Harmonic spirallike functions and harmonic strongly starlike functions

Lemma 2.1. Let $\lambda$ be a real number with $|\lambda| < \pi/2$ . Suppose that a function $f \in C^1(\mathbb{D})$ satisfies the conditions that f(z) = 0 if and only if z = 0, and that $J_f = |f_z|^2 - |f_{\bar{z}}|^2 > 0$ on $\mathbb{D}$ . Then f is injective on $\mathbb{D}$ and $f(\mathbb{D}_r)$ is $\lambda$ -spirallike for each 0 < r < 1 if and only if (2.1) $$\operatorname{Re}\left(e^{-i\lambda}\frac{Df(z)}{f(z)}\right) > 0, \quad z \in \mathbb{D} \setminus \{0\}.$$ For explanations, we recall a convenient quantity. For $w \in \mathbb{C} \setminus \{0\}$ , we will say that the $\lambda$ -argument of w is $\theta$ if w lies on the $\lambda$ -spiral $\gamma_{\lambda,\theta} = \{e^{i\theta} \exp(te^{i\lambda}) : t \in \mathbb{R}\}$ . We will write $\arg_{\lambda} w = \theta$ in this case. Note that the $\lambda$ -argument is determined up to an integer multiple of $2\pi$ and a more explicit expression is available as follows: <span id="page-2-1"></span> $$\arg_{\lambda} w = \arg w - (\tan \lambda) \log |w| \pmod{2\pi}.$$ This terminology was introduced in [12] but the same idea was essentially used in [3] and other papers earlier. Proof of Lemma 2.1. As we mentioned before, the "if" part was shown by Al-Amiri and Mocanu [3]. For completeness, we describe the essential ideas for this part. Let $C_r = f(\partial \mathbb{D}_r)$ for 0 < r < 1. Note that each $C_r$ does not pass through the origin by assumption. We will show that $\{C_r\}$ is a family of non-intersecting Jordan curves. Since $C_r$ has winding number 1 about the origin, we may take a continuous branch of $\phi(\theta) = \arg_{\lambda} f(re^{i\theta})$ with period relation $\phi(\theta + 2\pi) = \phi(\theta) + 2\pi$ . A straightforward computation (see [3, p. 63]) leads to <span id="page-3-0"></span>(2.2) $$\phi'(\theta) = \frac{1}{\cos \lambda} \operatorname{Re} \left( e^{-i\lambda} \frac{Df(z)}{f(z)} \right) > 0.$$ Hence, $\phi(\theta)$ is (strictly) increasing, which implies that f is injective on each circle |z| = r; in other words, $C_r$ is a Jordan curve, and that the inside of $C_r$ is a $\lambda$ -spirallike domain. Now, we need only to show that $C_r$ lies in the Jordan domain bounded by $C_{r'}$ for 0 < r < r' < 1. To this end, fix $\phi \in \mathbb{R}$ and we express the unique intersection point of $C_r$ and $\gamma_{\lambda,\phi}$ as $f(re^{i\theta}) = \exp(i\phi + te^{i\lambda})$ for $t = t(r) \in \mathbb{R}$ and $\theta = \theta(r) \in \mathbb{R}$ . Then, it suffices to check that t(r) < t(r') for 0 < r < r' < 1. By (12) in [3] or by a formal computation, we obtain the relation <span id="page-3-1"></span>(2.3) $$|f(z)|^2 \frac{dt}{dr} \operatorname{Re} \left( e^{-i\lambda} \frac{Df(z)}{f(z)} \right) = r J_f(z),$$ where $z = re^{i\theta}$ . Since $J_f > 0$ by assumption, we conclude that t = t(r) is increasing in 0 < r < 1. Thus we have shown the "if" part. Secondly, we show the "only if" part. Assume that f is univalent on $\mathbb{D}$ and that $f(\mathbb{D}_r)$ is $\lambda$ -spirallike for 0 < r < 1. Then the intersection of $C_r = \partial f(\mathbb{D}_r)$ with $\gamma_{\lambda,\phi}$ is connected for each 0 < r < 1 and $\phi \in \mathbb{R}$ so that $\phi(\theta) = \arg_{\lambda} f(re^{i\theta})$ is non-decreasing in $\theta$ . Also, t = t(r) defined above is non-decreasing in 0 < r < 1 and thus $dt/dr \ge 0$ . In view of (2.2) and (2.3), we obtain (2.1) because $J_f > 0$ by assumption. We restate the lemma in the case when f is harmonic. Corollary 2.2. Let $\lambda$ be a real number with $|\lambda| < \pi/2$ . Suppose that a function $f \in \mathcal{H}_0$ satisfies the conditions that $f(z) \neq 0$ for 0 < |z| < 1 and that $J_f = |f_z|^2 - |f_{\bar{z}}|^2 > 0$ on $\mathbb{D}$ . Then $f \in \mathcal{SP}_H(\lambda)$ if and only if the inequality (2.1) holds. In particular, by (1.4), we obtain the following characterization of hereditarily strongly starlike functions of order $\alpha$ . <span id="page-3-3"></span>Corollary 2.3. Let $\alpha$ be a real number with $|\alpha| < 1$ . Suppose that a function $f \in \mathcal{H}_0$ satisfies the conditions that $f(z) \neq 0$ for 0 < |z| < 1 and that $J_f = |f_z|^2 - |f_{\bar{z}}|^2 > 0$ on $\mathbb{D}$ . Then $f \in \mathcal{SS}_H(\alpha)$ if and only if <span id="page-3-2"></span>(2.4) $$\left| \arg \frac{Df(z)}{f(z)} \right| < \frac{\pi \alpha}{2}, \quad z \in \mathbb{D} \setminus \{0\}.$$ Proof of non-hereditary starlikeness of k(z). We now show that the harmonic Koebe function k(z) is not hereditarily starlike. By virtue of Lemma 2.1, it is enough to check that the function k does not satisfy the condition Re[Dk/k] > 0 on $\mathbb{D}$ . Here, $$Dk(z) = zh'(z) - \overline{zg'(z)} = \frac{z(1+z)}{(1-z)^4} - \frac{\overline{z}^2(1+\overline{z})}{(1-\overline{z})^4}.$$ Let $z_0 = (1+2i)/3 \in \mathbb{D}$ . Then, straightforward computations yield $k(z_0) = (-17+9i)/24$ and $Dk(z_0) = -15(1+2i)/16$ . Hence, we see that $Dk(z_0)/k(z_0) = 9(-1+43i)/148$ has negative real part. Let $r_1$ be the radius of hereditary starlikeness for the harmonic Koebe function k. Then, numerical computations suggest that $0.572154 < r_1 < 0.572155$ .

Lemma 3.1. Let $f \in \mathcal{S}_H$ . Then there is a point $w_0 \in \mathbb{C}$ with $|w_0| \leq \pi/2$ such that $w_0 \notin f(\mathbb{D})$ . The bound $\pi/2$ is sharp. Our result in this section is the following.

Theorem 3.2. Let $\alpha$ be a real number with $0 < \alpha < 1$ . For each $f \in \mathcal{SS}_{H}(\alpha)$ , the inequality $|f(z)| \leq N(\alpha), z \in \mathbb{D}$ , holds, where $$N(\alpha) = \frac{\pi}{2} \exp \left\{ \pi \tan(\pi \alpha/2) \right\}.$$ Proof. We define $f_r$ by $f_r(z) = f(rz)/r$ . Then $f_r \in \mathcal{SS}_{\mathrm{H}}(\alpha) \subset \mathcal{S}_{\mathrm{H}}$ for each 0 < r < 1. Let $\Omega_r = f_r(\mathbb{D})$ for 0 < r < 1. Then $\Omega_r$ is a strongly starlike domain of order $\alpha$ . For an arbitrary point $w \in \Omega_r \setminus \{0\}$ , we have $wV_\alpha \subset \Omega_r$ . On the other hand, by Lemma 3.1, there is a point $w_0 \in \mathbb{C} \setminus \Omega_r$ with $|w_0| \leq \pi/2$ . In view of the relation (1.1), we have $$|w| \exp\left(-\pi \tan(\pi\alpha/2)\right) \le |w_0| \le \frac{\pi}{2}$$ for $w \in \Omega_r$ , which implies $|w| \leq N(\alpha)$ . Since 0 < r < 1 was arbitrary, we have the expected conclusion. We exhibit the graph of $\log M(\alpha)$ and $\log N(\alpha)$ in Figure 1. Though $M(\alpha), N(\alpha) \to +\infty$ as $\alpha \to 1$ , the graph suggests that $\log N(\alpha) - \log M(\alpha)$ is bounded. Indeed, that is true. Consider the ratio $$\frac{N(\alpha)}{M(\alpha)} = 2\pi \exp\left\{\pi \tan(\pi \alpha/2) + \psi((1-\alpha)/2) + \gamma\right\} = 2\pi \exp\left\{\pi \cot(\pi t) + \psi(t) + \gamma\right\},$$  <span id="page-5-0"></span>FIGURE 1. The graph of $\log M(\alpha)$ and $\log N(\alpha)$ where $t = (1 - \alpha)/2$ . Since $\cot x = 1/x + O(x)$ and $\psi(x) = 1/x - \gamma + O(x)$ as $x \to 0$ , we have $\pi \cot(\pi t) + \psi(t) + \gamma = O(t)$ as $t \to 0$ . Hence, $$\lim_{\alpha \to 1} \frac{N(\alpha)}{M(\alpha)} = 2\pi.$$ By numerical computations, we observed that $N(\alpha) \leq 2\pi M(\alpha)$ for $0 < \alpha < 1$ . As an application of the boundedness, we establish quasiconformal extendability of hereditarily strongly starlike harmonic functions under a mild condition. First, we recall that a homeomorphism $f:\Omega\to\Omega'$ between plane domains is called K-quasiconformal if f belongs to the Sobolev class $W_{\rm loc}^{1,2}(\Omega)$ and if the inequality $|f_{\bar{z}}| \leq k|f_z|$ holds a.e. on $\Omega$ , where $k=(K-1)/(K+1)\in[0,1)$ . When $\Omega=\Omega'$ , we call f a K-quasiconformal endomorphism of $\Omega$ . It is well known [2] that $f_1\circ f_2$ is $K_1K_2$ -quasiconformal whenever $f_j$ is $K_j$ -quasiconformal for j=1,2. A bounded domain $\Omega$ is called a K-quasidisk if $\Omega=f(\mathbb{D})$ for a K-quasiconformal mapping $f:\mathbb{C}\to\mathbb{C}$ . Fait, Krzyż and Zygmunt [9] showed the following.

Lemma 3.3. Let $0 < \alpha < 1$ . A strongly starlike function in $SS(\alpha)$ extends to a $\cot^2 \frac{\pi(1-\alpha)}{4}$ -quasiconformal endomorphism of $\mathbb{C}$ . In particular, a strongly starlike domain of order $\alpha$ is a $\cot^2 \frac{\pi(1-\alpha)}{4}$ -quasidisk. We extend this result to the class $\mathcal{SS}_{H}(\alpha)$ of hereditarily strongly starlike harmonic functions of order $\alpha$ .

Theorem 3.4. Let $f = h + \bar{g} \in \mathcal{SS}_H(\alpha)$ for some $0 < \alpha < 1$ . Suppose that the second complex dilatation $\omega = g'/h'$ of f satisfies the inequality $|\omega| \leq (K-1)/(K+1)$ on $\mathbb D$ for a constant $K \geq 1$ . Then f extends to a $K \cot^2 \frac{\pi(1-\alpha)}{4}$ -quasiconformal endomorphism of $\mathbb C$ . Proof. Let $\Omega = f(\mathbb{D})$ . By definition, $\Omega$ is a strongly starlike domain of order $\alpha$ . Let $\mu = f_{\bar{z}}/f_z = \overline{g'}/h'$ be the complex dilatation of f. Then $|\mu| = |\omega| \le (K-1)/(K+1) < 1$ . Let $w : \mathbb{D} \to \mathbb{D}$ be a quasiconformal homeomorphism with w(0) = 0, w(1) = 1 and $w_{\bar{z}}/w_z = \mu$ a.e. on $\mathbb{D}$ . Note that existence of such a mapping is guaranteed by the measurable Riemann mapping theorem (see [2]). Moreover, the mapping w extends to a K-quasiconformal mapping of $\mathbb{C}$ with the property 1/w(1/z) = w(z) for $z \in \mathbb{D}$ . Then the composed mapping  FIGURE 2. The half-plane $H_{\lambda}$ and the point c <span id="page-6-0"></span> $F = f \circ w^{-1} : \mathbb{D} \to \Omega$ is analytic and satisfies F(0) = 0. Let a = F'(0) and G = F/a. Since the image $G(\mathbb{D}) = \Omega/a$ is strongly starlike of order $\alpha$ , we observe that $G \in \mathcal{SS}(\alpha)$ . Now Lemma 3.3 implies that G extends to a $\cot^2 \frac{\pi(1-\alpha)}{4}$ -quasiconformal endomorphism of $\mathbb{C}$ . Hence, $f = F \circ w$ extends to a $K \cot^2 \frac{\pi(1-\alpha)}{4}$ -quasiconformal endomorphism of $\mathbb{C}$ as required.

Lemma 4.1. For $n \geq 2$ , the following inequalities hold: $$(4.1) 2n\sin\frac{\pi\alpha}{2} < A_n(\alpha) < B_n(\alpha) (0 < \alpha < 1).$$

Theorem 4.2. Let $f = h + \bar{g} \in \mathcal{H}_0$ for $h(z) = z + a_2 z^2 + a_3 z^3 + \cdots$ and $g(z) = b_1 z + b_2 z^2 + b_3 z^3 + \cdots$ . Suppose that the inequality <span id="page-7-0"></span>(4.2) $$\sum_{n=2}^{\infty} A_n(\alpha)|a_n| + \sum_{n=1}^{\infty} B_n(\alpha)|b_n| \le 2\sin\frac{\pi\alpha}{2}$$ holds. Then $f \in \mathcal{SS}_{H}(\alpha)$ .

Theorem 4.3. Let $-\pi/2 < \lambda < \pi/2$ . Suppose that a function $f = h + \overline{g} \in \mathcal{H}_0$ satisfies $f(z) \neq 0$ for 0 < |z| < 1 and $J_f(z) > 0$ for $z \in \mathbb{D}$ . Then $f \in \mathcal{SP}_H(\lambda)$ if and only if $$(4.4) (f * \varphi_{\lambda,\zeta})(z) \neq 0 for z \in \mathbb{D} \setminus \{0\}, \ \zeta \in \mathbb{T} \setminus \{-1\},$$ where $\mathbb{T}$ denotes the unit circle $\partial \mathbb{D}$ and <span id="page-8-0"></span> $$\varphi_{\lambda,\zeta}(z) = \frac{(1 + e^{2i\lambda})z + (\zeta - e^{2i\lambda})z^2}{(1 - z)^2} + \frac{(-1 + e^{2i\lambda} - 2\zeta)\bar{z} + (\zeta - e^{2i\lambda})\bar{z}^2}{(1 - \bar{z})^2}.$$

Corollary 4.4. Let f be an orientation-preserving harmonic function in $\mathcal{H}_0$ satisfying the condition $f(z) \neq 0$ for 0 < |z| < 1. For $0 < \alpha < 1$ , $f \in \mathcal{SS}_{\mathcal{H}}(\alpha)$ if and only if $$(f\varphi_{\frac{\pi(1-\alpha)}{2},\zeta})(z)\neq 0 \quad and \quad (f\varphi_{-\frac{\pi(1-\alpha)}{2},\zeta})(z)\neq 0$$ for all $z \in \mathbb{D} \setminus \{0\}$ and $\zeta \in \mathbb{T} \setminus \{-1\}$ . As a simple application of the above results, we examine hereditary strong starlikeness of the harmonic function $f_{b,n}$ of the special form $$f_{b,n}(z) = z + b\overline{z}^n$$ for $b \in \mathbb{C}$ and $n = 1, 2, 3, \dots$

Proposition 4.5. Let $0 < \alpha < 1$ and set $\lambda = \pi(1 - \alpha)/2$ . Then the following are equivalent: - (i) $f_{hn} \in \mathcal{SS}_{H}(\alpha)$ ; - (ii) $f_{b,n} \in \mathcal{SP}_{\mathrm{H}}(\lambda)$ ; (iii) $$|b| \le C_n(\alpha)$$ , where $C_n(\alpha) = \frac{2\sin(\pi\alpha/2)}{n+1+|n+e^{i\pi\alpha}|}$ . Remark 1. Since $C_n(\alpha) = 2\sin(\pi\alpha/2)/B_n(\alpha)$ , Lemma 4.1 implies that $C_n(\alpha) < 1$ . Proof. (i) ⇒ (ii). It is obvious by the relation [\(1.3\)](#page-1-1). (ii) ⇒ (iii). Assume that fb,n ∈ SPH(λ). By Theorem [4.3,](#page-8-1) fb,n must satisfy the condition [\(4.4\)](#page-8-0); namely, $$(f_{b,n} * \varphi_{\lambda,\zeta})(z) = (1 - e^{2i\lambda})z - b[(n+1)\zeta + n + e^{2i\lambda}]\bar{z}^n \neq 0,$$ for 0 < |z| < 1 and |ζ| = 1 with ζ 6= −1. This implies $$|1 - e^{2i\lambda}| \ge |b| |(n+1)\zeta + n + e^{2i\lambda}|, \quad \zeta \in \mathbb{T} \setminus \{-1\}.$$ Hence, $$|b| \le \sup_{|\zeta|=1, \zeta \ne -1} \frac{|1 - e^{2i\lambda}|}{|(n+1)\zeta + n + e^{2i\lambda}|} = C_n(\alpha).$$ (iii) ⇒ (i). Condition (iii) means the inequality Bn(α)|b| ≤ 2 sin(πα/2). We now conclude that fb,n ∈ SSH(α) by Theorem [4.2.](#page-7-2)

Definition 1. Let $\lambda$ and $\alpha$ be real numbers with $|\lambda| < \pi/2$ and $0 < \alpha < 1$ . A harmonic function f in $\mathcal{H}_0$ is called hereditarily $\lambda$ -spirallike if f is orientation-preserving and univalent on $\mathbb{D}$ and if $f(\mathbb{D}_r)$ is $\lambda$ -spirallike for each 0 < r < 1. The class of such functions will be denoted by $\mathcal{SP}_{\mathrm{H}}(\lambda)$ . Similarly, a harmonic function $f \in \mathcal{H}_0$ is called hereditarily strongly starlike of order $\alpha$ if it is orientation-preserving and univalent on $\mathbb{D}$ and if $f(\mathbb{D}_r)$ is a strongly starlike domain of order $\alpha$ for each 0 < r < 1. We denote by $\mathcal{SS}_{\mathrm{H}}(\alpha)$ the class of such functions. In particular, the class $\mathcal{SP}_{H}(0)$ consists of hereditarily starlike harmonic functions on $\mathbb{D}$ . We would like to point out here that these classes are not considered in the literature though spirallike logharmonic mappings and spirallike $C^{1}$ -functions are studied by [1] and [3], respectively. As we saw in (1.3), a domain $\Omega$ with $0 \in \Omega \subset \mathbb{C}$ is strongly starlike of order $\alpha$ if and only if $\Omega$ is $\pm \pi (1 - \alpha)/2$ -spirallike at the same time. Therefore, we have similarly (1.4) $$SS_{H}(\alpha) = SP_{H}(\frac{\pi(1-\alpha)}{2}) \cap SP_{H}(-\frac{\pi(1-\alpha)}{2}).$$