Extreme points method and univalent harmonic mappings

Theorem 1. Let $h \in \mathcal{F}$ . Then for $\beta > 0$ and $r \in (0,1)$ one has $$I_{\beta}(r,f) = \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{d\theta}{|h'(re^{i\theta})|^{2\beta}} \le \frac{2^{6\beta}}{\pi} \mathbf{B} \Big( \frac{6\beta + 1}{2}, \frac{1}{2} \Big),$$ where $\mathbf{B}(.,.)$ denotes the Euler-beta function. The inequality is sharp. We present two different proofs of Theorem 1. One of the proofs relies on the method of extreme points (see for example [12]) while the other relies on the sub-ordination relation. The estimates of $I_1$ has received special attention in the field of planar fluid mechanics, where these functionals are participating in isoperimetric problems for moving phase domains, eg. [27] and [28]. In order to present the third consequence of our approach, we recall the following result which is a partial extension of the classical result of Alexander's theorem from conformal mappings to univalent harmonic mappings. Theorem D. ([11, p.108, Lemma]) Let $f = h + \overline{g}$ be a sense-preserving harmonic starlike mapping in $\mathbb{D}$ . If H and G are the analytic functions defined by the relations (2) $$zH'(z) = h(z), zG'(z) = -g(z), H(0) = G(0) = 0,$$ then $F = H + \overline{G}$ is a convex mapping in $\mathbb{D}$ . A generalization of Theorem D has been obtained by Ponnusamy and Sairam Kaliraj [24]. However, it is natural to ask what would be the conclusion if the assumption about f is replaced just by the analytic part h being starlike in $\mathbb{D}$ . We remark that the harmonic Koebe function (see [8, 11, 22]) K defined by $$K(z) = \frac{z - \frac{1}{2}z^2 + \frac{1}{6}z^3}{(1 - z)^3} + \overline{\left(\frac{\frac{1}{2}z^2 + \frac{1}{6}z^3}{(1 - z)^3}\right)} \quad \text{for } z \in \mathbb{D},$$ is starlike in $\mathbb{D}$ whereas its analytic part is not even univalent in $\mathbb{D}$ . Also, there are harmonic convex function whose analytic part is not necessarily starlike in $\mathbb{D}$ .

Theorem 2. Let $f = h + \overline{g}$ be a sense-preserving harmonic mapping in $\mathbb{D}$ , where $h \in \mathcal{S}^*$ and g(0) = 0. If H and G are the analytic functions defined by the relations (2), then for each $|\lambda| \leq 1$ , the harmonic function $F_{\lambda} = H + \lambda \overline{G}$ is sense-preserving and close-to-convex mapping in $\mathbb{D}$ . In particular, $F = H + \overline{G}$ is a close-to-convex mapping in $\mathbb{D}$ . We now state our next result whose proof follows similarly. So we omit its detail.

Theorem 3. Let $f = h + \overline{g}$ be a harmonic mapping in $\mathbb{D}$ , where $h \in \mathcal{S}^*(\beta)$ for some $\beta \in (-1/2, 0]$ , g(0) = 0 and $g'(z) = \omega(z)h'(z)$ in $\mathbb{D}$ for some $\omega : \mathbb{D} \to \mathbb{D}$ satisfying the condition $|\omega(z)| < \cos(\beta \pi)$ for $z \in \mathbb{D}$ . If H and G are the analytic functions defined by the relations (2), then for each $|\lambda| = 1$ , the harmonic function $F_{\lambda} = H + \lambda \overline{G}$ is sense-preserving and close-to-convex mapping in $\mathbb{D}$ . We remark that functions in $S^*(\beta)$ are not necessarily univalent in $\mathbb{D}$ if $\beta < 0$ . At the end of the article, Bshouty and Lyzzaik [6] expressed their interest in determining sufficient condition on h so that $g'(z) = e^{i\theta}zh'(z)$ implies that $f = h + \overline{g}$ is univalent in $\mathbb{D}$ . Several of the remaining results of this article motivate their desire by choosing h appropriately. Proof of Theorem 2 will be given in Section 2. For $\alpha \in (1,2)$ , let $CO_H(\alpha)$ denote the class of all harmonic mappings $f = h + \overline{g}$ defined on $\mathbb{D}$ , where $g'(z) = \omega(z)h'(z)$ with $|\omega(z)| < 1$ for $z \in \mathbb{D}$ and $h \in CO(\alpha)$ , the class of all concave univalent functions (see Section 2 for the precise definition). The class $CO(\alpha)$ has been extensively studied in the recent years and for a detailed discussion about concave functions, we refer to [2, 3, 5, 9] and the references therein. We now state our next result.

Theorem 4. For $\alpha \in (1,2)$ , let $f = h + \overline{g} \in CO_H(\alpha)$ . If the dilatation $\omega$ satisfies the conditions $|\omega(z)| < \sin(\frac{2-\alpha}{2})\pi$ for $z \in \mathbb{D}$ , then f is close-to-convex (univalent) in $\mathbb{D}$ . A simple consequence of Theorem 4 gives

Corollary 1. For $\alpha \in (1,2)$ , each harmonic mapping $f \in CO_H(\alpha)$ with the dilatation $\omega(z) = \left(\sin(\frac{2-\alpha}{2})\pi\right)e^{i\theta}z$ for $z \in \mathbb{D}$ is close-to-convex (univalent) in $\mathbb{D}$ . We conjecture that Corollary 1 is sharp in the sense that the number $\sin(\frac{2-\alpha}{2})\pi$ cannot be replaced by a larger one for a given $\alpha \in (1,2)$ . A function h analytic and locally univalent in $\mathbb{D}$ is said to have boundary rotation bounded by $K\pi$ , $K \geq 2$ , if for 0 < r < 1 (3) $$\int_0^{2\pi} \left| \operatorname{Re} \left( 1 + \frac{re^{i\theta}h''(re^{i\theta})}{h'(re^{i\theta})} \right) \right| d\theta \le K\pi.$$ Let $\mathcal{V}_K$ be the class of all analytic functions h in $\mathbb{D}$ (with the normalization h(0) = 0 = h'(0) - 1) having boundary rotation bounded by $K\pi$ . The reader is referred to Paatero [17] (see also [10, 13] and Section 2 for additional information about the class $\mathcal{V}_K$ ), where the study of these classes was initiated, for the geometric significance.

Theorem 5. Let $h \in \mathcal{V}_K$ with $2 \leq K \leq 4 - \delta$ for a fixed $\delta \in [0, 2]$ , and the dilatation satisfies the condition $|\omega(z)| < \sin(\frac{\delta \pi}{4})$ for $z \in \mathbb{D}$ . Then harmonic mapping $f = h + \overline{g}$ is close-to-convex and univalent in $\mathbb{D}$ . As an immediate corollary to this result, we have

Corollary 2. Let $f = h + \overline{g}$ be a harmonic mapping in $\mathbb{D}$ such that $h \in \mathcal{V}_K$ with $2 \leq K \leq 4 - \delta$ for a fixed $\delta \in [0,2]$ , and that $g'(z) = e^{i\theta} \sin(\frac{\delta \pi}{4})zh'(z)$ for $z \in \mathbb{D}$ . Then f is close-to-convex and univalent in $\mathbb{D}$ . We conjecture that Corollary 2 is sharp in the sense that the number $\sin(\frac{\delta\pi}{4})$ cannot be replaced by a larger one for a given $\delta < 2$ . Images of $\mathbb{D}$ under the close-to-convex mappings $f_{K,\delta}(z) = h(z) + \overline{g(z)}$ for certain values of $\delta$ and K with $2 \leq K \leq 4 - \delta$ , where $$h(z) = \frac{1}{K} \left[ \left( \frac{1+z}{1-z} \right)^{K/2} - 1 \right]$$ and $g'(z) = \sin\left( \frac{\delta \pi}{4} \right) z h'(z)$ for $z \in \mathbb{D}$ , are drawn in Figure 1(a)–(h) using mathematica as plots of the images of equally spaced radial segments and concentric circles of the unit disk. Finally, we consider the class $\mathcal{G}$ of functions $h \in \mathcal{A}$ such that (4) $$\operatorname{Re}\left(1 + \frac{zh''(z)}{h'(z)}\right) < \frac{3}{2} \text{ for } z \in \mathbb{D}.$$ Functions in $\mathcal{G}$ are known to be starlike in $\mathbb{D}$ . This class has been discussed recently, see for example [16] and the references therein. For this class we prove the following general result.

Theorem 6. Suppose that $h \in \mathcal{G}$ and satisfies the condition $g'(z) = \omega(z)h'(z)$ in $\mathbb{D}$ , where $\omega : \mathbb{D} \to \mathbb{D}$ is analytic, $\omega(0) = 0$ and $W(z) = z(1 + \omega(z))$ is starlike in $\mathbb{D}$ . Then the harmonic mapping $f = h + \overline{g}$ is close-to-convex and univalent in $\mathbb{D}$ .   (b) $$\delta = 1, K = 2.5$$      (a) $$\delta = 1.0 \ K = 2.05$$  (e) $$\delta = 1.9, K = 2.05$$ (f) $\delta = 0.1, K = 2.05$  FIGURE 1. The images of unit disk $\mathbb{D}$ under $f_{K,\delta}(z) = h(z) + \overline{g(z)}$ for certain values of $\delta$ and K

Corollary 3. Let $h \in \mathcal{G}$ and g be analytic in $\mathbb{D}$ such that $g'(z) = \lambda z^n h'(z)$ for some $n \in \mathbb{N}$ and $0 < |\lambda| \le 1/(n+1)$ . Then the harmonic mapping $f = h + \overline{g}$ is close-to-convex and univalent in $\mathbb{D}$ .

Lemma 1. If h ∈ VK, then there exists a sequence of functions {hn(z)} analytic in D such that (13) $$h'_{n}(z) = \frac{\prod_{k=1}^{n} (1 - \overline{x}_{k}z)^{\alpha_{k}}}{\prod_{k=1}^{n} (1 - \overline{y}_{k}z)^{\beta_{k}}},$$ where |xk| = 1, |yk| = 1, 0 ≤ αk, β^k ≤ 1 with (14) $$\sum_{k=1}^{n} \alpha_k = \frac{K}{2} - 1 \quad and \quad \sum_{k=1}^{n} \beta_k = \frac{K}{2} + 1,$$ and {hn} converges uniformly on compact subsets of D. That is, {hn(z)} is dense in the family VK. Proof of Theorem 5. Set F = h + ǫg, where |ǫ| = 1 and h ∈ VK. In view of Lemma 1, it suffices to choose h ∈ V^K so that $$h'(z) = \frac{\prod\limits_{k=1}^{n} (1 - \overline{x}_k z)^{\alpha_k}}{\prod\limits_{k=1}^{n} (1 - \overline{y}_k z)^{\beta_k}},$$ where |xk| = 1, |yk| = 1, 0 ≤ αk, β^k ≤ 1 satisfying the conditions (14). It is convenient to rewrite the last expression as $$h'(z) = \prod_{k=1}^{n} \left( \frac{1 - \overline{x}_k z}{1 - \overline{y}_k z} \right)^{\alpha_k} \cdot \prod_{k=1}^{n} (1 - \overline{y}_k z)^{t_k}, \quad t_k = \alpha_k - \beta_k.$$ Observe now that $\sum_{k=1}^{n} t_k = 2$ and thus, the function S defined by $$S(z) = \frac{\overline{c}z}{\prod_{k=1}^{n} (1 - \overline{y}_k z)^{t_k}} \quad (|c| = 1)$$ is starlike in $\mathbb{D}$ . Further, $$F'(z) = h'(z) + \epsilon g'(z) = (1 + \epsilon \omega(z))h'(z)$$ so that $$\frac{zF'(z)}{S(z)} = c(1 + \epsilon\omega(z)) \prod_{k=1}^{n} \left(\frac{1 - \overline{x}_k z}{1 - \overline{y}_k z}\right)^{\alpha_k}$$ where (by the hypothesis) $$\sum_{k=1}^{n} \alpha_k = \frac{K}{2} - 1 \le 1 - \frac{\delta}{2}.$$ Note that $|\arg(1+\epsilon\omega(z))| < \pi\delta/4$ . This observation shows that (with a suitably defined c on $\partial \mathbb{D}$ ) $$\left| \arg \left( \frac{zF'(z)}{S(z)} \right) \right| < \frac{\pi}{2} \left( \sum_{k=1}^{n} \alpha_k + \frac{\delta}{2} \right) = \frac{\pi}{2} \left( \frac{K}{2} - 1 + \frac{\delta}{2} \right) \le \frac{\pi}{2}$$ and thus, the function zF'(z)/S(z) has positive real part in $\mathbb{D}$ . It follows that $F(z) = h(z) + \epsilon g(z)$ is close-to-convex in $\mathbb{D}$ for each $|\epsilon| = 1$ and hence, by Lemma A, the harmonic function $f = h + \overline{g}$ is close-to-convex in $\mathbb{D}$ . 2.4. The class $\mathcal{G}$ . Now, we let $h \in \mathcal{G}$ . Then (4) holds. Clearly, (4) can be written as $$1 + \frac{zh''(z)}{h'(z)} \prec p(z) = \frac{1 - 2z}{1 - z} \text{ for } z \in \mathbb{D}$$ and thus, by the Hergtlotz representation for analytic functions with positive real part in the unit disk, it follows easily that $$\frac{h''(z)}{h'(z)} = -\int_{\partial \mathbb{D}} \frac{\overline{x}}{1 - \overline{x}z} d\mu(x) \text{ for } z \in \mathbb{D},$$ where $\mu$ is a probability measure on $\partial \mathbb{D}$ so that $\int_{\partial \mathbb{D}} d\mu(x) = 1$ . This means that $$h'(z) = \exp\left(\int_{\partial \mathbb{D}} \log(1 - \overline{x}z) \, d\mu(x)\right) \text{ for } z \in \mathbb{D}.$$ Thus, we have a sequence of functions $\{h_n(z)\}$ analytic in $\mathbb{D}$ such that (15) $$h'_n(z) = \prod_{k=1}^n (1 - \overline{x}_k z)^{\alpha_k}$$ where $|x_k| = 1$ , $0 \le \alpha_k \le 1$ for k = 1, 2, ..., n, $\sum_{k=1}^n \alpha_k = 1$ , and $h_n \to h$ uniformly on compact subsets of $\mathbb{D}$ . That is, $\{h_n(z)\}$ is dense in the family $\mathcal{G}$ . We observe that functions in $\mathcal{G}$ are bounded in $\mathbb{D}$ . Proof of Theorem 6. As in the proofs of previous theorems, we begin to set $F = h + \epsilon g$ , where $|\epsilon| = 1$ and $h \in \mathcal{G}$ . In view of the above discussion and (15), it suffices to prove the theorem for functions h of the form $$h'(z) = \prod_{k=1}^{n} (1 - \overline{x}_k z)^{\alpha_k}$$ where $|x_k| = 1$ , $0 \le \alpha_k \le 1$ for k = 1, 2, ..., n and $\sum_{k=1}^n \alpha_k = 1$ . Consequently, there exists a complex number c with |c| = 1 and such that $$\frac{czF'(z)}{W(z)} = c \prod_{k=1}^{n} (1 - \overline{x}_k z)^{\alpha_k}$$ has positive real part for $z \in \mathbb{D}$ , where W defined by $W(z) = z + \epsilon z \omega(z)$ is starlike for each $|\epsilon| = 1$ (by hypothesis). Thus, the harmonic function $f = h + \overline{g}$ is close-to-convex in $\mathbb{D}$ (by Lemma A).