right)\right| < 1\right\}.$$ In view of [14, Lemma 2.1], a simple observation shows that the above region will lie inside $\Omega_{Ne}$ provided $$\alpha + (1 - \alpha)e \le \frac{5}{3}.$$ This on further simplification gives $\alpha \ge 1 - 2/3(e - 1)$ and settles (e). (f): It is easy to verify that the function $\varphi_{L,\alpha}(z) = \alpha + (1-\alpha)\sqrt{1+z}$ maps $\mathbb D$ onto $$\Delta_{L,\alpha}^* := \left\{ w : \left| \left( \frac{w - \alpha}{1 - \alpha} \right)^2 - 1 \right| < 1 \right\}.$$ Again, in view of [14, Lemma 2.1], it follows that $\Delta_{L,\alpha}^*$ lies in the interior of the region $\Omega_{Ne}$ if $\alpha \geq 1/3$ , which establishes (f) and the proof is complete.  FIGURE 3. Boundary curves of best dominants and best subordinants of the nephroid. The following theorem is an immediate consequence of Theorem 3.1 obtained by using the Alexander type relation between $\mathcal{S}_{Ne}^*$ and $\mathcal{C}_{Ne}$ .

Theorem 3.2. The class $C_{Ne}$ satisfies the following properties: - (a) $C_{Ne} \subset C(\alpha)$ , whenever $0 \le \alpha \le 1 \frac{2\sqrt{2}}{3}$ . (b) $C_{Ne} \subset \mathcal{SC}(\beta)$ , whenever $\beta_0 \le \beta \le 1$ , where $\beta_0 \approx 0.929121$ . (c) $C(q_c) \subset C_{Ne}$ , whenever $0 < c \le \frac{8}{9}$ , where $C(q_c)$ is the convex class corresponding to the class $\mathcal{S}_{q_c}^*$ . - (d) $k UCV \subset \mathcal{C}_{Ne}$ , whenever $k \geq \frac{5}{2}$ . (e) $\mathcal{C}_{\alpha,e} \subset \mathcal{C}_{Ne}$ , whenever $\alpha \geq 1 \frac{2/3}{e-1}$ , where $\mathcal{C}_{\alpha,e}$ is the convex class corresponding to - (f) $\mathcal{CL}(\alpha) \subset \mathcal{C}_{Ne}$ , whenever $\alpha \geq \frac{1}{3}$ , where $\mathcal{CL}(\alpha)$ is the convex class corresponding to the class $\mathcal{S}_{I}^{*}(\alpha)$ . The constant in each part is best possible. For $-1 \leq B < A \leq 1$ , let $\mathcal{P}[A, B]$ be the totality of analytic functions $p: \mathbb{D} \to \mathbb{C}$ of the form $p(z) = 1 + \sum_{n=1}^{\infty} p_n z^n$ and satisfying the subordination $p(z) \prec (1 + Az)/(1 + Bz)$ . Further, we set $\mathcal{P}[1-2\alpha,-1]=\mathcal{P}(\alpha)$ $(0 \le \alpha < 1)$ and $\mathcal{P}(0)=\mathcal{P}$ . Also, recall [11] that the Janowski starlike class $\mathcal{S}^*[A,B]$ consists of functions $f \in \mathcal{A}$ satisfying $zf'(z)/f(z) \in$ $\mathcal{P}[A,B].$

Lemma 3.1. ([25, Lemma 2.1]). If $p \in \mathcal{P}[A, B]$ , then for |z| = r < 1 $$\left| p(z) - \frac{1 - ABr^2}{1 - B^2r^2} \right| \le \frac{(A - B)r}{1 - B^2r^2}.$$ For the particular case $p \in \mathcal{P}(\alpha)$ , we have $$\left| p(z) - \frac{1 + (1 - 2\alpha)r^2}{1 - r^2} \right| \le \frac{2(1 - \alpha)r}{1 - r^2}.$$

Theorem 3.3. Let $-1 < B < A \le 1$ . Then $S^*[A, B] \subset S^*_{Ne}$ , if the parameters A and B satisfy any one of the following conditions: $$(1 - B^2) < 3(1 - AB) < 3(1 - B^2)$$ and $(1 - B) < 3(1 - A)$ . (3a) $$3(1-B^2) \le 3(1-AB) < 5(1-B^2) \text{ and } 3(1+A) \le 5(1+B).$$ (3b)

Notation repaired from the source PDF text layer (arxiv_1912_05767.pdf, p. 13; pdftotext (poppler) 25.02.0). Show original OCR text

Theorem 3.3. Let  $-1 < B < A \le 1$ . Then  $S^[A, B] \subset S^_{Ne}$ , if the parameters A and B satisfy any one of the following conditions:

$$(1 - B^2) < 3(1 - AB) < 3(1 - B^2)$$
 and  $(1 - B) < 3(1 - A)$ . (3a)

$$3(1-B^2) \le 3(1-AB) < 5(1-B^2) \text{ and } 3(1+A) \le 5(1+B).$$
 (3b)

Corollary 3.1. Let $-1 < B < A \le 1$ . If either of the conditions (3a)-(3b) mentioned in Theorem 3.3 hold true, then the Janowski convex class C[A, B] is contained in $C_{Ne}$ .

Theorem 4.1. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be a member of the function class $\mathcal{S}_{Ne}^*$ . Then $$\sum_{n=2}^{\infty} \left( (3n)^2 - 7^2 \right) |a_n|^2 \le 40.$$

Corollary 4.1. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{S}_{Ne}^*$ . Then $$|a_n| \le \sqrt{\frac{40}{(3n)^2 - 7^2}}$$ for $n = 3, 4, \dots$ Using the fact that $f \in \mathcal{C}_{Ne} \iff zf'(z) \in \mathcal{S}_{Ne}^*$ , the following result is an immediate consequence of Theorem 4.1.

Theorem 4.2. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{C}_{Ne}$ . Then $$\sum_{n=2}^{\infty} n^2 \left( (3n)^2 - 7^2 \right) |a_n|^2 \le 40.$$

Corollary 4.2. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{C}_{Ne}$ . Then $$|a_n| \le \frac{1}{n} \sqrt{\frac{40}{(3n)^2 - 7^2}}$$ for $n = 3, 4, \dots$ Remark 3. It is clear that $\varphi_{Ne}(z) = 1 + z - z^3/3 \in \mathcal{H}^2$ , the Hardy class of analytic functions in $\mathbb{D}$ , defined as $$\mathcal{H}^2 = \left\{ \sum_{n=0}^{\infty} b_n z^n : \sum_{n=0}^{\infty} |b_n|^2 < \infty \right\}.$$ Therefore, it may be concluded from Ma and Minda [18, Theorem 4] that for a function $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{S}_{Ne}^*$ (or $\mathcal{C}_{Ne}$ ) the sharp order of growth is $|a_n| = O(1/n)$ (or $O(1/n^2)$ ).

Theorem 4.3. (i) Let $F_n(z) = z + a_n z^n$ (n = 2, 3, ...). Then $F_n \in \mathcal{S}_{Ne}^*$ if, and only if, $$|a_n| \le \frac{2}{3n-1}, \quad n = 2, 3, \dots$$ - (ii) The function $K_A(z) = z/(1-Az)^2$ belongs to $\mathcal{S}_{Ne}$ if, and only if, $|A| \leq 1/4$ . (iii) Let |b| < 1. Then $L_b(z) = ze^{bz} \in \mathcal{S}_{Ne}$ if, and only if, $|b| \leq 2/3$ .

Lemma 4.1. Let $p(z) = 1 + \sum_{n=1}^{\infty} p_n z^n \in \mathcal{P}$ . Then [18] $$|p_n| \le 2 \text{ for } n \ge 1, \tag{4.5}$$ and for any complex number $\xi$ , we have $$|p_2 - \xi p_1^2| \le 2 \max\{1, |2\xi - 1|\}. \tag{4.6}$$ The results are sharp for the functions $$p(z) = \frac{1+z}{1-z}, \quad p(z) = \frac{1+z^2}{1-z^2}.$$ (i) If $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots \in S_{Ne}^*$ , then $|a_2| \le 1$ , $|a_3| \le 1/2$ , and for any complex number $\mu$ , $$|a_3 - \mu a_2^2| \le \frac{1}{2} \max\{1, |2\mu - 1|\}.$$ In particular, $|a_3 - a_2^2| \le 1/2$ . Equality holds for the function $f_{Ne}(z)$ . (ii) If $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots \in \mathcal{C}_{Ne}$ , then $|a_2| \le 1/2$ and $|a_3| \le 1/6$ . Equality holds for the function $\hat{f}_{Ne}(z)$ . Further, for any complex number $\mu$ , $$|a_3 - \mu a_2^2| \le \frac{1}{6} \max \left\{ 1, \left| \frac{3}{2}\mu - 1 \right| \right\}.$$ (i) Since $f \in \mathcal{S}_{Ne}^*$ , we can find a Schwarz function $w(z) = w_1 z + w_2 z^2 + w_3 z^3 + \cdots$ such that $$\frac{zf'(z)}{f(z)} = \varphi_{Ne}(w(z)) = 1 + w(z) - \frac{1}{3}(w(z))^3.$$ (4.7) Taking w(z) = (p(z) - 1)/(p(z) + 1), where $p(z) = 1 + p_1 z + p_2 z^2 + p_3 z^3 + \dots \in \mathcal{P}$ , and doing some simple calculations we obtain $$w(z) = \frac{p_1 z}{2} + \left(\frac{p_2}{2} - \frac{p_1^2}{4}\right) z^2 + \frac{1}{8} \left(p_1^3 - 4p_2 p_1 + 4p_3\right) z^3$$ $$+\frac{1}{16}\left(-p_1^4+6p_2p_1^2-8p_3p_1-4p_2^2+8p_4\right)z^4+\cdots$$ This gives $$1 + w(z) - \frac{1}{3}(w(z))^{3} = 1 + \frac{p_{1}}{2}z + \left(\frac{p_{2}}{2} - \frac{p_{1}^{2}}{4}\right)z^{2} + \left(\frac{p_{1}^{3}}{12} - \frac{p_{1}p_{2}}{2} + \frac{p_{3}}{2}\right)z^{3} + \left(\frac{1}{4}p_{1}^{2}p_{2} - \frac{p_{1}p_{3}}{2} - \frac{p_{2}^{2}}{4} + \frac{p_{4}}{2}\right)z^{4} + \cdots$$ $$(4.8)$$ Also it is easy to check that for $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots$ , $$\frac{zf'(z)}{f(z)} = 1 + a_2 z + \left(2a_3 - a_2^2\right) z^2 + \left(a_2^3 - 3a_2 a_3 + 3a_4\right) z^3 + \left(-a_2^4 + 4a_2^2 a_3 - 4a_2 a_4 - 2a_3^2 + 4a_5\right) z^4 + \cdots$$ (4.9) Using (4.8) and (4.9) in the equation (4.7), and comparing coefficients we get $$a_2 = \frac{p_1}{2}, \quad a_3 = \frac{p_2}{4}.$$ (4.10) Applying (4.5) in (4.10), we obtain $|a_2| \le 1$ and $|a_3| \le 1/2$ . Also, from (4.10) and (4.6), we have $$|a_3 - \mu a_2^2| = \frac{1}{4}|p_2 - \mu p_1^2| \le \frac{1}{2}\max\{1, |2\mu - 1|\}.$$ (ii) Similarly, $f \in \mathcal{C}_{Ne}$ implies $$1 + \frac{zf''(z)}{f'(z)} = 1 + w(z) - \frac{1}{3}(w(z))^3. \tag{4.11}$$ Using (4.8) and the expansion $$1 + \frac{zf''(z)}{f'(z)} = 1 + 2a_2z + \left(6a_3 - 4a_2^2\right)z^2 + \left(8a_2^3 - 18a_3a_2 + 12a_4\right)z^3 + \left(-16a_2^4 + 48a_3a_2^2 - 32a_4a_2 - 18a_3^2 + 20a_5\right)z^4 + \cdots$$ in (4.11), and comparing we get $a_2 = p_1/4$ and $a_3 = p_2/12$ . Now, as in part (i), the results easily follow. Remark 4. From Theorem 3.1 (a), we have $S_{Ne}^ \subset S^(1-2\sqrt{2}/3)$ . A result in Schild [27] yields that for every function $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ in $S_{Ne}^*$ , we have $$|a_n| \le \frac{1}{(n-1)!} \prod_{k=2}^n \left(k-2 + \frac{4\sqrt{2}}{3}\right), \quad n \ge 2.$$ Further, Alexander type theorem shows that if $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{C}_{Ne}$ , then $$|a_n| \le \frac{1}{n!} \prod_{k=2}^n \left(k - 2 + \frac{4\sqrt{2}}{3}\right), \quad n \ge 2.$$

Lemma 5.1. ([21, Theorem 3.4h, p. 132]). Let q be univalent in $\mathbb{D}$ , and let $\psi$ and $\nu$ be analytic in a domain D containing $q(\mathbb{D})$ with $\psi(w) \neq 0$ when $w \in q(\mathbb{D})$ . Set $$Q(z) := zq'(z)\psi\Big(q(z)\Big) \quad \text{ and } \quad h(z) := \nu\Big(q(z)\Big) + Q(z).$$ Suppose that either - (i) h is convex, or - (ii) Q is starlike. In addition, assume that - (iii) $\operatorname{Re}\left(\frac{zh'(z)}{Q(z)}\right) > 0.$ If $p \in \mathcal{H}$ with p(0) = q(0), $p(\mathbb{D}) \subset D$ and $$\nu(p(z)) + zp'(z)\psi(p(z)) \prec \nu(q(z)) + zq'(z)\psi(q(z)),$$ then $p \prec q$ , and q is the best dominant. Implications of $1 + \beta z p'(z) \prec \varphi_{Ne}(z)$ .

Theorem 5.1. Let $p \in \mathcal{H}$ such that p(0) = 1, and let $1 + \beta z p'(z) \prec \varphi_{Ne}(z)$ . Then the following subordinations hold: - (a) $p(z) \prec \varphi_{\mathbb{C}}(z) = \sqrt{1+z^2} + z \text{ whenever } \beta \geq \frac{4}{9} (2 + \sqrt{2}) \approx 1.51743.$ - (a) p(z) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0) + p(0), p(0) + p(0) + p(0), p(0) + p(0) + p(0), p(0) + p(0) + p(0), p(0) + p(0) + p(0), p(0) + p(0) + p(0), p(0) + p(0) + p(0), p(0) + p(0) + p(0), p(0) + p(0) + p(0), p(0) + p(0) + p(0), p(0) + p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), p(0) + p(0), - (e) $p(z) \prec \varphi_s(z) = 1 + \sin z$ whenever $\beta \ge \frac{8}{9 \sin 1} \approx 1.05635$ . - (f) $p(z) \prec \varphi_0(z) = 1 + \frac{z}{k} \left( \frac{k+z}{k-z} \right)$ $(k = 1 + \sqrt{2})$ whenever $\beta \ge \frac{8}{9} \left( 3 + 2\sqrt{2} \right) \approx 5.18082$ . - (g) $p(z) \prec \varphi_{Ne}(z)$ whenever $\beta \geq \frac{4}{3}$ . The bounds on $\beta$ are sharp.

Corollary 5.1. If $f \in A$ satisfies $$1 + \beta \frac{zf'(z)}{f(z)} \left( 1 - \frac{zf'(z)}{f(z)} + \frac{zf''(z)}{f'(z)} \right) \prec \varphi_{Ne}(z),$$ (a) $$f \in \mathcal{S}_{\mathbb{Q}}^*(z)$$ whenever $\beta \geq \frac{8}{9(2-\sqrt{2})}$ . - (b) $f \in \mathcal{S}_C^*$ whenever $\beta \geq \frac{4}{3}$ . - (c) $f \in \mathcal{S}_{lim}$ whenever $\beta \geq \frac{\pi}{3}$ . (d) $f \in \mathcal{S}_{e}$ whenever $\beta \geq \frac{8e}{9(e-1)}$ . (e) $f \in \mathcal{S}_{S}^*$ whenever $\beta \geq \frac{8}{9\sin 1}$ . - (f) $f \in \mathcal{S}_R^*$ whenever $\beta \ge \frac{8}{9} \left( 3 + 2\sqrt{2} \right)$ . - (g) $f \in \mathcal{S}_{Ne}^*$ whenever $\beta \geq \frac{4}{3}$ . The bounds on $\beta$ are best possible.  $\Gamma_{Cr}: |w^2 - 1| = 2|w|, \operatorname{Re} w > 0.$ $\Gamma_{\beta}$ : Boundary curve of $q_{\beta}(\mathbb{D})$ , where $q_{\beta}(z) = 1 + \frac{1}{\beta} \left( z - \frac{z^3}{9} \right)$ with $\beta = 4(2 + \sqrt{2})/9$ . $$c = \sqrt{2} - 1.$$ $$e = \sqrt{2} + 1.$$ FIGURE 4. Sharpness of $\beta = \frac{4}{9} \left( 2 + \sqrt{2} \right)$ in Theorem 5.1(a). Implications of $1 + \beta \frac{zp'(z)}{p(z)} \prec \varphi_{Ne}(z)$ .

Theorem 5.2. Let $p \in \mathcal{H}$ satisfies p(0) = 1, and let $$1 + \beta \frac{zp'(z)}{p(z)} \prec \varphi_{Ne}(z).$$ Then the following subordinations hold: - (a) $p(z) \prec \varphi_s(z)$ whenever $\beta \ge \frac{8/9}{\log(1+\sin 1)} \approx 1.45585$ . (b) $p(z) \prec \varphi_0(z)$ whenever $\beta \ge -\frac{8/9}{\log(2\sqrt{2}-2)} \approx 4.72245$ . - (c) $p(z) \prec \varphi_{Ne}(z)$ whenever $\beta \geq \frac{8/9}{\log(5/3)}$ The bounds on $\beta$ are best possible.

Corollary 5.2. If $f \in A$ satisfies $$1 + \beta \left( 1 - \frac{zf'(z)}{f(z)} + \frac{zf''(z)}{f'(z)} \right) \prec \varphi_{Ne}(z),$$ then - (a) $f \in \mathcal{S}_S^*$ whenever $\beta \ge \frac{8/9}{\log(1+\sin 1)}$ - (b) $f \in \mathcal{S}_R^*$ whenever $\beta \ge -\frac{8/9}{\log(2\sqrt{2}-2)}$ . - (c) $f \in \mathcal{S}_{Ne}^*$ whenever $\beta \geq \frac{8/9}{\log(5/3)}$ The bounds on $\beta$ are best possible. Implications of $1 + \beta \frac{zp'(z)}{n^2(z)} \prec \varphi_{Ne}(z)$ .

Theorem 5.3. Let $p \in \mathcal{H}$ such that p(0) = 1, and let $$1 + \beta \frac{zp'(z)}{p^2(z)} \prec \varphi_{Ne}(z).$$ Then the following subordinations hold: - (a) $p(z) \prec \varphi_{\mathcal{O}}(z)$ whenever $\beta \geq \frac{4}{9} \left(2 + \sqrt{2}\right) \approx 1.51743$ . - (b) $p(z) \prec \varphi_C(z)$ whenever $\beta \geq \frac{4}{3}$ . - (c) $p(z) \prec \varphi_{lim}(z)$ whenever $\beta \geq \frac{8}{63} \left(5 + 4\sqrt{2}\right) \approx 1.35325$ . - (d) $p(z) \prec e^z$ whenever $\beta \geq \frac{8e}{9(e-1)} \approx 1.4062$ . (e) $p(z) \prec \varphi_s(z)$ whenever $\beta \geq \frac{8(1+\sin 1)}{9\sin 1} \approx 1.94524$ . (f) $p(z) \prec \varphi_0(z)$ whenever $\beta \geq \frac{16}{9} \left(1 + \sqrt{2}\right) \approx 4.29194$ . (g) $p(z) \prec \varphi_{Ne}(z)$ whenever $\beta \geq \frac{20}{9}$ . The bounds on $\beta$ are best possible.

Definition 5.1. ([21]). Let $\Psi : \mathbb{C}^2 \times \mathbb{D} \to \mathbb{C}$ be a complex function, and let $h : \mathbb{D} \to \mathbb{C}$ be univalent. If $p \in \mathcal{H}$ satisfies the first-order differential subordination $$\Psi(p(z), zp'(z); z) \prec h(z), \quad z \in \mathbb{D}, \tag{5.1}$$ then p is called a solution of the differential subordination (5.1). If $q: \mathbb{D} \to \mathbb{C}$ is univalent and $p \prec q$ for all p satisfying (5.1), then q(z) is said to be a dominant of (5.1). A dominant $\tilde{q}$ that satisfies $\tilde{q} \prec q$ for all dominants q of (5.1) is called the best dominant of (5.1). The best dominant is unique up to the rotations of $\mathbb{D}$ . Let p(z) be an analytic function satisfying p(0) = 1. Nunokawa et al. [22] proved that the differential subordination $1 + zp'(z) \prec 1 + z$ implies $p(z) \prec 1 + z$ and, as a result, gave a criterion for $f \in \mathcal{A}$ to be univalent. Ali et al. [2] replaced 1 + z by (1 + Dz)/(1 + Ez) and determined the conditions on $\beta \in \mathbb{R}$ in terms of $A, B, D, E \in [-1, 1]$ (B < A and E < D) so that the subordination $1 + \beta zp'(z)/p^j(z) \prec (1 + Dz)/(1 + Ez)$ (j = 0, 1, 2) implies $p(z) \prec (1 + Az)/(1 + Bz)$ and, further, established some sufficient conditions for Janowski starlikeness. Ali et al. [3] determined the conditions on $\beta$ so that the subordination $p(z) \prec \sqrt{1 + z}$ holds true whenever $1 + \beta zp'(z)/p^j(z) \prec \sqrt{1 + z}$ , j = 0, 1, 2. Kumar and Ravichandran [16] gave sharp bounds for $\beta$ in order to guarantee the following subordination implication: $$1 + \beta \frac{zp'(z)}{p^j(z)} \prec \varphi_0(z), \sqrt{1+z}, 1 + \sin z \Longrightarrow p(z) \prec e^z.$$ As a consequence, they also established some sufficient conditions for the function class $S_e^*$ . Recently, Ahuja et al. [1] obtained sharp bounds on $\beta$ so that the differential subordination $1 + \beta z p'(z)/p^j(z) \prec \sqrt{1+z}$ (j=0,1,2) implies $p(z) \prec$ : $\sqrt{1+z}$ , $\varphi_0(z)$ , $1 + \sin z$ , $\sqrt{1+z^2}+z$ , $1+\frac{4}{3}z+\frac{2}{3}z^2$ , $\frac{1+Az}{1+Bz}$ , where -1 < A < B < 1. Similar problems have been considered in [5, 8, 30]. In this section, we will discuss certain implications of the first order differential subordination $1 + \beta z p'(z)/p^j(z) \prec \varphi_{N_e}(z) = 1 + z - z^3/3$ , j = 0, 1, 2. As a consequence, we obtain sufficient conditions for $f \in \mathcal{A}$ to belong to some of the early mentioned Ma-Minda type function classes. To prove our results, we will make use of the following lemma.