On the Univalence of Polyharmonic Mappings

Theorem 1. Let $f(z) = |z|^{2(p-1)}G(z) + K(z)$ , where $G \in C^1(\mathbb{D})$ (not necessarily harmonic) and K harmonic. If K is univalent and $K(\mathbb{D})$ is a linearly connected domain with constant M, and if $$\frac{2(p-1)|G| + \Lambda_G}{|\lambda_K|} < \frac{1}{M},$$ then f(z) is univalent. Moreover, if $$\frac{2(p-1)|G|+\Lambda_G}{|\lambda_K|} \le C < \frac{1}{M},$$ then $f(\mathbb{D})$ is a linearly connected domain

Corollary 1. Let $f(z) = |z|^2 G(z) + K(z)$ , where $G \in C^1(D)$ (not necessarily harmonic). If K is univalent harmonic and $K(\mathbb{D})$ is a linearly connected domain with constant M, and if $$\frac{2|G| + \Lambda_G}{|\lambda_K|} < \frac{1}{M},$$ then f(z) is univalent. Moreover, if $$\frac{2|G| + \Lambda_G}{|\lambda_K|} \le C < \frac{1}{M},$$ then $f(\mathbb{D})$ is a linearly connected domain. We are now ready to prove the theorem for polyharmonic functions. In fact, we will state the theorem in its most general form and the polyharmonic functions will be a special case.

Corollary 2. Let $$f(z) = \sum_{k=1}^{p} |z|^{2(k-1)} G_{p-k+1}(z)$$ where $G_{p-k+1} \in C^1(D)$ (not necessarily harmonic) $k = \{2, ..., p\}$ (In particular f is polyharmonic). If $G_p$ is univalent harmonic and $G_p(\mathbb{D})$ is a a convex domain, and if $$\frac{\sum_{k=1}^{p-1} 2k |G_{p-k}| + (k-1)\Lambda_{G_{p-k}}}{|\lambda_{G_p}|} < 1,$$ then f(z) is univalent. Moreover, if $$\frac{\sum_{k=1}^{p-1} 2k |G_{p-k}| + (k-1)\Lambda_{G_{p-k}}}{|\lambda_{G_p}|} \le C < 1,$$ then $f(\mathbb{D})$ is a convex domain. In [6], the following theorem that allows to conclude the univalence of K from the univalence of f was proved:

Theorem 3. ([6],theorem 3). Let $f(z) = |z|^2 G(z) + K(z)$ be a biharmonic function in the unit disk $\mathbb{D}$ . Suppose f is univalent and $f(\mathbb{D})$ is a linearly connected domain—with constant M and satisfies $$\frac{2|G| + \Lambda_G}{|\lambda_f|} < \frac{1}{M},$$ then K(z) is univalent. Moreover, if $$\frac{2|G| + \Lambda_G}{|\lambda_f|} \le C < \frac{1}{M},$$ then $K(\mathbb{D})$ is a linearly connected domain. More generally, we have

Theorem 4. Let $f(z) = |z|^{2(p-1)}G(z) + K(z)$ , where $G \in C^1(\mathbb{D})$ (not necessarily harmonic) and K harmonic. If K is univalent and $K(\mathbb{D})$ is a linearly connected domain with constant M, and if $$\frac{2(p-1)|G| + \Lambda_G}{|\lambda_f|} < \frac{1}{M},$$ then k(z) is univalent. Moreover, if $$\frac{2(p-1)|G| + \Lambda_G}{|\lambda_f|} \le C < \frac{1}{M},$$ then $k(\mathbb{D})$ is a linearly connected domain

Theorem 5. Let $$f(z) = \sum_{k=1}^{p} |z|^{2(k-1)} G_{p-k+1}(z)$$ where $G_{p-k+1} \in C^1(\mathbb{D})$ (not necessarily harmonic), $k = \{2, ..., p\}$ . Suppose f is univalent and $f(\mathbb{D})$ is a linearly connected domain—with constant M and satisfies $$\frac{\sum_{k=1}^{p-1} 2k|G_{p-k}| + (2(k-1))(|\Lambda_{G_{p-k}}|)}{|\lambda_f|} < \frac{1}{M},$$ then $G_p(z)$ is univalent. Moreover, if $$\frac{\sum_{k=1}^{p-1} 2k |G_{p-k}| + (2(k-1))(|\Lambda_{G_{p-k}}|)}{|\lambda_f|} \le C < \frac{1}{M},$$ then $G_p(\mathbb{D})$ is a linearly connected domain.

Theorem 6. Let $f(z) = |z|^{2(p-1)}G(z) + K(z)$ , where $G \in C^1(\mathbb{D})$ (not necessarily harmonic) and K harmonic. If f is univalent and $f(\mathbb{D})$ is a linearly connected domain with constant M, and if $$\frac{2(p-1)|G| + \Lambda_G}{|\lambda f|} < \frac{1}{2M},$$ then $f_a(z) = a|z|^{2(p-1)}G(z) + K(z)$ is univalent for any a, such that |a| < 1. Moreover, if $$\frac{2(p-1)|G| + \Lambda_G}{|\lambda_f|} \le C < \frac{1}{2M},$$ then $f_a(\mathbb{D})$ is a a linearly connected domain.

Corollary 4. Let $f(z) = |z|^{2(p-1)}G(z) + K(z)$ , where $G \in C^1(\mathbb{D})$ (not necessarily harmonic) and K harmonic. If f is univalent and $f(\mathbb{D})$ is a convex domain, and if $$\frac{2(p-1)|G| + \Lambda_G}{|\lambda_f|} < \frac{1}{2},$$ then $f_a(z) = a|z|^{2(p-1)}G(z) + K(z)$ is univalent for any a, such that |a| < 1. Moreover, if $$\frac{2(p-1)|G| + \Lambda_G}{|\lambda_f|} \le C < \frac{1}{2},$$ then $f_a(\mathbb{D})$ is a a convex domain.

Definition 1. A domain $\Omega \subset \mathbb{C}$ is linearly connected if there exists a constant $M < \infty$ such that any two points $w_1, w_2 \in \Omega$ are joined by a path $\gamma, \gamma \subset \Omega$ , of length $\ell(\gamma) \leq M|w_1 - w_2|$ . Such a domain is necessarily a Jordan domain, and for piecewise smoothly bounded domains, linear connectivity is equivalent to the boundary having no inward-pointing cusps. In [16], Chuaqui and Hermandez, considered the relationship between the harmonic mapping $f = h + \overline{g}$ and its analytic factor h on linearly connected domains. They show that if h is an analytic univalent function, then every harmonic mapping $f = h + \overline{g}$ with dilatation $|\omega| < c$ is univalent if and only if $h(\mathbb{D})$ is linearly connected. In [6], Abdulhadi and El Hajj showed analogous results for biharmonic functions. In this paper we generalise these results for polyharmonic mappings. Moreover some results are obtained for logpolyharmonic mappings. Recently properties of logbiharmonic and logpolyharmonic mappings have been investigated (See [11,15]). Many physical problems are modelled by logbiharmonic mappings particulary those arising from fluid flow theory. The property of stable univalence is also considered.