Sufficiency for Nephroid Starlikeness using Hypergeometric Functions

Lemma 2.1. (Ma and Minda [34, 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.

Lemma 2.2. (Küstner [26, Theorem 1 (a)]). If $0 < a \le b \le c$ , then $$1 - \frac{ab}{b+c} \le \sigma \left( zF(a,b;c;z) \right) \le 1 - \frac{ab}{2c}.$$

Lemma 2.3. (Miller and Mocanu [34, p. 236]). If $c \le 1 + N(a-1)$ , where $$N(a) = \begin{cases} |a| + \frac{1}{2} & \text{if } |a| \ge \frac{1}{3}, \\ \frac{3(a)^2}{2} + \frac{2}{3} & \text{if } |a| \le \frac{1}{3}, \end{cases}$$ then $z\Phi(a;c;z)$ is starlike in $\mathbb{D}$ .

Theorem 3.1. Let $p: \mathbb{D} \to \mathbb{C}$ be analytic and satisfies p(0) = 1. Let $\varphi_L(z) := \sqrt{1+z}$ and $$p(z) + \beta z p'(z) \prec \varphi_L(z), \qquad \beta > 0.$$ If $\beta \geq \beta_L \approx 0.158379$ , then $p(z) \prec \varphi_{Ne}(z)$ , where $\beta_L$ is the unique root of <span id="page-4-2"></span> $$\frac{3}{\Gamma(-\frac{1}{2})} \sum_{j=0}^{\infty} \frac{\Gamma(-\frac{1}{2}+j)}{j! (1+j\beta)} - 1 = 0.$$ (3.1) The estimate on $\beta$ is best possible.

Theorem 3.2. Let the analytic p satisfies p(0) = 1 and let $$p(z) + \beta z p'(z) \prec 1 + z, \qquad \beta > 0.$$ Then $p \prec \varphi_{Ne}$ whenever $\beta \geq 1/2$ and this estimate on $\beta$ is sharp.

Theorem 3.3. Let p be analytic in $\mathbb{D}$ satisfying p(0) = 1. For $\varphi_e(z) := e^z$ , let the differential subordination $$p(z) + \beta z p'(z) \prec \varphi_e(z), \qquad \beta > 0.$$ holds. Then $p(z) \prec \varphi_{Ne}(z)$ whenever $\beta \geq \beta_e \approx 1.14016$ , where $\beta_e$ is the unique solution of $$\sum_{i=0}^{\infty} \frac{1}{j! (1+j\beta)} - \frac{5}{3} = 0.$$ The estimate on $\beta$ can not be improved further.

Definition 1.1. Let $\Lambda : \mathbb{C}^2 \times \mathbb{D} \to \mathbb{C}$ be analytic, and let u be univalent. The analytic function p is said to satisfy the first-order differential subordination if $$\Lambda(p(z), zp'(z); z) \prec u(z), \qquad z \in \mathbb{D}.$$ (1.1) If $q: \mathbb{D} \to \mathbb{C}$ is univalent and $p \prec q$ for all p satisfying (1.1), then q is said to be a dominant of the differential subordination (1.1). A dominant $\tilde{q}$ that satisfies $\tilde{q} \prec q$ for all dominants q of (1.1) is called the best dominant of (1.1). If $\tilde{q}_1$ and $\tilde{q}_2$ are two best dominants of (1.1), then $\tilde{q}_2(z) = \tilde{q}_1(e^{i\theta}z)$ for some $\theta \in \mathbb{R}$ . For further details related to differential subordinations, we refer to the monograph of Miller and Mocanu [34] (see also Bulboacă [9]). Due to its straightforward consequences, the theory of differential subordinations (a complex analogue of differential inequalities) developed by Miller and ²⁰¹⁰ Mathematics Subject Classification. 30C45, 30C80, 33C05, 33C15. Key words and phrases. Differential Subordination, Starlike functions, Hypergeometric Functions, Nephroid, Bernoulli Lemniscate. Mocanu [34] is being extensively used in studying the analytic and geometric properties of univalent functions. For some recent works, see [5, 13–15, 17, 21, 37, 58]. Following [3, 12, 16, 18–20, 23, 31, 32, 40–42, 63], the authors in [60, 61] introduced and studied the geometric properties of the function $\varphi_{N_e}(z) := 1 + z - z^3/3$ and the associated Ma-Minda type (see [28, 44, 49]) function family $\mathcal{S}_{N_e}^*$ given by $$\mathcal{S}_{Ne}^* := \left\{ f \in \mathcal{A} : \frac{zf'(z)}{f(z)} \prec \varphi_{Ne}(z) \right\}.$$ It was proved by Wani and Swaminathan [61] that the function $\varphi_{Ne}(z)$ maps the boundary $\partial \mathbb{D}$ of the unit disk $\mathbb{D}$ univalently onto the nephroid, a 2-cusped kidney-shaped curve (see Figure 1), given by <span id="page-1-1"></span> $$\left( (u-1)^2 + v^2 - \frac{4}{9} \right)^3 - \frac{4v^2}{3} = 0.$$ (1.2) <span id="page-1-0"></span>Geometrically, a nephroid is the locus of a point fixed on the circumference of a circle of radius $\rho$ that rolls (without slipping) on the outside of a fixed circle having radius $2\rho$ . First studied by Huygens and Tschirnhausen in 1697, the nephroid curve was shown to be the catacaustic (envelope of rays emanating from a specified point) of a circle when the light source is at infinity. In 1692, Jakob Bernoulli had shown that the nephroid is the catacaustic of a cardioid for a luminous cusp. However, the word nephroid was first used by Richard A. Proctor in 1878 in his book 'The Geometry of Cycloids'. For further details related to the nephroid curve, we refer to [27,62].  Figure 1. Nephroid: The Boundary curve of $\varphi_{N_e}(\mathbb{D})$ . Thus $f \in \mathcal{S}_{Ne}$ if, and only if, all the values taken by the expression zf'(z)/f(z) lie in the region $\Omega_{Ne}$ bounded by the nephroid curve (1.2). Since $\mathcal{S}_{Ne}^ \subset \mathcal{S}$ , we call $f \in \mathcal{S}_{Ne}$ a nephroid starlike function. In this paper, we employ the differential subordination techniques and use the geometric properties of Gaussian and confluent hypergeometric functions to establish conditions which ensure that the analytic function $f \in \mathcal{A}$ is nephroid starlike in $\mathbb{D}$ . More specifically, we determine the best possible bounds on the real $\beta$ so that, for some analytic p satisfying p(0) = 1, the following implication holds: $$p(z) + \beta z p'(z) \prec \begin{cases} \sqrt{1+z}; \\ 1+z; \\ e^z; \end{cases} \implies p(z) \prec \varphi_{Ne}(z).$$ Replacing p(z) by the expression zf'(z)/f(z) for any $f \in \mathcal{A}$ , we obtain conditions that are sufficient to imply that the function f is nephroid starlike in $\mathbb{D}$ . Although similar type of differential subordination implication problems have been studied for several other function families (for instance see [2,4,6,8,11,14,21,22,24,29,36,37]), the approach of utilizing the properties of hypergeometric functions to arrive at the desired implication is totally new. In addition, this paper verifies analytically certain crucial facts which some of the above cited authors have concluded geometrically without providing any analytic clarification. However, graphical illustrations are also provided in this manuscript for enhancing the clarity of the results to the reader and competing with the existing related literature. In the sequel, it is always assumed that $z \in \mathbb{D}$ unless stated otherwise.

Definition 2.1. (Gaussian hypergeometric function). Let $a, b \in \mathbb{C}$ and $c \in \mathbb{C} \setminus \{0, -1, -2, \ldots\}$ . Define <span id="page-2-1"></span><span id="page-2-0"></span> $$F(a,b;c;z) := {}_{2}F_{1}(a,b;c;z) = \sum_{j=0}^{\infty} \frac{(a)_{j}(b)_{j}}{j! (c)_{j}} z^{j}, \quad z \in \mathbb{D},$$ (2.1) where $(x)_i$ is the Pochhammer symbol given by $$(x)_j = \begin{cases} 1, & j = 0 \\ x(x+1)(x+2)\cdots(x+j-1), & j \in \{1, 2, \ldots\}. \end{cases}$$ (2.2) The analytic function F(a,b;c;z) given in (2.1) is called the Gaussian hypergeometric function. Prior to the use of hypergeometric functions in the proof of Bieberbach's conjecture by de Branges [10], there has been little known connections between the univalent function theory and the theory of special functions. This surprising use of hypergeometric functions has given function theorists a renewed interest to study the interrelatedness of these two concepts and, as a result, a number of papers have been published in this direction. For instance, see [1,7,25,26,30,33,35,39,43,50,52-57,59]. The function F(a,b;c;z) defined in (2.1) has many interesting properties among which the following will be used to prove our results. For further details, we refer to Rainville [38]. (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 <span id="page-3-5"></span> $$\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} dt, \quad \operatorname{Re}(z) > 0$$ as $$F(a,b;c;z) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} \sum_{i=0}^{\infty} \frac{\Gamma(a+j)\Gamma(b+j)}{j! \Gamma(c+j)} z^{j}.$$ (2.3) (iii) F(a, b; c; z) satisfies <span id="page-3-3"></span><span id="page-3-2"></span> $$F'(a,b;c;z) = \frac{ab}{c}F(a+1,b+1;c+1;z)$$ (2.4) (iv) If $\operatorname{Re} c > \operatorname{Re} b > 0$ , then F(a, b; c; z) has the following integral representation $$F(a,b;c;z) = \frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} \int_0^1 \frac{t^{b-1}(1-t)^{c-b-1}}{(1-tz)^a} dt, \quad z \in \mathbb{D}.$$ (2.5) We hereby mention that the function zF(a,b;c;z) given by $$zF(a,b;c;z) = z_2F_1(a,b;c;z) = z + \sum_{j=2}^{\infty} \frac{(a)_{j-1}(b)_{j-1}}{(j-1)!(c)_{j-1}} z^j, \quad z \in \mathbb{D},$$ is known as normalized or shifted Gaussian hypergeometric function. Order of Starlikeness. Let $f \in \mathcal{A}$ . The order of starlikeness (with respect to zero) of the function f(z) is defined to be the number $\sigma(f)$ given by $$\sigma(f) := \inf_{z \in \mathbb{D}} \operatorname{Re}\left(\frac{zf'(z)}{f(z)}\right) \in [-\infty, 1].$$ (2.6) In terms of $\sigma(f)$ , we observe that $f \in \mathcal{A}$ is starlike if, and ony if, $\sigma(f) \geq 0$ , or precisely, <span id="page-3-4"></span><span id="page-3-0"></span> $$f \in \mathcal{S}^* \iff \sigma(f) > 0.$$ <span id="page-3-1"></span>Related to the order of starlikeness of the modified Gaussian hypergeometric function zF(a,b;c;z), Küstner [25,26] proved the following result.

Definition 2.2. (Confluent hypergeometric function). Let $a \in \mathbb{C}$ and $c \in \mathbb{C} \setminus \{0, -1, -2, \ldots\}$ . The confluent (or Kummer) hypergeometric function is defined as the convergent power series $$\Phi(a;c;z) := {}_{1}F_{1}(a;c;z) = \sum_{j=0}^{\infty} \frac{(a)_{j}}{(c)_{j}} \frac{z^{j}}{j!}, \quad z \in \mathbb{D},$$ (2.7) where $(x)_i$ is the Pochhammer symbol defined in (2.2). The function $\Phi(a;c;z)$ is analytic in $\mathbb{C}$ and satisfies the Kummer's differential equation $$zw''(z) + (c - z)w'(z) - aw(z) = 0.$$ If we replace b by $1/\varrho$ and z by $z\varrho$ in the series (2.1) and allow $\varrho \to 0$ , we obtain the series (2.7). Below, we mention certain well-known properties of $\Phi(a; c; z)$ given by (2.7). <span id="page-4-6"></span> $$\Phi(a;c;z) = \frac{\Gamma(c)}{\Gamma(a)} \sum_{j=0}^{\infty} \frac{\Gamma(a+j)}{\Gamma(c+j)} \frac{z^j}{j!},$$ (2.8) <span id="page-4-5"></span><span id="page-4-4"></span> $$\Phi'(a;c;z) = -\frac{a}{c}\Phi(a+1;c+1;z),$$ (2.9) and $$\Phi(a;c;z) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)} \int_0^1 t^{a-1} (1-t)^{c-a-1} e^{tz} dt, \quad (\text{Re } c > \text{Re } a > 0).$$ (2.10) Further, the function $$z\Phi(a;c;z) = z_1 F_1(a;c;z) = z + \sum_{j=2}^{\infty} \frac{(a)_{j-1}}{(c)_{j-1}} \frac{z^j}{(j-1)!}, \quad z \in \mathbb{D},$$ <span id="page-4-3"></span>is the normalised (shifted) confluent hypergeometric function. The following result related to the starlikeness of $z\Phi(a;c;z)$ will be used to prove Theorem 3.3.