In this section, we propose an extension of the integral transforms (1) and (2) to non-vanishing logharmonic mappings in \(\mathbb{D}\). To this end, we will use the fact that if *f* is a non-vanishing logharmonic mapping in \(\mathbb{D}\), there is a branch of log*f*, which is harmonic in \(\mathbb{D}\) with dilatation \(\omega _{\log f} = \omega _{f}\), and apply to it the theory that we developed in [8]. Note that \(\omega _{\log f} = \omega _{f}\) implies that log*f* is a sense-preserving harmonic mapping and therefore \(J_{f}\) is positive in \(\mathbb{D}\).

As was mentioned in Sect. 2, if *f* is a non-vanishing logharmonic mapping in \(\mathbb{D}\) with dilatation *ω*, then \(f=h\overline{g}\), where *h*, *g* are non-vanishing analytic functions in \(\mathbb{D}\). In this case \(\omega =g'h/gh'\), and if we assume the normalization \(h(0)=g(0)=1\), we can choose branches of log*f*, log*h*, and log*g* satisfying

$$ \log f(0)=\log h(0)=\log g(0)=0 \quad \text{and}\quad \log f=\log h+ \overline{\log g}. $$

The following lemma is a general tool that appears as a natural version of Theorem 1 in [12] for logharmonic mappings. We will use this lemma in the next subsections.

### Lemma 5.1

*Let* \(f=h\overline{g}\) *be a non*-*vanishing logharmonic mapping in* \(\mathbb{D}\) *with dilatation* *ω*, *and suppose that* \(\psi =\log h/g\) *is a univalent mapping such that* \(\Omega :=\psi (\mathbb{D})\) *is* *M*-*linearly connected*. *Then* *f* *is univalent in* \(\mathbb{D}\) *if* \(\|\omega \|<1/(2M+1)\).

### Proof

Suppose that there are \(z_{1}\neq z_{2}\) in \(\mathbb{D}\) such that \(f(z_{1})=f(z_{2})\), and let *S* be a path in Ω joining \(\psi (z_{1})\) to \(\psi (z_{2})\) such that \(\ell (S)\leq M |\psi (z_{1})-\psi (z_{2})|\). So,

$$ e^{\psi (z_{1})} \bigl\vert g(z_{1}) \bigr\vert ^{2}=e^{\psi (z_{2})} \bigl\vert g(z_{2}) \bigr\vert ^{2}, $$

and in consequence,

$$ \bigl\vert \psi (z_{1})-\psi (z_{2}) \bigr\vert \leq \bigl\vert 2\log g(z_{1})-2\log g(z_{2}) \bigr\vert \leq 2 \int _{\gamma } \biggl\vert \frac{g'(\xi )}{g(\xi )} \biggr\vert \vert \,d\xi \vert , $$

where \(\gamma =\psi ^{-1}(S)\). From here and the equality

$$ \frac{g'}{g}=\omega \frac{h'}{h}=\frac{\omega }{1-\omega }\psi ', $$

it follows that

$$ \bigl\vert \psi (z_{1})-\psi (z_{2}) \bigr\vert \leq 2\frac{ \Vert \omega \Vert }{1- \Vert \omega \Vert } \int _{\gamma } \bigl\vert \psi '(\xi ) \bigr\vert \vert \,d\xi \vert \leq 2 \frac{ \Vert \omega \Vert }{1- \Vert \omega \Vert }{\ell }(S)< \bigl\vert \psi (z_{1})- \psi (z_{2}) \bigr\vert , $$

if *ω* satisfies \(\|\omega \|<1/(2M+1)\). This contradiction ends the proof. □

### 5.1 Integral transform of the first type for non-vanishing logharmonic mappings

We consider a non-vanishing logharmonic mapping \(f=h\overline{g}\) defined in \(\mathbb{D}\), with dilatation \(\omega =g'h/h'g\), and normalized by \(h(0)=g(0)=1\). We suppose, moreover, that \(\varphi =\log h-\log g\) is zero only at \(z=0\). Given \(\alpha \in \overline{\mathbb{D}}\), we define the integral transform of the first type of *f* by

$$ f_{\alpha }=e^{H} \overline{e^{G}}= \exp \{H+\overline{G}\}, $$

(14)

where *H* and *G* satisfy the system

$$ H(z)-G(z)=\varphi _{\alpha }(z)= \int _{0}^{z} \biggl( \frac{\varphi (\zeta )}{\zeta } \biggr)^{\alpha }\,d\zeta \quad \text{and} \quad \frac{G'}{H'}=\alpha \omega , $$

with the initial conditions \(H(0)=G(0)=0\). In other words, \(f_{\alpha }\) is defined in such a way that \(\log f_{\alpha }=H+\overline{G}\) is a harmonic branch of the logarithm of \(f_{\alpha }\) in \(\mathbb{D}\), which is the horizontal shear of \(\varphi _{\alpha }\) with dilatation *αω*. The reader can find the details of the shear construction of harmonic mappings in [13].

### Theorem 5.1

*Let* \(f=h\overline{g}\) *be a non*-*vanishing logharmonic mapping in* \(\mathbb{D}\) *with dilatation* *ω*, *and let* \(f_{\alpha }\) *be defined by equation* (14). *If* \(\varphi =\log h/g\) *is a univalent function and* \(|\alpha |\leq 0.165\), *then* \(f_{\alpha }\) *is univalent*.

### Proof

For \(|\lambda |=1\), we define \(\Psi _{\lambda }=H+\lambda G\). A direct calculation shows that

$$ \frac{H''(z)}{H'(z)}=\alpha \biggl(\frac{\varphi '(z)}{\varphi (z)}- \frac{1}{z} \biggr)+\frac{\omega '_{\alpha }(z)}{1-\omega _{\alpha }(z)} $$

and

$$ \begin{aligned} \frac{z\Psi ''_{\lambda }(z)}{\Psi '_{\lambda }(z)}&= \frac{zH''(z)}{H'(z)}+ \frac{\lambda z\omega '_{\alpha }(z)}{1+\lambda \omega _{\alpha }(z)} \\ &=\alpha \biggl(\frac{z\varphi '(z)}{\varphi (z)}-1 \biggr)+ \frac{(1+\lambda )z\omega '_{\alpha }(z)}{(1-\omega _{\alpha }(z))(1+\lambda \omega _{\alpha }(z))}, \end{aligned} $$

(15)

from which we see that

$$\begin{aligned} \bigl(1- \vert z \vert ^{2} \bigr) \biggl\vert \frac{z\Psi ''_{\lambda }(z)}{\Psi '_{\lambda }(z)} \biggr\vert &\leq \vert \alpha \vert \biggl[ \bigl(1- \vert z \vert ^{2} \bigr) \biggl\vert \frac{z\varphi '(z)}{\varphi (z)}-1 \biggr\vert +2 \frac{(1- \vert z \vert ^{2}) \vert \omega '(z) \vert }{(1- \vert \alpha \vert \vert \omega (z) \vert )^{2}} \biggr] \\ & \leq \vert \alpha \vert \biggl[4+2 \frac{(1- \vert \omega (z) \vert ^{2}) \Vert \omega ^{*} \Vert }{(1- \vert \alpha \vert \vert \omega (z) \vert )^{2}} \biggr] \\ &\leq 2 \vert \alpha \vert \biggl(2+\frac{1}{1- \vert \alpha \vert ^{2}} \biggr), \end{aligned}$$

being the last inequality a consequence of

$$ \max_{z\in \mathbb{D}} \frac{(1- \vert \omega (z) \vert ^{2})}{(1- \vert \alpha \vert \vert \omega (z) \vert )^{2}}\leq \frac{1}{1- \vert \alpha \vert ^{2}} \quad \text{and}\quad \bigl\Vert \omega ^{*} \bigr\Vert \leq 1. $$

It follows from Becker’s criterion that \(\Psi _{\lambda }\) is univalent if \(2|\alpha | (2+\frac{1}{1-|\alpha |^{2}} )\leq 1\), whence \(H+\overline{G}\) is stable harmonic univalent if \(|\alpha |\leq 0.165\). In consequence, \(f_{\alpha }\) is univalent for these values of *α*. □

### Proposition 5.1

*Let* \(f=h\overline{g}\) *be a non*-*vanishing logharmonic mapping in* \(\mathbb{D}\) *with dilatation* *ω*, *and let* \(f_{\alpha }\) *be defined by equation* (14). *If* \(\varphi =\log h/g\) *is a starlike function and* \(\alpha \in (-0.303,0.707)\), *then* \(f_{\alpha }\) *is a univalent logharmonic mapping in* \(\mathbb{D}\).

### Proof

Since *φ* is starlike, we have \(\operatorname{Re} \lbrace z\varphi '(z)/\varphi (z) \rbrace >0\) for all \(z\in \mathbb{D}\). So, for \(\alpha >0\) and \(\Psi _{\lambda }\) defined as in the previous theorem, we get from (15) that

$$\begin{aligned} & \int _{\theta _{1}}^{\theta _{2}}\operatorname{Re} \biggl\lbrace 1+ z \frac{\Psi _{\lambda }''(z)}{\Psi _{\lambda }'(z)} \biggr\rbrace \,d \theta \\ &\quad = \int _{\theta _{1}}^{\theta _{2}} \biggl[1-\alpha +\alpha \operatorname{Re} \biggl\lbrace z\frac{\varphi '(z)}{\varphi (z)} + \frac{\lambda z\omega '(z)}{1+\lambda \omega _{\alpha }(z)} + \frac{z\omega '}{1-\omega _{\alpha }(z)} \biggr\rbrace \biggr]\,d \theta \\ &\quad \geq ( 1-\alpha ) (\theta _{2}-\theta _{1})+\operatorname{Arg} \biggl\lbrace \frac{1+\lambda \alpha \omega (re^{i\theta _{2}})}{1+\lambda \alpha \omega (re^{i\theta _{1}})} \cdot \frac{1-\alpha \omega (re^{i\theta _{2}})}{1-\alpha \omega (re^{i\theta _{1}})} \biggr\rbrace \\ &\quad > -4\arcsin (\alpha ) \end{aligned}$$

for all \(0\leq \theta _{2}-\theta _{1}\leq 2\pi \). Therefore,

$$ \int _{\theta _{1}}^{\theta _{2}}\operatorname{Re} \biggl\lbrace 1+ z \frac{\Psi _{\lambda }''(z)}{\Psi _{\lambda }'(z)} \biggr\rbrace \,d \theta >-\pi \quad \text{if } 0\leq \alpha \leq \sqrt{2}/2, $$

from where \(\Psi _{\lambda }\) is a close to convex mapping in \(\mathbb{D}\) if \(0\leq \alpha \leq \sqrt{2}/2\).

On the other hand, since *φ* is starlike, then \(\vert \operatorname{Arg} \lbrace \varphi (z)/z \rbrace \vert \leq 2\arcsin (|z|)\) and, in consequence,

$$\begin{aligned} \bigl\vert \operatorname{Arg} \bigl\lbrace \Psi _{\lambda }'(z) \bigr\rbrace \bigr\vert &= \biggl\vert \operatorname{Arg} \biggl\lbrace \biggl( \frac{\varphi (z)}{z} \biggr)^{\alpha } \frac{1+\lambda \alpha \omega (z)}{1-\alpha \omega (z)} \biggr\rbrace \biggr\vert \\ &\leq 2 \vert \alpha \vert \arcsin (r)+2\arcsin \bigl(r \vert \alpha \vert \bigr),\quad r= \vert z \vert . \end{aligned}$$

Hence, by a straightforward calculation, we have \(\vert \operatorname{Arg} \lbrace \Psi _{\lambda }'(z) \rbrace \vert < \pi /2\) if \(|\alpha |< 0.303\), which implies that \(\operatorname{Re} \lbrace \Psi _{\lambda }'(z) \rbrace >0\) for these values of *α* and hence \(\Psi _{\lambda }\) is a close to convex mapping in \(\mathbb{D}\), when \(|\alpha |< 0.303\). From this and the discussion above, it follows that \(\log f_{\alpha }=H+\overline{G}\) is stable harmonic close-to-convex if \(\alpha \in (-0.303, 0.707 )\), whence \(f_{\alpha }\) is univalent in \(\mathbb{D}\). □

### Proposition 5.2

*Let* \(f=h\overline{g}\) *be a non*-*vanishing logharmonic mapping defined in* \(\mathbb{D}\) *with* \(\|\omega \|<1/3\), *and let* \(f_{\alpha }\) *be defined by equation* (14). *If* \(\varphi =\log h/g\) *is a convex function and* \(\alpha \in [0,2]\), *then* \(f_{\alpha }\) *is univalent*.

### Proof

The proof follows as a direct application of Lemma 5.1 and the fact that \(\varphi _{\alpha }\) is a convex mapping for \(\alpha \in [0,2]\), the case in which \(\varphi _{\alpha }(\mathbb{D})\) is a *M*-linearly connected domain with \(M=1\). □

### 5.2 Integral transform of the second type for non-vanishing logharmonic mappings

The definition of the integral transform of the second type for non-vanishing logharmonic mappings is completely analogous to that given in the previous subsection: let \(f=h\overline{g}\) be a non-vanishing logharmonic mapping in \(\mathbb{D}\) with dilatation \(\omega =g'h/h'g\) and normalized by \(h(0)=g(0)=1\). Note that from the condition \(|\omega (z)|<1\) for all \(z\in \mathbb{D}\), it follows that \(\varphi =\log h-\log g\) is locally univalent in \(\mathbb{D}\). We define the logharmonic mapping \(F_{\alpha }=e^{H}\overline{e^{G}}\), where *H*, *G* satisfy the system

$$ H(z)-G(z)=\Phi _{\alpha }(z)= \int _{0}^{z} \bigl(\varphi '( \zeta ) \bigr)^{\alpha }\,d\zeta\quad \text{and}\quad \omega _{F_{\alpha }}= \alpha \omega $$

(16)

with the initial conditions \(H(0)=G(0)=0\).

### Theorem 5.2

*Let* \(f=h\overline{g}\) *be a non*-*vanishing logharmonic mapping in* \(\mathbb{D}\) *with dilatation* *ω*, *and let* \(F_{\alpha }\) *be defined by equation* (16). *If* *φ* *is a univalent function and* \(|\alpha |\leq 0.125\), *then* \(F_{\alpha }\) *is univalent*.

### Proof

For \(|\lambda |=1\), we define \(\Psi _{\lambda }=H+\lambda G\). Using (16), we obtain by a direct calculation

$$ \frac{H''(z)}{H'(z)}=\alpha \biggl(\frac{\varphi '(z)}{\varphi (z)}- \frac{1}{z} \biggr)+\frac{\omega '_{\alpha }(z)}{1-\omega _{\alpha }(z)} $$

and

$$ \frac{z\Psi ''_{\lambda }(z)}{\Psi '_{\lambda }(z)}=\frac{zH''(z)}{H'(z)}+ \frac{\lambda z\omega '_{\alpha }(z)}{1+\lambda \omega _{\alpha }(z)}= \alpha \biggl(\frac{z\varphi '(z)}{\varphi (z)}-1 \biggr)+ \frac{(1+\lambda )z\omega '_{\alpha }(z)}{(1-\omega _{\alpha }(z))(1+\lambda \omega _{\alpha }(z))}. $$

It follows from the univalence of *φ* and

$$ \max_{z\in \mathbb{D}} \frac{(1- \vert \omega (z) \vert ^{2})}{(1- \vert \alpha \vert \vert \omega (z) \vert )^{2}}\leq \frac{1}{1- \vert \alpha \vert ^{2}} $$

that

$$\begin{aligned} & \bigl(1- \vert z \vert ^{2} \bigr) \biggl\vert \frac{z\Psi ''_{\lambda }(z)}{\Psi '_{\lambda }(z)} \biggr\vert \\ &\quad \leq \vert \alpha \vert \biggl( \biggl\vert \bigl(1- \vert z \vert ^{2} \bigr) \frac{\varphi ''(z)}{\varphi '(z)}-2\overline{z} \biggr\vert +2 \vert z \vert ^{2}+2 \frac{(1- \vert \omega (z) \vert ^{2}) \Vert \omega ^{*} \Vert }{(1- \vert \alpha \vert \vert \omega (z) \vert )^{2}} \biggr) \\ &\quad \leq 2 \vert \alpha \vert \biggl(3+\frac{1}{1- \vert \alpha \vert ^{2}} \biggr). \end{aligned}$$

Consequently, we conclude from Becker’s criterion that \(\Psi _{\lambda }\) is univalent if \(|\alpha |\leq 0.125\), and therefore \(\log F_{\alpha }\) is stable harmonic univalent for these values of *α*. This completes the proof of the theorem. □

### Proposition 5.3

*Let* \(f=h\overline{g}\) *be a non*-*vanishing logharmonic mapping defined in* \(\mathbb{D}\) *with dilatation* *ω*, *and let* \(F_{\alpha }\) *be defined by equation* (16). *If* *φ* *is a convex function and* \(\alpha \in (-0.303,0.5605)\), *then* \(F_{\alpha }\) *is a univalent logharmonic mapping in* \(\mathbb{D}\).

### Proof

The proof is almost the same as that of Proposition 5.1. Indeed, with the same notation, one sees that

$$\begin{aligned} & \int _{\theta _{1}}^{\theta _{2}}\operatorname{Re} \biggl\lbrace 1+ z \frac{\Psi _{\lambda }''(z)}{\Psi _{\lambda }'(z)} \biggr\rbrace \,d \theta \\ &\quad = \int _{\theta _{1}}^{\theta _{2}} \biggl[1-\alpha +\alpha \operatorname{Re} \biggl\lbrace z\frac{\varphi ''(z)}{\varphi '(z)} + \frac{\lambda z\omega '(z)}{1+\lambda \omega _{\alpha }(z)} + \frac{z\omega '}{1-\omega _{\alpha }(z)} \biggr\rbrace \biggr]\,d \theta \\ &\quad \geq ( 1-2\alpha ) (\theta _{2}-\theta _{1})+ \operatorname{Arg} \biggl\lbrace \frac{1+\lambda \alpha \omega (re^{i\theta _{2}})}{1+\lambda \alpha \omega (re^{i\theta _{1}})} \cdot \frac{1-\alpha \omega (re^{i\theta _{2}})}{1-\alpha \omega (re^{i\theta _{1}})} \biggr\rbrace \\ &\quad > -4\arcsin (\alpha )>-\pi \end{aligned}$$

for all \(0\leq \theta _{2}-\theta _{1}\leq 2\pi \) and \(0\leq \alpha < 1/2\), which implies that \(\Psi _{\lambda }\) is a close to convex mapping in the unit disk if \(0\leq \alpha < 1/2\). The same conclusion is obtained if we assume \(1/2\leq \alpha <0.5605\) since in this case

$$ \int _{\theta _{1}}^{\theta _{2}}\operatorname{Re} \biggl\lbrace 1+ z \frac{\Psi _{\lambda }''(z)}{\Psi _{\lambda }'(z)} \biggr\rbrace \,d \theta \geq 2\pi (1-2\alpha )-4\arcsin ( \alpha )>-\pi $$

for all \(0\leq \theta _{2}-\theta _{1}\leq 2\pi \). On the other hand,

$$\begin{aligned} \bigl\vert \operatorname{Arg} \bigl\lbrace \Psi _{\lambda }'(z) \bigr\rbrace \bigr\vert &= \biggl\vert \operatorname{Arg} \biggl\lbrace \bigl(\varphi '(z) \bigr)^{\alpha } \frac{1+\lambda \alpha \omega (z)}{1-\alpha \omega (z)} \biggr\rbrace \biggr\vert \\ &\leq \pi \vert \alpha \vert +2\arcsin \bigl(r \vert \alpha \vert \bigr),\quad r= \vert z \vert \\ & < \pi /2 \end{aligned}$$

if \(|\alpha |< 0.303\). The proof is completed by proceeding as at the end of the proof of Proposition 5.1.

Here we have used the fact that 0.5605 and 0.303 are the approximate roots of \(3\pi -4\pi x-4\arcsin (x)=0\) and \(\pi x-\pi /2+2\arcsin (x)=0\), respectively. □

The proof of the following proposition is essentially the same as that of Proposition 5.2; so we omit its proof.

### Proposition 5.4

*Let* \(f=h\overline{g}\) *be a non*-*vanishing logharmonic mapping defined in the unit disk with* \(\|\omega \|<1/3\), *and let* \(F_{\alpha }\) *be defined by equation* (16). *If* *φ* *is a convex function*, *then* \(F_{\alpha }\) *is univalent for* \(\alpha \in [0,1]\).