Differential Subordinations for Starlike Functions Associated With A Nephroid Domain

Lemma 1.1. ([20, Theorem 3.4h, p. 132]). Let $q : \mathbb{D} \to \mathbb{C}$ be univalent, and let $\lambda$ and $\vartheta$ be analytic in a domain $\Omega \supseteq q(\mathbb{D})$ with $\lambda(\xi) \neq 0$ whenever $\xi \in q(\mathbb{D})$ . Define $$\Theta(z) := zq'(z) \lambda(q(z))$$ and $h(z) := \vartheta(q(z)) + \Theta(z),$ $z \in \mathbb{D}.$ Suppose that either - (i) h(z) is convex, or - (ii) $\Theta(z)$ is starlike. In addition, assume that (iii) Re $(zh'(z)/\Theta(z)) > 0$ in $\mathbb{D}$ . If $p \in \mathcal{H}$ with p(0) = q(0), $p(\mathbb{D}) \subset \Omega$ and $$\vartheta(p(z)) + zp'(z)\lambda(p(z)) \prec \vartheta(q(z)) + zq'(z)\lambda(q(z)), \qquad z \in \mathbb{D},$$ then $p \prec q$ , and q is the best dominant. In the sequel, it is always assumed that $z \in \mathbb{D}$ unless stated otherwise.

Theorem 2.1. Let $p \in \mathcal{H}$ satisfies p(0) = 1, and let $\varphi_L(z) := \sqrt{1+z}$ , where the branch of the square root is chosen in order that $\varphi_L(0) = 1$ . Then each of the following subordinations imply $p(z) \prec \varphi_{N_e}(z) := 1 + z - z^3/3$ . - (a) $1 + \beta z p'(z) \prec \varphi_L(z)$ for $\beta \ge 3(1 \log 2) \approx 0.920558$ . (b) $1 + \beta \left(\frac{z p'(z)}{p(z)}\right) \prec \varphi_L(z)$ for $\beta \ge \frac{2\left(\sqrt{2} + \log 2 1 \log(1 + \sqrt{2})\right)}{\log(5/3)} \approx 0.884792$ . (c) $1 + \beta \left(\frac{z p'(z)}{p^2(z)}\right) \prec \varphi_L(z)$ for $\beta \ge 5\left(\sqrt{2} + \log 2 1 \log(1 + \sqrt{2})\right) \approx 1.12994$ . Each estimate on $\beta$ is sharp.

Corollary 2.1. Let $f \in A$ , and let $$\mathcal{G}(z) := 1 - \frac{zf'(z)}{f(z)} + \frac{zf''(z)}{f'(z)}, \quad z \in \mathbb{D}.$$ $$(2.3)$$ Then each of the following is sufficient to imply $f \in \mathcal{S}_{Ne}^*$ . (a) $$1 + \beta \mathcal{G}(z) \left( \frac{zf'(z)}{f(z)} \right) \prec \varphi_L(z)$$ for $\beta \ge 3(1 - \log 2)$ , (b) $$1 + \beta \mathcal{G}(z) \prec \varphi_L(z) \text{ for } \beta \ge \frac{2(\sqrt{2} + \log 2 - \log(1 + \sqrt{2}) - 1)}{\log(5/3)}$$ , (b) $$1 + \beta \mathcal{G}(z) \prec \varphi_L(z)$$ for $\beta \geq \frac{2\left(\sqrt{2} + \log 2 - \log(1 + \sqrt{2}) - 1\right)}{\log(5/3)}$ , (c) $1 + \beta \mathcal{G}(z) \left(\frac{zf'(z)}{f(z)}\right)^{-1} \prec \varphi_L(z)$ for $\beta \geq 5\left(\sqrt{2} + \log 2 - \log(1 + \sqrt{2}) - 1\right)$ . The bounds on $\beta$ are best possible.  $\gamma_{Ne}$ : Boundary curve of $\varphi_{Ne}(\mathbb{D})$ . $\hat{\gamma}_{\beta}$ : Boundary curve of $\hat{q}_{\beta}(\mathbb{D})$ with $\beta = \frac{2(\sqrt{2}-1+\log 2-\log(1+\sqrt{2}))}{\log(5/3)}$ . $\tilde{\gamma}_{\beta}$ : Boundary curve of $\tilde{q}_{\beta}(\mathbb{D})$ with $\beta = 5 \left( \sqrt{2} - 1 + \log 2 - \log \left( 1 + \sqrt{2} \right) \right).$ Figure 4. The functions $\hat{q}_{\beta}$ and $\tilde{q}_{\beta}$ are, respectively, the functions defined in (b) and (c) of Theorem 2.1.

Corollary 2.2. Let $f \in \mathcal{A}$ and let $\mathcal{G}(z)$ be given by (2.3). Then each of the following conditions imply $f \in \mathcal{S}_{Ne}^*$ . (a) $$1 + \beta \mathcal{G}(z) \left( \frac{zf'(z)}{f(z)} \right) \prec \varphi_{RL}(z) \text{ for } \beta \ge -\frac{3\left(2(\sqrt{2}-1)\log\left(\frac{1}{2}\left(\sqrt{1-2(\sqrt{2}-1)}+\sqrt{2}\right)\right)+g_0(-1)\right)}{2},$$ (b) $$1 + \beta \mathcal{G}(z) \prec \varphi_{RL}(z) \text{ for } \beta \geq \frac{2(\sqrt{2}-1)\log(\frac{1}{2}\sqrt{2(\sqrt{2}-1)+1}) + g_0(1)}{\log(\frac{5}{3})},$$ (c) $$1 + \beta \mathcal{G}(z) \left(\frac{zf'(z)}{f(z)}\right)^{-1} \prec \varphi_{RL}(z) \text{ for } \beta \ge \frac{5\left(2\left(\sqrt{2}-1\right)\log\left(\frac{1}{2}\sqrt{2\left(\sqrt{2}-1\right)+1}\right)+g_0(1)\right)}{2}$$ . The bounds on $\beta$ are sharp.

Theorem 2.3. Let $\varphi_{SG}(z) := 2/(1 + e^{-z})$ , and let $$\ell(z) = \int_0^z \frac{e^t - 1}{t(e^t + 1)} dt.$$ Then, for $p \in \mathcal{H}$ with p(0) = 1, each of the following differential subordinations is sufficient for the subordination $p \prec \varphi_{Ne}$ : - (a) $1 + \beta z p' \prec \varphi_{SG}(z)$ for $\beta \ge 3\ell(1)/2 \approx 0.730333$ , - (b) $1 + \beta z p'/p \prec \varphi_{SG}(z)$ for $\beta \geq \ell(1)/\log(5/3) \approx 0.953141$ , (c) $1 + \beta z p'/p^2 \prec \varphi_{SG}(z)$ for $\beta \geq 5\ell(1)/2 \approx 1.21722$ . The bounds on $\beta$ can not be improved further.

Theorem 2.4. Let $\varphi_e(z) := e^z$ be the exponential function, and let p(z) be analytic such that p(0) = 1. If any one of the following differential subordinations hold true, then $p \prec \varphi_{Ne}$ . Each estimate on $\beta$ is sharp. - (a) $1 + \beta z p' \prec \varphi_e(z)$ for $\beta \geq \sum_{n=1}^{\infty} \frac{3}{2n(n!)} \approx 1.97685$ , (b) $1 + \beta z p'/p \prec \varphi_e(z)$ for $\beta \geq \frac{\sum_{n=1}^{\infty} \frac{1}{n(n!)}}{\log(5/3)} \approx 2.57995$ , (c) $1 + \beta z p'/p^2 \prec \varphi_e(z)$ for $\beta \geq \sum_{n=1}^{\infty} \frac{5}{2n(n!)} \approx 3.29476$ .

Theorem 3.1. Let $p \in \mathcal{H}$ with p(0) = 1, and let $\varphi_{\mathcal{C}}(z) := z + \sqrt{1 + z^2}$ . If any of the following differential subordinations hold true, then $p \prec \varphi_{Ne}$ . $$\begin{array}{l} \text{(a)} \ 1+\beta z p'(z) \prec \varphi_{\mathbb{C}}\left(z\right) \ for \ \beta \geq \frac{3\left(\sqrt{2}-\log\left(1+\sqrt{2}\right)+\log 2\right)}{2} \approx 1.83898, \\ \text{(b)} \ 1+\beta\left(\frac{z p'(z)}{p(z)}\right) \prec \varphi_{\mathbb{C}}\left(z\right) \ for \ \beta \geq \frac{\sqrt{2}+\log(2)-\log\left(1+\sqrt{2}\right)}{\log\left(\frac{5}{3}\right)} \approx 2.40001, \end{array}$$ (b) $$1 + \beta \left(\frac{zp'(z)}{p(z)}\right) \prec \varphi_{\mathbb{C}}(z) \text{ for } \beta \geq \frac{\sqrt{2} + \log(2) - \log\left(1 + \sqrt{2}\right)}{\log\left(\frac{5}{3}\right)} \approx 2.40001,$$ (c) $$1 + \beta \left(\frac{zp'(z)}{p^2(z)}\right) \prec \varphi_{\mathfrak{C}}(z)$$ for $\beta \geq \frac{5\left(\sqrt{2} - \log\left(1 + \sqrt{2}\right) + \log 2\right)}{2} \approx 3.06497$ . Each estimate is sharp.

Corollary 3.1. Let $f \in A$ and let G(z) be defined by (2.3). Then each of the following conditions sufficiently implies that f is a member of $\mathcal{S}_{Ne}^*$ . (a) $$1 + \beta \mathcal{G}(z)(zf'/f) \prec \varphi_{\mathfrak{C}}(z)$$ for $\beta \geq \frac{3(\sqrt{2} - \log(1 + \sqrt{2}) + \log 2)}{2}$ (a) $$1 + \beta \mathcal{G}(z)(zf'/f) \prec \varphi_{\mathbb{C}}(z)$$ for $\beta \geq \frac{3(\sqrt{2} - \log(1 + \sqrt{2}) + \log 2)}{2}$ , (b) $1 + \beta \mathcal{G}(z) \prec \varphi_{\mathbb{C}}(z)$ for $\beta \geq \frac{\sqrt{2} + \log(2) - \log(1 + \sqrt{2})}{\log(\frac{5}{3})}$ , (c) $$1 + \beta \mathcal{G}(z) (zf'/f)^{-1} \prec \varphi_{\mathfrak{C}}(z)$$ for $\beta \geq \frac{5(\sqrt{2} - \log(1 + \sqrt{2}) + \log 2)}{2}$ . Each estimate on $\beta$ is sharp.  1: Boundary of $$q_{\beta}(\mathbb{D})$$ with $\beta = \frac{3(\sqrt{2} - \log(1 + \sqrt{2}) + \log 2)}{2}$ . 2: Boundary of $$\hat{q}_{\beta}(\mathbb{D})$$ with $\beta = \frac{\sqrt{2} + \log(2) - \log(1 + \sqrt{2})}{\log(\frac{5}{3})}$ . 3: Boundary of $$\tilde{q}_{\beta}(\mathbb{D})$$ with $\beta = \frac{5(\sqrt{2} - \log(1 + \sqrt{2}) + \log 2)}{2}$ . Figure 6. The functions $q_{\beta}$ , $\hat{q}_{\beta}$ and $\tilde{q}_{\beta}$ are defined in (a), (b) and (c) of Theorem 3.1.

Theorem 3.2. Let $\varphi_C(z) := 1 + 4z/3 + 2z^2/3$ . Then, for $p \in \mathcal{H}$ with p(0) = 1, each of the following subordinations is sufficient to imply that $p \prec \varphi_{Ne}$ . Moreover, each estimate on $\beta$ is sharp. - (a) $1 + \beta z p' \prec \varphi_C(z)$ for $\beta \geq 5/2$ , - (b) $1 + \beta z p'/p \prec \varphi_C(z)$ for $\beta > 5/3 \log(5/3) \approx 3.26269$ , - (c) $1 + \beta z p'/p^2 \prec \varphi_C(z)$ for $\beta > 25/6$ .

Theorem 3.3. Let $p \in \mathcal{H}$ with p(0) = 1, and let $$\varphi_0(z) := 1 + \frac{z}{k} \left( \frac{k+z}{k-z} \right), \quad k = \sqrt{2} + 1.$$ If any one of the following differential subordinations hold true, then $p \prec \varphi_{Ne}$ . Each of the respective bounds on $\beta$ is best possible. (a) $$1 + \beta z p' \prec \varphi_0 \text{ for } \beta \ge 3 \log \left(1 + \frac{1}{\sqrt{2}}\right) - \frac{3}{2} \left(\sqrt{2} - 1\right) \approx 0.98308,$$ (b) $$1 + \beta z p'/p \prec \varphi_0 \text{ for } \beta \ge \frac{2\left(\log\left(\frac{1+\sqrt{2}}{\sqrt{2}}\right) - \frac{1}{2(1+\sqrt{2})}\right)}{\log\left(\frac{5}{3}\right)} \approx 1.28299,$$ (c) $$1 + \beta z p'/p^2 \prec \varphi_0 \text{ for } \beta \ge 5 \left( \log \left( \frac{1 + \sqrt{2}}{\sqrt{2}} \right) - \frac{1}{2(1 + \sqrt{2})} \right) \approx 1.63847.$$

Corollary 3.2. Let $f \in A$ and G(z) be given by (2.3). If any one of the following (a) $$\left| \left( \frac{zf'(z)}{f(z)} \right) \mathcal{G}(z) \right| \le \frac{3 - 2\sqrt{2}}{3 \log\left( \frac{1 + \sqrt{2}}{\sqrt{2}} \right) - \frac{3}{2(1 + \sqrt{2})}} \approx 0.174526$$ (b) $$|\mathcal{G}(z)| \le \frac{\left(3 - 2\sqrt{2}\right)\log(5/3)}{2\left(\log\left(\frac{1+\sqrt{2}}{\sqrt{2}}\right) - \frac{1}{2\left(1+\sqrt{2}\right)}\right)} \approx 0.133728$$ (b) $$|\mathcal{G}(z)| \le \frac{\left(3 - 2\sqrt{2}\right)\log(5/3)}{2\left(\log\left(\frac{1 + \sqrt{2}}{\sqrt{2}}\right) - \frac{1}{2(1 + \sqrt{2})}\right)} \approx 0.133728$$ (c) $\left|\left(\frac{zf'(z)}{f(z)}\right)^{-1}\mathcal{G}(z)\right| \le \frac{3 - 2\sqrt{2}}{5\left(\log\left(\frac{1 + \sqrt{2}}{\sqrt{2}}\right) - \frac{1}{2(1 + \sqrt{2})}\right)} \approx 0.104716.$ holds true, then $f \in \mathcal{S}_{Ne}^*$ .

Theorem 3.4. Let $\varphi_s(z) := 1 + \sin z$ , and let $p \in \mathcal{H}$ satisfies p(0) = 1. Each of the following subordinations imply $p \prec \varphi_{Ne}$ : (a) $$1 + \beta z p' \prec \varphi_s(z)$$ for $\beta \ge \frac{3}{2} \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)! \times (2n+1)} \approx 1.41912$ , (a) $$1 + \beta z p' \prec \varphi_S(z)$$ for $\beta \ge \frac{3}{2} \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)! \times (2n+1)} \approx 1.41912$ , (b) $1 + \beta z p' / p \prec \varphi_S(z)$ for $\beta \ge \frac{\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)! \times (2n+1)}}{\log(5/3)} \approx 1.85207$ , (c) $1 + \beta z p' / p^2 \prec \varphi_S(z)$ for $\beta \ge \frac{5}{2} \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)! \times (2n+1)} \approx 2.36521$ . (c) $$1 + \beta z p'/p^2 \prec \varphi_S(z)$$ for $\beta \ge \frac{5}{2} \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)! \times (2n+1)} \approx 2.36521$ . The estimates on $\beta$ cannot be improved.

Theorem 4.1. Let $-1 < B < A \le 1$ , $B \ne 0$ , and let $p \in \mathcal{H}$ satisfies p(0) = 1. Then each of the following differential subordinations sufficiently ensures the subordination $p \prec \varphi_{N_e}$ . Moreover, the respective estimates on the real $\beta$ are best possible. (a) $1 + \beta z p'(z) \prec \frac{1+Az}{1+Bz}$ for $\beta \geq \max{\{\beta_1, \beta_2\}}$ , where $$\beta_1 = \frac{A - B}{2B} \log(1 - B)^{-3} \text{ and } \beta_2 = \frac{A - B}{2B} \log(1 + B)^3.$$ (b) $1 + \beta\left(\frac{zp'(z)}{p(z)}\right) \prec \frac{1+Az}{1+Bz}$ for $\beta \geq \max\{\beta_1, \beta_2\}$ , where $$\beta_1 = \frac{A - B}{B \log 3} \log(1 - B)^{-1} \text{ and } \beta_2 = \frac{A - B}{B \log(5/3)} \log(1 + B).$$ (c) $$1 + \beta\left(\frac{zp'(z)}{p^2(z)}\right) \prec \frac{1+Az}{1+Bz}$$ for $\beta \geq \max\{\beta_1, \beta_2\}$ , where $$\beta_1 = \frac{A - B}{2B} \log(1 - B)^{-1} \text{ and } \beta_2 = \frac{A - B}{2B} \log(1 + B)^5.$$

Corollary 4.1. Let $f \in A$ and $\mathcal{G}(z)$ be defined as in (2.3). If any one of the following conditions hold true, then $f \in \mathcal{S}_{Ne}^*$ . (a) $$\left| \left( \frac{zf'(z)}{f(z)} \right) \mathcal{G}(z) \right| \le \frac{A-B}{1+|B|} \left( \max \left\{ \beta_1, \beta_2 \right\} \right)^{-1}$$ , where $$\beta_1 = \frac{A - B}{2B} \log(1 - B)^{-3} \text{ and } \beta_2 = \frac{A - B}{2B} \log(1 + B)^3.$$ (b) $|\mathcal{G}(z)| \leq \frac{A-B}{1+|B|} (\max \{\beta_1, \beta_2\})^{-1}$ , where $$\beta_1 = \frac{A - B}{B \log 3} \log(1 - B)^{-1} \text{ and } \beta_2 = \frac{A - B}{B \log(5/3)} \log(1 + B).$$ (c) $$\left| \left( \frac{zf'(z)}{f(z)} \right)^{-1} \mathcal{G}(z) \right| \leq \frac{A-B}{1+|B|} \left( \max \left\{ \beta_1, \beta_2 \right\} \right)^{-1}$$ , where $$\beta_1 = \frac{A-B}{2B} \log(1-B)^{-1}$$ and $\beta_2 = \frac{A-B}{2B} \log(1+B)^5$ . 5. Subordination Results Using Hypergeometric Functions Let $a, b \in \mathbb{C}$ and $c \in \mathbb{C} \setminus \{0, -1, -2, \ldots\}$ . Define $$F(a,b;c;z) = {}_{2}F_{1}(a,b;c;z) := \sum_{j=0}^{\infty} \frac{(a)_{j}(b)_{j}}{j! (c)_{j}} z^{j}, \quad z \in \mathbb{D},$$ (5.1) where $(x)_i$ is the Pochhammer symbol given by $$(x)_j = \begin{cases} 1, & j = 0 \\ x(x+1)(x+2)\cdots(x+j-1), & j \in \mathbb{N} := \{1, 2, \ldots\}. \end{cases}$$ The analytic function F(a, b; c; z) given in (5.1) is called the Gaussian hypergeometric function. The following properties of F(a, b; c; z) will be used to prove our results. For further details, we refer to [24]. (i) F(a, b; c; z) is a solution of the differential equation $$z(1-z)w''(z) + (c - (a+b+1)z)w'(z) - abw(z) = 0.$$ (ii) F(a, b; c; z) has a representation in terms of the gamma function $$\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} dt, \quad \text{Re}(z) > 0$$ as $$F(a,b;c;z) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} \sum_{j=0}^{\infty} \frac{\Gamma(a+j)\Gamma(b+j)}{j! \Gamma(c+j)} z^{j}.$$ (5.2) (iii) F(a, b; c; z) satisfies $$F'(a,b;c;z) = \frac{ab}{c}F(a+1,b+1;c+1;z)$$ (5.3) (iv) If Re c > Re b > 0, then F(a, b; c; z) has the following integral representation $$F(a,b;c;z) = \frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} \int_0^1 \frac{t^{b-1}(1-t)^{c-b-1}}{(1-tz)^a} dt, \quad z \in \mathbb{D}.$$ (5.4) We hereby mention that the function zF(a,b;c;z) given by $$zF(a,b;c;z) = z_2F_1(a,b;c;z) = z + \sum_{i=2}^{\infty} \frac{(a)_{j-1}(b)_{j-1}}{(j-1)!(c)_{j-1}} z^j, \quad z \in \mathbb{D},$$ is known as normalized or shifted Gaussian hypergeometric function. Order of Starlikeness. Let f ∈ A. The order of starlikeness (with respect to zero) of the function f(z) is defined to be the number σ(f) given by $$\sigma(f) := \inf_{z \in \mathbb{D}} \operatorname{Re} \left( \frac{zf'(z)}{f(z)} \right) \in [-\infty, 1].$$ In terms of σ(f), we observe that f ∈ A is starlike if, and ony if, σ(f) ≥ 0, or precisely, $$f \in \mathcal{S}^* \iff \sigma(f) \ge 0.$$ Related to the order of starlikeness of the modified Gaussian hypergeometric function zF(a, b; c; z), K¨ustner [\[14,](#page-20-22) [15\]](#page-20-23) proved the following result.

Lemma 5.1. (K¨ustner [\[15,](#page-20-23) Theorem 1 (a)]). If 0 < a ≤ b ≤ c, then $$1 - \frac{ab}{b+c} \le \sigma \left( zF(a,b;c;z) \right) \le 1 - \frac{ab}{2c}.$$ In this section, we use Lemma [5.1](#page-17-0) along with Lemma [1.1](#page-2-1) to find sharp bounds on β so that the first-order differential subordination $$p(z) + \beta z p'(z) \prec \sqrt{1+z}$$ , or $1+z$ implies the subordination p ≺ ϕNe.

Theorem 5.1. Let p ∈ H satisfies p(0) = 1, and let $$p(z) + \beta z p'(z) \prec \varphi_L(z) = \sqrt{1+z}, \quad \beta > 0.$$ If β ≥ βL, then p ≺ ϕNe, where β^L is the unique root of $$\frac{3}{\Gamma(-\frac{1}{2})} \sum_{j=0}^{\infty} \frac{\Gamma(-\frac{1}{2}+j)}{j! (1+j\beta)} - 1 = 0.$$ The estimate on β is best possible.

Corollary 5.1. Let $\mathcal{G}(z)$ be defined as in (2.3), and let $f \in \mathcal{A}$ satisfies $$(1 + \beta \mathcal{G}(z)) \frac{zf'(z)}{f(z)} \prec \varphi_{\scriptscriptstyle L}(z).$$ Then $f \in \mathcal{S}_{N_e}^*$ for $\beta \geq \beta_L$ .

Theorem 5.2. Let $p(z) + \beta z p'(z) \prec 1 + z$ , where $p \in \mathcal{H}$ satisfies p(0) = 1 and $\beta > 0$ . Then $p \prec \varphi_{Ne}$ for $\beta \geq 1/2$ . The estimate on $\beta$ is sharp.