On a subclass of close-to-convex harmonic mappings

Lemma 1.1. If harmonic mapping $f = h + \overline{g} : \mathbb{D} \to \mathbb{C}$ satisfies |g'(0)| < |h'(0)| and the function $F_{\lambda} = h + \lambda g$ is close-to-convex for every $|\lambda| = 1$ , then f is close-to-convex function. An analytic function $\varphi$ is said to subordinate to the analytic function $\psi$ and written by $\varphi(z) \prec \psi(z)$ , if there exists a function w analytic in $\mathbb D$ with w(0) = 0, and |w(z)| < 1 for all $z \in \mathbb D$ , such that $\varphi(z) = \psi(w(z))$ , $z \in \mathbb D$ . Furthermore, if the function $\psi$ is univalent in $\mathbb D$ , then we have the following equivalence: $$\varphi(z) \prec \psi(z) \iff [\varphi(0) = \psi(0) \text{ and } \varphi(\mathbb{D}) \subset \psi(\mathbb{D})].$$ In this article, we shall use the following known result of subordination.

Lemma 1.2. (see [17, Ponnusamy Eq. 16]). Let $\mathcal{P}$ be an analytic function such that $\mathcal{P}(0) = 1$ . Then for real $\alpha$ such that $\alpha > -1$ , we have $$\mathcal{P}(z) + \alpha z \mathcal{P}'(z) \prec 1 + \lambda z \Rightarrow \mathcal{P}(z) \prec 1 + \frac{\lambda}{\alpha + 1} z, \qquad z \in \mathbb{D}.$$ For two analytic functions $\psi_1$ and $\psi_2$ in $\mathbb{D}$ , given by $\psi_1(z) = \sum_{n=0}^{\infty} a_n z^n$ and $\psi_2(z) = \sum_{n=0}^{\infty} b_n z^n$ , the convolution (or Hardamard product) is defined by $(\psi_1 \psi_2)(z) = \sum_{n=0}^{\infty} a_n b_n z^n$ , $z \in \mathbb{D}$ . Analogously, for harmonic functions $f_1 = h_1 + \overline{g_1}$ and $f_2 = h_2 + \overline{g_2}$ in $\mathcal{H}$ , the convolution of $f_1$ and $f_2$ is defined as $f_1 f_2 = h_1 h_2 + \overline{g_1 g_2}$ . Clunie and Sheil-Small [3] proved that, if f is harmonic convex function, and $\phi$ is an analytic convex function, then $f (\phi + \alpha \overline{\phi})$ is a harmonic close-to-convex function for all $\alpha$ such that $|\alpha| < 1$ . Clearly the space $\mathcal{H}$ is closed under the convolution, i.e. $\mathcal{H} \mathcal{H} \subset \mathcal{H}$ . We refer [4, 9, 11, 14] for more information concerning convolution of harmonic mappings and in the case of analytic functions, we refer for examples [16, 24] and the references therein. Let $\mathcal{A}$ denote the class of analytic functions in the unit disk $\mathbb{D}$ and are normalized by f(0) = f'(0) - 1 = 0. We also denote by $\mathcal{S}$ , the subclass of $\mathcal{A}$ consisting of univalent functions. Let $\mathcal{S}^*$ and $\mathcal{K}$ denote the subclasses of $\mathcal{A}$ , which consists the starlike and convex functions, respectively. A function $f \in \mathcal{A}$ is close-to-convex in $\mathbb{D}$ , if there exists a convex analytic function $\phi$ in $\mathbb{D}$ , not necessarily normalized, such that $\Re(f'(z)/\phi'(z)) > 0$ in $\mathbb{D}$ . Ponnusamy and Singh [21] have studied a subclass $\mathcal{B}(\alpha, \beta)$ of close-to-convex functions $f \in \mathcal{A}$ which satisfy the condition <span id="page-2-0"></span> $$|zf''(z) + \alpha(f'(z) - 1)| < \beta, \qquad z \in \mathbb{D},$$ where $\alpha > -1$ and $\beta > 0$ . Also, they proved that functions in the class $\mathcal{B}(\alpha, \beta)$ are convex in $\mathbb{D}$ , if $\alpha > -1$ and (1.2) $$0 < \beta \le \begin{cases} \frac{1-\alpha}{2+\alpha} & \text{for} & -1 < \alpha \le \sqrt{5} - 2, \\ \frac{1+\alpha}{\sqrt{5}} & \text{for} & \sqrt{5} - 2 \le \alpha \le 1, \\ \frac{1+\alpha}{\alpha\sqrt{5}} & \text{for} & 1 \le \alpha \le \frac{2}{\sqrt{5} - 1}, \\ \frac{1+\alpha}{2+\alpha} & \text{for} & \frac{2}{\sqrt{5} - 1} \le \alpha \le 2, \\ \frac{1+\alpha}{2\alpha} & \text{for} & \alpha \ge 2, \end{cases}$$ (see [21, Corollary 4]); and stralike in $\mathbb{D}$ , if $\alpha > -1$ and <span id="page-2-1"></span>(1.3) $$0 < \beta \le \begin{cases} \frac{2(1+\alpha)}{2+\alpha^{2/(1-\alpha)}} & \text{for} & -1 < \alpha \ne 1 < \infty, \\ \frac{4e^2}{1+e^2} & \text{for} & \alpha = 1, \end{cases}$$ (see [21, Theorem 1.14]). Further, we deduce the conditions for the univalency of functions in the class $\mathcal{B}(\alpha, \beta)$ by taking p(z) = f'(z) - 1, $k = 1/\alpha$ ( $\alpha > -1$ ) and $J = \beta/|\alpha|$ in [17, Ponnusamy Eq. 16]. This provides, if $f \in \mathcal{A}$ and $|zf''(z) + \alpha(f'(z) - 1)| < \beta$ ( $z \in \mathbb{D}$ ), then $|f'(z) - 1| < \beta/(1 + \alpha)$ ( $z \in \mathbb{D}$ ). Therefore, the functions in the $\mathcal{B}(\alpha, \beta)$ are close-to-convex (hence univalent) in $\mathbb{D}$ , if $\alpha > -1$ and $\beta \in (0, 1 + \alpha]$ . Now we define harmonic analogue of the class $\mathcal{B}(\alpha, \beta)$ . For $\alpha > -1$ and $\beta > 0$ , let $\mathcal{B}^0_{\mathcal{H}}(\alpha, \beta)$ be a subclass of $\mathcal{H}^0$ which is defined by $$\mathcal{B}_{\mathcal{H}}^{0}(\alpha,\beta) = \left\{ f = h + \overline{g} \in \mathcal{H}^{0} : |zh''(z) + \alpha(h'(z) - 1)| \le \beta - |zg''(z) + \alpha g'(z)|, \ z \in \mathbb{D} \right\}.$$ Note that, for $\alpha = 0$ , the class $\mathcal{B}^0_{\mathcal{H}}(\alpha, \beta)$ reduces to the class $\mathcal{B}^0_{\mathcal{H}}(\beta)$ , which was studied recently by Ghosh and Vasudevaro [5]. Further $\mathcal{B}^0_{\mathcal{H}}(\alpha, \beta)$ reduces to $\mathcal{B}(\alpha, \beta)$ , if the coanalytic part of f in $\mathcal{B}^0_{\mathcal{H}}(\alpha, \beta)$ is zero. In this article, we prove that the functions in $\mathcal{B}^0_{\mathcal{H}}(\alpha,\beta)$ are close-to-convex in $\mathbb{D}$ . We also prove that functions in $\mathcal{B}^0_{\mathcal{H}}(\alpha,\beta)$ are stable harmonic univalent, stable harmonic convex and stable harmonic starlike in $\mathbb{D}$ for different values of its parameters. Further, the coefficient estimates, growth results, area theorem, boundary behaviour, convolution and convex combination properties of the class $\mathcal{B}^0_{\mathcal{H}}(\alpha,\beta)$ of harmonic mapping are obtained. Finally, we consider the harmonic mappings which involve hypergeometric functions and obtain conditions on its parameters such that it belongs to the class $\mathcal{B}^0_{\mathcal{H}}(\alpha,\beta)$ .

Theorem 2.1. For $\alpha > -1$ and $\beta > 0$ , the harmonic mapping $f = h + \overline{g} \in \mathcal{B}^0_{\mathcal{H}}(\alpha, \beta)$ if and only if $F_{\lambda} = h + \lambda g \in \mathcal{B}(\alpha, \beta)$ for all $\lambda(|\lambda| = 1|)$ .

Theorem 2.2. For $\alpha > -1$ and $\beta \in (0, 1 + \alpha]$ , the harmonic mappings in $\mathcal{B}^0_{\mathcal{H}}(\alpha, \beta)$ are close-to-convex in $\mathbb{D}$ .

Theorem 2.3. Let $\alpha > -1$ and $\beta \in (0, 1 + \alpha]$ . If $f = h + \overline{g} \in \mathcal{B}^0_{\mathcal{H}}(\alpha, \beta)$ , then $$|z| - \frac{\beta}{2(1+\alpha)}|z|^2 \le |f(z)| \le |z| + \frac{\beta}{2(1+\alpha)}|z|^2, \qquad z \in \mathbb{D}.$$ Both the inequalities are sharp for the functions $$f_1(z) = z + \frac{\beta}{2(1+\alpha)}z^2$$ and $f_2(z) = z + \frac{\beta}{2(1+\alpha)}\overline{z}^2$ , and their rotations. Proof. If $f \in \mathcal{B}^0_{\mathcal{H}}(\alpha, \beta)$ , then $F_{\lambda} = h + \lambda g \in \mathcal{B}(\alpha, \beta)$ for all $\lambda(|\lambda| = 1)$ . Hence $zF''_{\lambda}(z) + \alpha F'_{\lambda}(z) \prec \alpha + \beta z, \quad z \in \mathbb{D}$ . Using Lemma 1.2, we obtain $$F'_{\lambda}(z) \prec 1 + \frac{\beta}{1+\alpha} z, \quad z \in \mathbb{D}.$$ Therefore $$1 - \frac{\beta}{1 + \alpha} |z| \le |F'_{\lambda}(z)| = |h'(z) + \lambda g'(z)| \le 1 + \frac{\beta}{1 + \alpha} |z|.$$ Since $\lambda(|\lambda|=1)$ is arbitrary, it follows that $$|h'(z)| + |g'(z)| \le 1 + \frac{\beta}{1+\alpha}|z|$$ and $$|h'(z)| - |g'(z)| \ge 1 - \frac{\beta}{1 + \alpha} |z|.$$ If $\Gamma$ is the radial segment from 0 to z, then $$|f(z)| = \left| \int_{\Gamma} \frac{\partial f}{\partial \xi} d\xi + \frac{\partial f}{\partial \overline{\xi}} d\overline{\xi} \right| \le \int_{\Gamma} (|h'(\xi)| + |g'(\xi)|) |d\xi|$$ $$\le \int_{0}^{|z|} \left( 1 + \frac{\beta}{1+\alpha} t \right) dt = |z| + \frac{\beta}{2(1+\alpha)} |z|^{2},$$ and $$|f(z)| = \left| \int_{\Gamma} \frac{\partial f}{\partial \xi} d\xi + \frac{\partial f}{\partial \overline{\xi}} d\overline{\xi} \right| \ge \int_{\Gamma} (|h'(\xi)| - |g'(\xi)|) |d\xi|$$ $$\ge \int_{0}^{|z|} \left( 1 - \frac{\beta}{1 + \alpha} t \right) dt = |z| - \frac{\beta}{2(1 + \alpha)} |z|^{2},$$ which completes the proof of the theorem. The following theorem provides sharp coefficient bounds for functions in $\mathcal{B}^0_{\mathcal{H}}(\alpha,\beta)$ .

Theorem 2.4. Let $f = h + \overline{g} \in \mathcal{B}^0_{\mathcal{H}}(\alpha, \beta)$ be given by (1.1), then for $n \geq 2$ , $$|a_n| \le \frac{\beta}{n(n+\alpha-1)}$$ and $|b_n| \le \frac{\beta}{n(n+\alpha-1)}$ . Both the inequalities are sharp.

Theorem 2.5. Let $\alpha > -1$ and $\beta > 0$ . If $f = h + \overline{g} \in \mathcal{H}^0$ be given by (1.1) and (2.4) $$\sum_{n=2}^{\infty} n(n+\alpha-1)(|a_n|+|b_n|) \le \beta,$$ then $f \in \mathcal{B}^0_{\mathcal{U}}(\alpha, \beta)$ .

Theorem 2.6. For real $\alpha$ and $\beta$ such that $\alpha > -1$ and $\beta \in (0, 1 + \alpha]$ , each function in $\mathcal{B}^0_{\mathcal{H}}(\alpha, \beta)$ maps the $\mathbb{D}$ onto a domain which is bounded by a rectifiable Jordan curve.

Lemma 2.1. (see [25]). Let p be an analytic function in $\mathbb{D}$ , with p(0) = 1 and $\Re(p(z)) > 1/2$ in $\mathbb{D}$ . Then for any analytic function f in $\mathbb{D}$ , the function p \* f takes values in the convex hull of the image of $\mathbb{D}$ under f.

Theorem 2.7. The class $\mathcal{B}^0_{\mathcal{H}}(\alpha,\beta)$ is closed under convex combination.

Theorem 2.8. Let $f \in \mathcal{B}^0_{\mathcal{H}}(\alpha, \beta)$ and $\phi \in \mathcal{K}$ . Then $f * (\phi + \lambda \overline{\phi}) \in \mathcal{B}^0_{\mathcal{H}}(\alpha, \beta)$ for all $\lambda$ ( $|\lambda| = 1$ ).

Lemma 3.1. (see [20]). Let $a, b \in \mathbb{R} \setminus \{0\}$ and c is a positive real number. Then the following holds (a) For c > a + b + 1, $$\sum_{n=0}^{\infty} \frac{(n+1)(a)_n(b)_n}{(c)_n n!} = \frac{\Gamma(c)\Gamma(c-a-b-1)}{\Gamma(c-a)\Gamma(c-b)} (ab+c-a-b-1).$$ (b) For c > a + b + 2, $$\sum_{n=0}^{\infty} \frac{(n+1)^2 (a)_n (b)_n}{(c)_n n!} = \frac{\Gamma(c) \Gamma(c-a-b)}{\Gamma(c-a) \Gamma(c-b)} \left( \frac{(a)_2 (b)_2}{(c-a-b-2)_2} + \frac{3ab}{c-a-b-1} + 1 \right).$$ (c) For $a \neq 1, b \neq 1$ and $c \neq 1$ with $c > max\{0, a + b + 1\}$ , $$\sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_n(n+1)!} = \frac{1}{(a-1)(b-1)} \left( \frac{\Gamma(c)\Gamma(c-a-b-1)}{\Gamma(c-a)\Gamma(c-b)} - (c-1) \right).$$ Below we use the ideas used by [1, 18] for the univalency of harmonic mappings involving the Gaussian hypergeometric functions. The first result in this section is given by

Theorem 3.1. Let $a, b \in \mathbb{R} \setminus \{0\}$ and c is a positive real number. Suppose that $f_1(z) = z + \overline{z^2 F(a, b; c; z)}, f_2(z) = z + \overline{z} (F(a, b; c; z) - 1)$ and $f_3(z) = z + \overline{z} \int_0^z F(a, b; c; t) dt$ , then the following holds (a) If c > a + b + 2 and <span id="page-9-0"></span>(3.9) $$\frac{(a)_2(b)_2}{(c-a-b-2)_2} + \frac{ab(\alpha+4)}{c-a-b-1} + 2(1+\alpha) \le \frac{\beta}{\Lambda},$$ then $f_1 \in \mathcal{B}^0_{\mathcal{H}}(\alpha, \beta)$ . (b) If c > a + b + 2 and <span id="page-9-1"></span>(3.10) $$\frac{ab(ab+c-1)}{(c-a-b-2)_2} + \frac{ab(1+\alpha)}{c-a-b-1} + \alpha \le \frac{\beta-\alpha}{\Lambda},$$ then $f_2 \in \mathcal{B}^0_{\mathcal{H}}(\alpha, \beta)$ . (c) If $a \neq 1, b \neq 1$ and $c \neq 1$ with $c > max\{0, a + b + 1\}$ and <span id="page-9-2"></span> $$(3.11) \quad \Lambda\left(\frac{ab}{c-a-b-1} + \frac{\alpha}{(a-1)(b-1)(c-a-b-1)} + \alpha\right) - \frac{\alpha(c-1)}{(a-1)(b-1)} \le \beta,$$ then $f_3 \in \mathcal{B}^0_{\mathcal{H}}(\alpha, \beta)$ . Proof. (a) Let f1(z) = z + P^∞ ⁿ=2 Cnz ⁿ, where Cⁿ = (a)n−2(b)n−2 (c)n−2(n−2)! (n ≥ 2). Using Lemma [3.1](#page-8-1) and Gauss formula, we have $$\sum_{n=2}^{\infty} n(n+\alpha-1)|C_n| = \sum_{n=2}^{\infty} n(n+\alpha-1) \frac{(a)_{n-2}(b)_{n-2}}{(c)_{n-2}(n-2)!}$$ $$= \sum_{n=0}^{\infty} (n+1)^2 \frac{(a)_n(b)_n}{(c)_n n!} + (1+\alpha) \sum_{n=0}^{\infty} (n+1) \frac{(a)_n(b)_n}{(c)_n n!}$$ $$+ \alpha \sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_n n!}$$ $$= \Lambda \left( \frac{(a)_2(b)_2}{(c-a-b-2)_2} + \frac{ab(\alpha+4)}{c-a-b-1} + 2(1+\alpha) \right).$$ Now if [\(3.9\)](#page-9-0) holds, then P^∞ ⁿ=2 n(n+α−1)|Cn| ≤ β. Now using Theorem [2.5,](#page-5-1) we conclude that f¹ ∈ B⁰ ^H(α, β). (b) Let f2(z) = z + P^∞ ⁿ=2 Dnz ⁿ, where Dⁿ = (a)n−1(b)n−¹ (c)n−1(ⁿ ⁻ 1)! (ⁿ ^≥ 2). Using Lemma [3.1](#page-8-1) and Gauss formula, we have $$\sum_{n=2}^{\infty} n(n+\alpha-1)|D_n| = \sum_{n=2}^{\infty} n(n+\alpha-1) \frac{(a)_{n-1}(b)_{n-1}}{(c)_{n-1}(n-1)!}$$ $$= \sum_{n=0}^{\infty} (n+1) \frac{(a)_{n+1}(b)_{n+1}}{(c)_{n+1} n!} + (1+\alpha) \sum_{n=0}^{\infty} \frac{(a)_{n+1}(b)_{n+1}}{(c)_{n+1} n!}$$ $$+ \alpha \sum_{n=0}^{\infty} \frac{(a)_{n+1}(b)_{n+1}}{(c)_{n+1}(n+1)!}$$ $$= \Lambda \left[ \frac{ab(ab+c-1)}{(c-a-b-2)_2} + \frac{ab(1+\alpha)}{c-a-b-1} + \alpha \right] - \alpha.$$ Now if [\(3.10\)](#page-9-1) holds, then in view of Theorem [2.5,](#page-5-1) we have f² ∈ B⁰ H(α, β). (c) Let $f_3(z) = z + \overline{\sum_{n=2}^{\infty} E_n z^n}$ , where $E_n = \frac{(a)_{n-2}(b)_{n-2}}{(c)_{n-2}(n-1)!}$ $n \ge 2$ . Therefore in view of Lemma 3.1 and Gauss formula, we have $$\sum_{n=2}^{\infty} n(n+\alpha-1)|E_n| = \sum_{n=2}^{\infty} n(n+\alpha-1) \frac{(a)_{n-2}(b)_{n-2}}{(c)_{n-2}(n-1)!}$$ $$= \sum_{n=0}^{\infty} (n+1) \frac{(a)_n(b)_n}{(c)_n n!} + (1+\alpha) \sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_n n!}$$ $$+ \alpha \sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_n (n+1)!}$$ $$= \frac{\Gamma(c)\Gamma(c-a-b-1)}{\Gamma(c-a)\Gamma(c-b)}(ab+c-a-b-1) + (1+\alpha)\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)} + \frac{\alpha}{(a-1)(b-1)}\left(\frac{\Gamma(c)\Gamma(c-a-b-1)}{\Gamma(c-a)\Gamma(c-b)} - (c-1)\right).$$ If (3.11) holds, then by Theorem 2.5, we have $f_3 \in \mathcal{B}^0_{\mathcal{H}}(\alpha, \beta)$ . Note that for $\eta \in \mathbb{C} \setminus \{-1, -2, \dots\}$ and $n \in \mathbb{N} \cup \{0\}$ , we have $$\frac{(-1)^n(-\eta)_n}{n!} = \binom{\eta}{n} = \frac{\Gamma(\eta+1)}{n!\Gamma(\eta-n+1)}.$$ In particular, when $\eta = m \ (m \in \mathbb{N}, m > n)$ , we have $$(-m)_n = \frac{(-1)^n m!}{(m-n)!}.$$ Using this relation in Theorem 3.1, we can obtain harmonic univalent polynomials that belong to the class $\mathcal{B}^0_{\mathcal{H}}(\alpha,\beta)$ . Corollary 3.1. Let $m \in \mathbb{N}$ , c be a positive real numbers. Let $$F_1(z) = z + \sum_{n=0}^{m} {m \choose n} \frac{(m-n+1)_n}{(c)_n} z^{n+2}, \quad F_2(z) = z + \sum_{n=0}^{m} {m \choose n} \frac{(m-n+1)_n}{(c)_n} z^{n+1}$$ and $$F_3(z) = z + \sum_{n=0}^{m} {m \choose n} \frac{(m-n+1)_n}{(c)_n} \frac{z^{n+2}}{n+1}.$$ Then the following holds. $$(a)$$ If $$\frac{m^2(m-1)^2}{(c+2m-1)(c+2m-2)} + \frac{m^2(\alpha+4)}{c+2m-1} + 2(1-\alpha) \le \frac{\beta \left[\Gamma(c+m)\right]^2}{\Gamma(c)\Gamma(c+2m)},$$ then $F_1 \in \mathcal{B}^0_{\mathcal{H}}(\alpha, \beta)$ . (b) If $$\frac{m^2(c+m^2-1)}{(c+2m-2)(c+2m-1)} + \frac{m^2(1+\alpha)}{c+2m-1} + \alpha \le \frac{(\beta-\alpha)\left[\Gamma(c+m)\right]^2}{\Gamma(c)\Gamma(c+2m)},$$ then $F_2 \in \mathcal{B}^0_{\mathcal{H}}(\alpha, \beta)$ . (b) If $$\frac{\Gamma(c)\Gamma(c+2m)}{(\Gamma(c+2m)]^2} \left[ \frac{m^2}{c+2m-1} + \frac{\alpha}{(m+1)^2(c+2m-1)} + \alpha \right) - \frac{\alpha(c-1)}{(m+1)^2} \le \beta,$$ then $F_3 \in \mathcal{B}^0_{\mathcal{H}}(\alpha, \beta)$ .