Some Geometric Characterizations of a Certain Class of Log-Harmonic Mappings

Abstract

This article investigates structural properties of a class of log-harmonic mappings associated with starlike analytic functions in the unit disk. Beginning with a general representation of the log-harmonic mappings, we use sharp inequalities using analytic and dilatation functions to determine growth and distortion bounds for the mappings and their complex derivatives. The exact region where the log harmonic mappings of the form $f\left(z\right)=zh\left(z\right)\overline{h^{\prime }\left(z\right)}$, with h being starlike analytic, give the starlike image is determined. Subordination relations for logarithmic derivatives are established, connecting the mappings with the Schwarz lemma and the Carathéodory class. Furthermore, we obtain the growth estimates for the underlying starlike functions h and their derivatives, as well as accurate inequalities governing the arclength of the circle image under log-harmonic mappings. These findings contribute to the geometric function theory of log-harmonic mappings.

1. Motivation

Potential theory is a well-established subject in mathematical analysis that studies the properties of harmonic functions, which are solutions of Laplace’s equation,

$$\Delta u(x,y)={\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}=0.$$

(1)

For a complex-valued function $f=u+iv$, we say that f is harmonic if both u and v are harmonic. The study of harmonic mappings serves as a link between the scalar potentials of classical physics and the geometric framework of complex analysis. Physically, a harmonic mapping represents two potential functions simultaneously: the real part can be interpreted as a velocity potential, and the imaginary part corresponds to a stream function in fluid dynamics [1]. Harmonic mappings generalize analytic functions because every analytic function is harmonic, while the converse is not true [2]. Harmonic functions and their properties have been extensively explored in the literature [3].

Harmonic functions are closely connected to quasiconformal mappings and minimal surfaces and play a crucial role in both mathematics and physics, including elasticity, fluid flow, and image processing [4,5].

While studying the representation of a harmonic mapping as analytic functions, we note that a complex-valued harmonic mapping f in the open unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ can be written in the canonical form

$$f\left(z\right)=h\left(z\right)+\overline{g\left(z\right)},$$

where h and g are analytic functions in $\mathbb{D}$.

The function

$$\omega \left(z\right)={\displaystyle \frac{g^{\prime }\left(z\right)}{h^{\prime }\left(z\right)}}$$

is called the (second complex) dilatation of f. A harmonic mapping f is said to be sense-preserving in $\mathbb{D}$ if and only if $\left|\omega \right(z\left)\right|<1$ for all $z\in \mathbb{D}$, in which case, the Jacobian of f satisfies

$$J_f\left(z\right)=|h^{\prime }{\left(z\right)|}^2-|g^{\prime }{\left(z\right)|}^2={\left|h^{\prime }\left(z\right)\right|}^2\left(1-{\left|\omega \left(z\right)\right|}^2\right)>0.$$

The dilatation $\omega$ plays a central role in determining the geometric behavior of harmonic mappings. In particular, bounds on $\left|\omega \right(z\left)\right|$ control univalence, distortion, and quasiconformality of f. If ${\parallel \omega \parallel }_{\infty }\le k<1$, then f is K-quasiconformal with

$$K={\displaystyle \frac{1+k}{1-k}}.$$

Many subclasses of harmonic mappings are defined by imposing analytic or geometric restrictions on the dilatation, such as constant dilatation, rotationally symmetric dilatation, or dilatations subordinate to Carathéodory functions. Distortion theorems describe how the size, derivatives, and area of harmonic mappings are controlled inside the unit disk. For univalent harmonic mappings, growth and distortion inequalities provide upper and lower bounds for $\left|f\right(z\left)\right|$ and its derivatives in terms of $\left|z\right|$. These results extend the classical distortion theorems for univalent analytic functions to the harmonic setting and show that the distortion increases as z approaches the boundary of $\mathbb{D}$. The presence of the co-analytic part introduces additional flexibility compared to the analytic case, leading to richer geometric behavior. We refer [6,7] for more details on structural studies of various subclasses of planar harmonic mappings.

The dilatation plays a central role in distortion estimates. If the dilatation is bounded by a constant $k<1$, then f is quasiconformal and the distortion bounds depend explicitly on k. Moreover, the Jacobian of a sense-preserving harmonic mapping remains positive and admits explicit estimates that connect area distortion to the size of the dilatation. Significant progress has been made in the study of geometric properties of harmonic mappings. Growth, distortion, and covering theorems have been established for various subclasses of univalent and quasiconformal harmonic mappings, extending classical results from the theory of univalent analytic functions. Sharp coefficient bounds and extremal problems have been investigated for harmonic starlike, convex, and close-to-convex mappings, as well as for subclasses defined by differential operators and subordination conditions. (see also [4,8]).

Normal family theory and function space properties have also been extended to harmonic mappings. Recent works establish criteria for normality, compactness, and boundedness of families of harmonic mappings in terms of growth conditions, dilatations, and pre-Schwarzian norms. In addition, Landau–Bloch-type theorems for harmonic and quasiregular mappings have been developed, yielding sharp lower bounds for injectivity radii and image domains. Recent studies have also emphasized the role of distortion and function-space properties in harmonic mapping theory. Das and Rasila analyzed harmonic quasiregular mappings in Bergman spaces and obtained norm estimates linking quasiregularity with integrability conditions. On the other hand, Mateljević, Salimov, and Sevost’yanov established Hölder and Lipschitz continuity results for harmonic mappings in Orlicz–Sobolev classes and derived distortion estimates under finite-distortion assumptions. These contributions highlight the deep interplay between geometric properties and functional analytic frameworks in the study of harmonic mappings [8,9].

Beyond the planar case, harmonic map theory continues to advance through new existence, regularity, and rigidity results for harmonic maps between Riemannian manifolds. These developments demonstrate strong interactions between geometric analysis, partial differential equations, and classical complex function theory.

Despite these advances, many open problems remain, particularly concerning sharp estimates, extremal functions, and the precise influence of dilatations on the geometric behavior of harmonic mappings. These challenges continue to motivate ongoing research in this area.

Next, we emphasize the concept of Log-harmonic mappings that are a generalization of harmonic mappings in which the logarithm of the modulus of a function is harmonic, rather than the function itself [10]. This modification allows potential theory to shift from an additive to a multiplicative framework. These mappings have practical applications in physics and image processing due to their ability to model nonlinear structures and enhance features in complex domains.

In physics, log-harmonic mappings are often used to describe potential functions and flow patterns in fluid dynamics, particularly when radial or logarithmic symmetry is present. They also provide a mathematical framework for analyzing problems involving logarithmic potentials, which naturally occur in electrostatics, gravitation, and fluid flow [11]. Their invariance properties under conformal-type deformations make them effective in modeling quasi-symmetric physical systems.

In image processing, log-harmonic mappings contribute to local contrast enhancement and feature preservation. They form the foundation for nonlinear image transformations where pixel intensities are modified in a multiplicative or logarithmic manner. This is especially significant in medical imaging, remote sensing, and low-light enhancement tasks [12]. Many well-known techniques, such as Retinex-based enhancement, adaptive histogram equalization, and contrast-limited methods, can be interpreted within a log-harmonic framework, ensuring that fine details are preserved while global visibility is improved. Thus, log-harmonic mappings establish a strong connection between abstract geometric function theory and applied computational methodologies.

Beyond theoretical importance, log-harmonic mappings have found applications in diverse scientific domains. In mathematical physics, they generalize harmonic functions that already describe incompressible fluid flow, potential theory, and elasticity, with the logarithmic factor allowing for more accurate modeling of physical systems [4,13]. They appear naturally in analytic–anti-analytic interactions and iterated function systems in complex dynamics, offering insights into domain deformations and the formation of fractal geometries.

Their orientation-preserving nature and controlled distortion have also made them useful in computer graphics and image processing, including texture mapping, image warping, and shape analysis. In medical imaging, their resemblance to quasiconformal mappings enables the modeling of elastic deformations in biological tissues—an essential step in MRI and CT image registration and analysis [12]. Finally, log-harmonic mappings have been applied in cartography and geometric modeling to create nearly distortion-free projections, maintaining both angular and area-preserving properties.

2. Introduction

In the mid-1990s, Abdulhadi and Khuri [14] introduced log harmonic mappings in relation to the nonlinear elliptic partial differential equation.

Consider the nonlinear elliptic partial differential equation,

$$\overline{\left({\displaystyle \frac{f_{\overline{z}}\left(z\right)}{f\left(z\right)}}\right)}=w\left(z\right){\displaystyle \frac{f_z\left(z\right)}{f\left(z\right)}},$$

(2)

where $f_z={\displaystyle \frac{\partial f}{\partial z}}$, $f_{\overline{z}}={\displaystyle \frac{\partial f}{\partial \overline{z}}}$ and $w\left(z\right)$ is called the second dilatation of f, $\left|w\right(z\left)\right|<1$ for $\left|z\right|<1.$

Definition 1 (Log harmonic mapping).

A logarithmic harmonic mapping is defined as a complex-valued function that satisfies the nonlinear elliptic partial differential Equation (2) on the unit disk $\mathbb{D}$.

Definition 2 (Starlike functions and Starlike log harmonic mappings).

Let $\mathcal{A}$ represent the family of analytic functions of the form

$$h\left(z\right)=z+\sum _{n=2}^{\infty }{a}_n{z}^n,z\in \mathbb{D},$$

which are normalized so that $h\left(0\right)=0$ and $h^{\prime }\left(0\right)=1$. When the image of an analytic function $h\in \mathcal{A}$ is starlike with regard to the origin, it is called starlike. If

$$\Re \left\{{\displaystyle \frac{z{h}^{\prime }\left(z\right)}{h\left(z\right)}}\right\}>0,z\in \mathbb{D},$$

then h is equivalently a member of the starlike class ${\mathcal{S}}^{*}$.

Similar to this analytic context, a log-harmonic mapping in the unit disk $\mathbb{D}$ of the form

$$f\left(z\right)=zh\left(z\right)\overline{g\left(z\right)},z\in \mathbb{D},$$

is defined as a starlike log-harmonic mapping if

$$\Re \left({\displaystyle \frac{z{f}_z\left(z\right)-\overline{z}{f}_{\overline{z}}\left(z\right)}{f\left(z\right)}}\right)>0,z\in \mathbb{D}\setminus \left\{0\right\},$$

where h and g are analytic in $\mathbb{D}$ with $h\left(0\right)=g\left(0\right)=1$.

Definition 3 (Subordination).

Let p and q be analytic functions in $\mathbb{D}$. We say that p is subordinate to q, written $p\left(z\right)\prec q\left(z\right)$, if there exists a Schwarz function $\omega \left(z\right)$, analytic in $\mathbb{D}$ with $\omega \left(0\right)=0$ and $\left|\omega \right(z\left)\right|<1$, such that

$$p\left(z\right)=q\left(\omega \right(z\left)\right),z\in \mathbb{D}.$$

Definition 4 (Carathéodory class).

Let $\mathcal{P}$ denote the Carathéodory class consisting of all analytic functions p in the unit disk $\mathbb{D}$ of the form

$$p\left(z\right)=1+p_{1}z+p_2{z}^2+\dots ,$$

satisfying

$$\Re \left(p\right(z\left)\right)>0,z\in \mathbb{D}.$$

The following lemma highlights some the basic properties of the Carathéodory class.

Lemma 1.

(i) Let $p\in \mathcal{P}$. The coefficients satisfy sharp bounds, specifically $|p_n|\le 2$, $n\ge 1.$

(ii) 

If ϕ is starlike analytic function, then for some $p\in \mathcal{P}, p\left(z\right)={\displaystyle \frac{z{\varphi }^{\prime }\left(z\right)}{\varphi \left(z\right)}}.$

(iii) 

The function $p\left(z\right)$ satisfies ${\displaystyle \frac{1-r}{1+r}}\le \Re \left(p\left(z\right)\right)\le \left|p\left(z\right)\right|\le {\displaystyle \frac{1+r}{1-r}}$ for $\left|z\right|=r.$

(iv) 

For $p\in \mathcal{P}$ and $\left|z\right|=r<1$, the sharp bound on the derivative is

$$|p^{\prime }\left(z\right)|\le {\displaystyle \frac{2\Re \left(p\left(z\right)\right)}{1-r^2}}\le {\displaystyle \frac{2(1+r)}{{(1-r)}^3}}.$$

We can note that the Jacobian of a non-constant logharmonic mapping f defined by $J_f:=|f_z{|}^2{(1-|w|}^2)>0$ in the open unit disk $\mathbb{D}.$ Therefore, it is always sense-preserving in $\mathbb{D}$.

A univalent logharmonic mapping f that vanishes only at zero can be represented with the help of two analytic functions h and g as

$$f\left(z\right)={z\left|z\right|}^{2\beta }h\left(z\right)\overline{g\left(z\right)},$$

where

$$\beta =\overline{w\left(0\right)}{\displaystyle \frac{1+w\left(0\right)}{1-{\left|w\left(0\right)\right|}^2}}$$

and $Re\beta >-{\displaystyle \frac{1}{2}}$. The restriction guarantees the positivity of the Jacobian $J_f$ in a neighbourhood of the origin and therefore ensures that the log-harmonic mapping is sense-preserving and locally univalent. The logharmonic mapping that is not zero at zero has the form $f\left(z\right)=h\left(z\right)\overline{g\left(z\right)}$. There is a close link between logharmonic and harmonic mapping, since for logharmonic mapping f, $\log f\left(e^z\right)$ is a univalent harmonic mapping of the left half-plane $\{z:Re(z)<0\}.$

Abdulhadi and Khuri’s (1996, 1999) [14] groundbreaking work established the essential features of log-harmonic mappings, such as existence and uniqueness results, structural representations, and Jacobian behavior. Since then, this type of mapping has received more attention in geometric function theory [2,5,6,15,16,17,18].

Such mappings admit a canonical representation in terms of two analytic functions and exhibit rich geometric behavior. In recent years, significant progress has been made in studying the geometric properties of various subclasses of log-harmonic mappings. Growth, distortion, and covering theorems have been established for log-harmonic mappings associated with starlike and spirallike analytic functions. In particular, sharp bounds for $\left|f\right(z\left)\right|$, $|f_z\left(z\right)|$, and the Jacobian $J_f\left(z\right)$ have been obtained for univalent log-harmonic mappings related to the starlike class $S^{*}$, extending classical results from analytic function theory to the log-harmonic setting.

Another active direction of research concerns coefficient problems and radius estimates. Several authors have investigated the radius of starlikeness, spirallikeness, and convexity for different families of log-harmonic mappings, including close-to-starlike and symmetric subclasses. Bohr-type inequalities and improved Bohr radii for log-harmonic mappings have also attracted recent attention, reflecting growing interest in quantitative geometric bounds. More recently, the notions of pre-Schwarzian and Schwarzian derivatives have been introduced for non-vanishing log-harmonic mappings. These operators provide useful criteria for univalence, normality, and boundedness, paralleling the corresponding theories for analytic and harmonic functions. Normal family criteria and compactness properties for families of log-harmonic mappings have further been developed, with applications to iteration theory and complex dynamics.

Furthermore, generalizations such as log-p-harmonic mappings and log-harmonic Bloch-type spaces have been explored, indicating the continued expansion of the theory. Despite these advances, several problems remain open, particularly concerning the sharpness of bounds, extremal functions, and the influence of analytic dilatations on the geometric behavior of log-harmonic mappings. These challenges continue to motivate further investigation in this area.

Log-harmonic mappings of the form

$$f\left(z\right)=zh\left(z\right)\overline{g\left(z\right)},$$

where h and g are analytic in the unit disk $\mathbb{D}$ and satisfy certain normalizing conditions, constitute an important subclass of complex mappings. In the analytic case, such functions serve as natural analogues of classical univalent starlike and convex mappings. The class $S_{LH}^{*}$ was introduced to investigate log-harmonic functions whose images are starlike with respect to the origin. A log-harmonic mapping f defined in the unit disk $\mathbb{D}$ is said to belong to the class $S_{LH}^{*}$ if f is sense-preserving, locally univalent, and satisfies

$$\Re \left({\displaystyle \frac{z{f}_z\left(z\right)-\overline{z}{f}_{\overline{z}}\left(z\right)}{f\left(z\right)}}\right)>0,z\in \mathbb{D}.$$

Functions belonging to this class satisfy growth and distortion estimates that parallel, yet differ significantly from, those in the standard theory of analytic univalent functions [15,19].

Substantial progress was made during the 2000s in understanding subclasses of log-harmonic mappings, particularly regarding univalence, starlikeness, convexity, and close-to-convexity. Several authors have derived sharp estimates for the modulus of log-harmonic mappings and their complex partial derivatives $f_z$ and $f_{\overline{z}}$ [19]. More recently, the theory has evolved in several directions. Coefficient problems such as the Fekete–Szegő functional, radius results, and Bohr-type inequalities have been explored for various subclasses of log-harmonic functions [14]. These studies have deepened the connections between log-harmonic mappings and broader themes in modern geometric function theory, including subordination theory, Bloch-type space investigations, and Bohr radius phenomena. Moreover, the applicability of log-harmonic mappings has extended to higher-dimensional and applied contexts, underscoring their versatility and significance beyond classical function theory.

3. Main Results

The behavior of $f\left(z\right)=zh\left(z\right)\overline{h^{\prime }\left(z\right)}$ is dependent on the relationship between $h\in S^{*}$, the class of starlike analytic functions and an associated Carathéodory function. Therefore, a representation formula is necessary. As a result, we present the lemma below, which will be used frequently in subsequent studies.

Lemma 2. 

Suppose $f\left(z\right)=zh\left(z\right)\overline{h^{\prime }\left(z\right)}$ is a logarithmic harmonic mapping with respect to dilation $w\in B$, where B is the unit ball of analytic functions and $h\in S^{*}$. Then there exists a function $p\in \mathcal{P}$ (the Carathéodory class) such that

$$f\left(z\right)={\displaystyle \frac{z^2}{p\left(z\right)}}\exp \left(2Re\int {\displaystyle \frac{w\left(z\right)}{1-w\left(z\right)}}\left({\displaystyle \frac{2}{z}}-{\displaystyle \frac{p^{\prime }\left(z\right)}{p\left(z\right)}}\right)dz\right).$$

Proof. 

Let $f\left(z\right)=zh\left(z\right)\overline{h^{\prime }\left(z\right)}$ be a logarithmic harmonic mapping with respect to $w\in B$, and suppose that $h\in S^{*}$. Then, define

$$p\left(z\right)={\displaystyle \frac{z{h}^{\prime }\left(z\right)}{h\left(z\right)}}\in \mathcal{P}.$$

This implies

$$h\left(z\right)={\displaystyle \frac{z{h}^{\prime }\left(z\right)}{p\left(z\right)}}.$$

(3)

Logarithmic differentiation of (3) yields

$${\displaystyle \frac{1}{z}}+{\displaystyle \frac{h^{″}\left(z\right)}{h^{\prime }\left(z\right)}}={\displaystyle \frac{h^{\prime }\left(z\right)}{h\left(z\right)}}+{\displaystyle \frac{p^{\prime }\left(z\right)}{p\left(z\right)}}.$$

This provides

$${\displaystyle \frac{h^{\prime }\left(z\right)}{h\left(z\right)}}={\displaystyle \frac{1}{z}}+{\displaystyle \frac{h^{″}\left(z\right)}{h^{\prime }\left(z\right)}}-{\displaystyle \frac{p^{\prime }\left(z\right)}{p\left(z\right)}}.$$

(4)

$$f\left(z\right)=z·\left({\displaystyle \frac{z{h}^{\prime }\left(z\right)}{p\left(z\right)}}\right)·\overline{h^{\prime }\left(z\right)}={\displaystyle \frac{z^2{h}^{\prime }\left(z\right)}{p\left(z\right)}}·\overline{h^{\prime }\left(z\right)}.$$

Therefore,

$$f\left(z\right)={\displaystyle \frac{z^2}{p\left(z\right)}}{\left|h^{\prime }\left(z\right)\right|}^2.$$

(5)

From the definition of log harmonic mapping, we know that for $f_z={\displaystyle \frac{\partial f}{\partial z}}, f_{\overline{z}}={\displaystyle \frac{\partial f}{\partial \overline{z}}}$, and a dilatation $w\left(z\right)$, a measurable function on $\mathbb{D}$ with $\left|w\right(z\left)\right|<1$, $z\in \mathbb{D}$, the logharmonic mapping f satisfies

$${\displaystyle \frac{f_{\overline{z}}}{f}}=w\left(z\right)·{\displaystyle \frac{f_z}{f}},\mathrm{where},$$

(6)

$$f_z=\overline{h^{\prime }\left(z\right)}(h\left(z\right)+z{h}^{\prime }\left(z\right)),f_{\overline{z}}=zh\left(z\right)\overline{h^{″}\left(z\right)}.$$

Substitution of these values in (6) gives

$${\displaystyle \frac{h^{″}\left(z\right)}{h^{\prime }\left(z\right)}}={\displaystyle \frac{w\left(z\right)}{z}}+w\left(z\right){\displaystyle \frac{h^{\prime }\left(z\right)}{h\left(z\right)}}.$$

(7)

Now substituting the value of ${\displaystyle \frac{h^{″}\left(z\right)}{h^{\prime }\left(z\right)}}$ in (4), it gives

$${\displaystyle \frac{1}{w}}{\displaystyle \frac{h^{″}\left(z\right)}{h^{\prime }\left(z\right)}}={\displaystyle \frac{2}{z}}+{\displaystyle \frac{h^{″}\left(z\right)}{h^{\prime }\left(z\right)}}-{\displaystyle \frac{p^{\prime }\left(z\right)}{p\left(z\right)}}.$$

Therefore,

$${\displaystyle \frac{h^{″}\left(z\right)}{h^{\prime }\left(z\right)}}={\displaystyle \frac{w}{1-w}}\left({\displaystyle \frac{2}{z}}-{\displaystyle \frac{p^{\prime }\left(z\right)}{p\left(z\right)}}\right).$$

After integrating the above expression, it results in

$$h^{\prime }\left(z\right)=\exp \left(\int \left({\displaystyle \frac{w\left(z\right)}{1-w\left(z\right)}}·\left({\displaystyle \frac{2}{z}}-{\displaystyle \frac{p^{\prime }\left(z\right)}{p\left(z\right)}}\right)\right)dz\right).$$

Substituting the above equation into (5), We obtain the expression

$$f\left(z\right)={\displaystyle \frac{z^2}{p\left(z\right)}}\exp \left(2Re\int {\displaystyle \frac{w\left(z\right)}{1-w\left(z\right)}}\left({\displaystyle \frac{2}{z}}-{\displaystyle \frac{p^{\prime }\left(z\right)}{p\left(z\right)}}\right)dz\right).$$

□

To comprehend the geometric behavior of the log-harmonic mapping $f\left(z\right)=zh\left(z\right)\overline{h^{\prime }\left(z\right)}$, precise estimations of its magnitude, derivative, and Jacobian are necessary. Such constraints are crucial for describing distortion, growth, and covering properties, all of which are essential components of log-harmonic mapping theory. Building on the preceding representation formula, we now derive explicit inequalities that regulate the behavior of $f\left(z\right)$ in the unit disk. The following theorem summarizes all of these.

Theorem 1. 

Let $f\left(z\right)=zh\left(z\right)\overline{h^{\prime }\left(z\right)}$ be a log harmonic mapping with regard to $w\in B,$ where $h\in S^{*}$. Consequently, for $\left|z\right|=r<1$,

(i) 

$\left|f\left(z\right)\right|\le {\displaystyle \frac{{\left|z\right|}^2(1+|z\left|\right)}{{(1-|z\left|\right)}^4}}\exp \left({\displaystyle \frac{2\left|z\right|}{1-\left|z\right|}}\right),$

(ii) 

$|f_z\left(z\right)|\le {\displaystyle \frac{2\left|z\right|(1+|z\left|\right)}{{(1-|z\left|\right)}^5}}\exp {\displaystyle \frac{2\left|z\right|}{1-\left|z\right|}},$

(iii) 

$|J_f\left(z\right)|\le {\displaystyle \frac{{4\left|z\right|}^2{(1+|z\left|\right)}^2}{{(1-|z\left|\right)}^{10}}}\exp \left({\displaystyle \frac{4\left|z\right|}{1-\left|z\right|}}\right).$

Proof. 

Lemma 2 gives a general form of a log-harmonic mapping,

$$f\left(z\right)={\displaystyle \frac{z^2}{p\left(z\right)}}\exp \left(2Re\int {\displaystyle \frac{w\left(z\right)}{1-w\left(z\right)}}\left({\displaystyle \frac{2}{z}}-{\displaystyle \frac{p^{\prime }\left(z\right)}{p\left(z\right)}}\right)dz\right).$$

We have used the following well-known inequalities for $\left|z\right|=r<1$

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{p^{\prime }\left(z\right)}{p\left(z\right)}}\right|& \le {\displaystyle \frac{2}{1-r^2}},\hfill \\ \hfill \left|{\displaystyle \frac{w\left(z\right)}{1-w\left(z\right)}}\right|& \le {\displaystyle \frac{r}{1-r}}.\hfill \end{array}$$

By combining all the above boundaries, we get

$$\begin{array}{cc}\hfill \left|f\right(z\left)\right|& \le {\displaystyle \frac{r^2(1+r)}{1-r}}\exp \left(2{\int }_0^r{\displaystyle \frac{t}{1-t}}\left({\displaystyle \frac{2}{t}}+{\displaystyle \frac{2}{1-t^2}}\right)dt\right)\hfill \\ \hfill & ={\displaystyle \frac{r^2(1+r)}{1-r}}\exp \left(3\pi i-\log {(r-1)}^3+{\displaystyle \frac{2r}{1-r}}\right)\hfill \\ \hfill & ={\displaystyle \frac{r^2(1+r)}{{(1-r)}^4}}\exp \left({\displaystyle \frac{2r}{1-r}}\right).\hfill \end{array}$$

Regarding portion (ii), we observe that

$$|f_z\left(z\right)|=\left(h\left(z\right)+z{h}^{\prime }\left(z\right)\right)·\overline{h^{\prime }\left(z\right)},$$

and

$${\displaystyle \frac{|f_z\left(z\right)|}{\left|f\right(z\left)\right|}}={\displaystyle \frac{1}{\left|z\right|}}+{\displaystyle \frac{|h^{\prime }\left(z\right)|}{\left|h\right(z\left)\right|}}.$$

Hence,

$$|f_z\left(z\right)|=\left({\displaystyle \frac{1}{\left|z\right|}}+{\displaystyle \frac{|h^{\prime }\left(z\right)|}{\left|h\right(z\left)\right|}}\right)\left|f\left(z\right)\right|=\left({\displaystyle \frac{1}{\left|z\right|}}+{\displaystyle \frac{\left|p\right(z\left)\right|}{\left|z\right|}}\right)\left|f\left(z\right)\right|.$$

Using portion (i) and the inequality

$$\left|p\left(z\right)\right|\le {\displaystyle \frac{1+r}{1-r}},$$

we get

$$|f_z\left(z\right)|\le \left({\displaystyle \frac{1}{r}}+{\displaystyle \frac{1+r}{r(1-r)}}\right)·{\displaystyle \frac{r^2(1+r)}{{(1-r)}^4}}\exp \left({\displaystyle \frac{2r}{1-r}}\right)={\displaystyle \frac{2r(1+r)}{{(1-r)}^5}}\exp {\displaystyle \frac{2r}{1-r}}.$$

Regarding section (iii), as f is log harmonic, we have

$$J_f\left(z\right)=|f_z{\left(z\right)|}^2-|f_{\overline{z}}{\left(z\right)|}^2={(1-|w\left(z\right)|}^2\left)\right|f_z{\left(z\right)|}^2.$$

Since $\left|w\right(z\left)\right|<1$, we have

$$J_f\left(z\right)\le {\left|f_z\left(z\right)\right|}^2.$$

Using the bound for $|f_z\left(z\right)|$,

$$|f_z\left(z\right)|\le {\displaystyle \frac{2r(1+|r\left|\right)}{{(1-|r\left|\right)}^5}}\exp \left({\displaystyle \frac{2\left|r\right|}{1-\left|r\right|}}\right),$$

By squaring both sides, we get

$$|f_z{\left(z\right)|}^2\le {\left(2r{\displaystyle \frac{1+r}{{(1-r)}^5}}\right)}^2\exp \left({\displaystyle \frac{4r}{1-r}}\right)={\displaystyle \frac{4r^2{(1+r)}^2}{{(1-r)}^{10}}}\exp \left({\displaystyle \frac{4r}{1-r}}\right).$$

Therefore,

$$J_f\left(z\right)\le {\displaystyle \frac{4r^2{(1+r)}^2}{{(1-r)}^{10}}}\exp \left({\displaystyle \frac{4r}{1-r}}\right).$$

□

Remark 1 (Sharpness).

We show that the estimates in parts (i)–(iii) of Theorem 1 are sharp. Consider the classical extremal starlike function

$$h\left(z\right)={\displaystyle \frac{z}{{(1-z)}^2}},z\in D,$$

which belongs to the class $S^{*}$. Define the associated log-harmonic mapping by

$$f\left(z\right)=zh\left(z\right)\overline{h^{\prime }\left(z\right)},$$

with dilatation $w\in \mathcal{B}$. Such mappings are known to belong to the class $S_{LH}^{*}$; hence, all the hypotheses of Theorem 1 are satisfied.

For points on the circle $\left|z\right|=r<1$, the extremal behavior of starlike functions implies that the maximum values of $\left|h\right(z\left)\right|$ and $|h^{\prime }\left(z\right)|$ are attained along the positive real axis $z=r$. Consequently, the quantities $\left|f\right(z\left)\right|$ and $|f_z\left(z\right)|$ and the Jacobian $J_f\left(z\right)$ also attain their maximal values at $z=r$.

A direct computation for this extremal choice yields

$$\left|f\left(z\right)\right|={\displaystyle \frac{r^2(1+r)}{{(1-r)}^4}}\exp \left({\displaystyle \frac{2r}{1-r}}\right),$$

$$|f_z\left(z\right)|={\displaystyle \frac{2r(1+r)}{{(1-r)}^5}}\exp \left({\displaystyle \frac{2r}{1-r}}\right),$$

and

$$|J_f\left(z\right)|={\displaystyle \frac{4r^2{(1+r)}^2}{{(1-r)}^{10}}}\exp \left({\displaystyle \frac{4r}{1-r}}\right).$$

These expressions coincide with the right-hand sides of the inequalities in parts (i), (ii), and (iii) of Theorem 1. Hence, equality is attained, which shows that the bounds in Theorem 1 are best possible.

The geometric theory of analytic and log-harmonic mappings heavily relies on the concept of starlikeness. A function is said to be starlike if it maps the unit disk to a domain that is starlike about the origin, preserving radial segments as simple arcs. For log-harmonic mappings of the type $f\left(z\right)=zh\left(z\right)\overline{h^{\prime }\left(z\right)},$ where h is starlike analytic in the unit disk, it becomes simple to calculate the greatest subdisk in which the function itself retains starlikeness. Defining such a radius is significant since it quantifies the extent to which geometric properties are retained. The theorem below reveals the mapping’s sharp radius of starlikeness.

Theorem 2. 

Let $f\left(z\right)=zh\left(z\right)\overline{h^{\prime }\left(z\right)}$ be a logharmonic mapping concerning $w\in B$, where $h\in S^{*}$ is a starlike analytic function in the unit disk $\mathbb{D}$. Then f is starlike in the disk

$$\mathsf{\Omega }=\left\{\mathrm{z}\in \mathbb{C}:\left|\mathrm{z}\right|<\rho ={\displaystyle \frac{\sqrt{5}-1}{2}}=0.6180\right\}.$$

Proof. 

Since $h\in S^{*}$, the function

$$p\left(z\right):={\displaystyle \frac{z{h}^{\prime }\left(z\right)}{h\left(z\right)}}$$

belongs to the class $\mathcal{P}$, the class of analytic functions with positive real part in $\mathbb{D}$.

Consider the logharmonic mapping

$$f\left(z\right)=zh\left(z\right)\overline{h^{\prime }\left(z\right)}.$$

We calculate the partial derivatives with respect to z and $\overline{z}$:

$$f_z\left(z\right)=h\left(z\right)\overline{h^{\prime }\left(z\right)}+z{h}^{\prime }\left(z\right)\overline{h^{\prime }\left(z\right)},$$

$$f_{\overline{z}}\left(z\right)=zh\left(z\right)\overline{h^{″}\left(z\right)}.$$

The starlikeness of f is characterized by the positivity of the real part of the expression

$$Q\left(z\right):={\displaystyle \frac{z{f}_z\left(z\right)-\overline{z}{f}_{\overline{z}}\left(z\right)}{f\left(z\right)}}.$$

Substituting the expressions for f, $f_z$, and $f_{\overline{z}}$, we get

$$Q\left(z\right)={\displaystyle \frac{z\left(h\left(z\right)\overline{h^{\prime }\left(z\right)}+z{h}^{\prime }\left(z\right)\overline{h^{\prime }\left(z\right)}\right)-\overline{z}·zh\left(z\right)\overline{h^{″}\left(z\right)}}{zh\left(z\right)\overline{h^{\prime }\left(z\right)}}}=1+{\displaystyle \frac{z{h}^{\prime }\left(z\right)}{h\left(z\right)}}-\overline{z}{\displaystyle \frac{\overline{h^{″}\left(z\right)}}{\overline{h^{\prime }\left(z\right)}}}.$$

Now,

$${\displaystyle \frac{z{h}^{″}\left(z\right)}{h^{\prime }\left(z\right)}}=1-{\displaystyle \frac{z{p}^{\prime }\left(z\right)}{p\left(z\right)}}-p\left(z\right).$$

$$Re\left(Q\left(z\right)\right)=Re\left({\displaystyle \frac{z{h}^{\prime }\left(z\right)}{h\left(z\right)}}-{\displaystyle \frac{z{h}^{″}\left(z\right)}{h^{\prime }\left(z\right)}}+1\right)=Re\left(p\left(z\right)-{\displaystyle \frac{z{p}^{\prime }\left(z\right)}{p\left(z\right)}}+2\right)=Re\left(2-{\displaystyle \frac{z{p}^{\prime }\left(z\right)}{p\left(z\right)}}\right).$$

$$Re\left(2-{\displaystyle \frac{z{p}^{\prime }\left(z\right)}{p\left(z\right)}}\right)\ge 2-{\displaystyle \frac{2\left|z\right|}{1-{\left|z\right|}^2}}={\displaystyle \frac{2-{2\left|z\right|}^2-2\left|z\right|}{1-{\left|z\right|}^2}}.$$

Thus,

$$Re\left({\displaystyle \frac{z{f}_z\left(z\right)-\overline{z}{f}_{\overline{z}}\left(z\right)}{f\left(z\right)}}\right)>0,\mathrm{whenever}2-{2\left|z\right|}^2-2\left|z\right|>0.$$

That is, if $1-{\left|z\right|}^2-\left|z\right|>0$. Hence, the radius of starlikeness is the smallest positive root of ${\rho }^2+\rho -1=0$, which is ${\displaystyle \frac{\sqrt{5}-1}{2}}.$ This validates the theorem. □

To analyze the image of log-harmonic mappings, it is helpful to relate their analytic and geometric behavior using auxiliary functions that capture the classical structure, like starlikeness and convexity. The function $\phi \left(z\right):={\displaystyle \frac{zh\left(z\right)}{h^{\prime }\left(z\right)}}$ is a suitable candidate for the mapping $f\left(z\right)=zh\left(z\right)\overline{h^{\prime }\left(z\right)},$ where h is a starlike analytic function in the unit disk D. When $\phi$ belongs to the class of starlike functions $S^{*}$, the interaction between f and $\phi$ reveals more about the structural features of f. The relationship between their logarithmic derivatives creates subordination criteria that closely restrict the analytic and co-analytic elements of the mapping. The theory below explains how the auxiliary function $\phi$ describes the behavior of the log-harmonic mapping.

Theorem 3. 

Let $f:\mathsf{\Omega }\to \mathbb{C}$ be the log-harmonic mapping

$$f\left(z\right)=zh\left(z\right)\overline{h^{\prime }\left(z\right)},$$

where h is starlike and nonvanishing in $\mathbb{D}$. Define

$$\phi \left(z\right):={\displaystyle \frac{zh\left(z\right)}{h^{\prime }\left(z\right)}}.$$

If $\phi \in S^{*}$, then for every $z\in \mathsf{\Omega }$ the subordinations results

$${\displaystyle \frac{\overline{f_{\overline{z}}}/\overline{f}}{{\phi }^{\prime }\left(z\right)/\phi \left(z\right)}}\prec {\displaystyle \frac{z}{1-z}},{\displaystyle \frac{{\phi }^{\prime }\left(z\right)/\phi \left(z\right)}{f_z/f}}\prec 1-z$$

hold in $\mathbb{D}$.

Proof. 

Consider the log-harmonic mapping

$$f\left(z\right)=zh\left(z\right)\overline{h^{\prime }\left(z\right)},$$

where $\mathbb{D}$ denotes the open unit disk, and h is analytic in $\mathbb{D}$ with normalization $h\left(0\right)=h^{\prime }\left(0\right)=1.$

Since $f\in S_{LH}^{*}$, it is starlike and satisfies the log-harmonic differential equation

$$\overline{f_{\overline{z}}}=w\left(z\right)f_z,$$

where $w\left(z\right)$ is the second complex dilatation of f.

From the representation of f, we compute the logarithmic derivatives

$${\displaystyle \frac{f_z}{f}}={\displaystyle \frac{1}{z}}+{\displaystyle \frac{h^{\prime }\left(z\right)}{h\left(z\right)}},\mathrm{and}{\displaystyle \frac{\overline{f_{\overline{z}}}}{\overline{f}}}={\displaystyle \frac{h^{″}\left(z\right)}{h^{\prime }\left(z\right)}}.$$

Therefore, the dilatation can be expressed as

$$w\left(z\right)={\displaystyle \frac{\frac{\overline{f_{\overline{z}}}}{\overline{f}}}{\frac{f_z}{f}}}={\displaystyle \frac{\frac{h^{″}\left(z\right)}{h^{\prime }\left(z\right)}}{\frac{1}{z}+\frac{h^{\prime }\left(z\right)}{h\left(z\right)}}}.$$

For simplicity of computation, we write

$$u\left(z\right)=zh\left(z\right).$$

Because $f\left(z\right)=u\left(z\right)\overline{h^{\prime }\left(z\right)}$, we compute the logarithmic derivatives directly:

$${\displaystyle \frac{f_z}{f}}={\displaystyle \frac{u^{\prime }\left(z\right)\overline{h^{\prime }\left(z\right)}}{u\left(z\right)\overline{h^{\prime }\left(z\right)}}}={\displaystyle \frac{u^{\prime }\left(z\right)}{u\left(z\right)}},{\displaystyle \frac{\overline{f_{\overline{z}}}}{\overline{f}}}={\displaystyle \frac{h^{″}\left(z\right)}{h^{\prime }\left(z\right)}}.$$

Thus, the dilatation w (coming from $\overline{f_{\overline{z}}}=w{f}_z$) is

$$w\left(z\right)={\displaystyle \frac{\overline{f_{\overline{z}}}/\overline{f}}{f_z/f}}={\displaystyle \frac{h^{″}\left(z\right)/h^{\prime }\left(z\right)}{u^{\prime }\left(z\right)/u\left(z\right)}}.$$

Compute the logarithmic derivative of $\phi :$

$${\displaystyle \frac{{\phi }^{\prime }\left(z\right)}{\phi \left(z\right)}}={\displaystyle \frac{u^{\prime }\left(z\right)}{u\left(z\right)}}-{\displaystyle \frac{h^{″}\left(z\right)}{h^{\prime }\left(z\right)}}.$$

From these identities we obtain the algebraic equalities

$${\displaystyle \frac{w\left(z\right)}{1-w\left(z\right)}}={\displaystyle \frac{\frac{h^{″}\left(z\right)}{h^{\prime }\left(z\right)}}{\frac{u^{\prime }\left(z\right)}{u\left(z\right)}-\frac{h^{″}\left(z\right)}{h^{\prime }\left(z\right)}}}={\displaystyle \frac{\overline{f_{\overline{z}}}/\overline{f}}{{\phi }^{\prime }\left(z\right)/\phi \left(z\right)}},$$

and

$${\displaystyle \frac{1-w\left(z\right)}{w\left(z\right)}}={\displaystyle \frac{\frac{u^{\prime }\left(z\right)}{u\left(z\right)}-\frac{h^{″}\left(z\right)}{h^{\prime }\left(z\right)}}{\frac{h^{″}\left(z\right)}{h^{\prime }\left(z\right)}}}={\displaystyle \frac{{\phi }^{\prime }\left(z\right)/\phi \left(z\right)}{f_z/f}}.$$

Because $f\in S_{LH}^{*}$, we have $\left|w\right(z\left)\right|<1$ on $\mathbb{D}$ and $w\left(0\right)=0$.

Schwarz’s lemma (equivalently the standard subordination for functions with $w\left(0\right)=0$ and $\left|w\right|<1$) gives

$${\displaystyle \frac{w\left(z\right)}{1-w\left(z\right)}}\prec {\displaystyle \frac{z}{1-z}},\mathrm{and}1-w\left(z\right)\prec 1-z.$$

Combining these subordinations with the algebraic identities above yields the claimed subordinations

$${\displaystyle \frac{\overline{f_{\overline{z}}}/\overline{f}}{{\phi }^{\prime }\left(z\right)/\phi \left(z\right)}}\prec {\displaystyle \frac{z}{1-z}},\mathrm{and}{\displaystyle \frac{{\phi }^{\prime }\left(z\right)/\phi \left(z\right)}{f_z/f}}\prec 1-z.$$

This completes the proof. □

The lemma below offers crucial inequalities for partial derivatives of log-harmonic mappings of the $f\left(z\right)$. These estimations, in particular, allow us to regulate the growth of the analytic factor h and its derivative.

Lemma 3. 

Let $f\left(z\right)=zh\left(z\right)\overline{g\left(z\right)}$ be a log-harmonic mapping in the class $S_{LH}^{*}$, where h and g are analytic functions defined on the unit disk $\mathbb{D}$. For each $z\in \mathbb{D}$ with modulus $\left|z\right|=r<1$, the complex partial derivatives of f satisfy the following bounds:

$$e^{-{\displaystyle \frac{4r}{1+r}}}{\displaystyle \frac{1-r}{{(1+r)}^2}}\le \left|f_z\left(z\right)\right|\le e^{{\displaystyle \frac{4r}{1-r}}}{\displaystyle \frac{1+r}{{(1-r)}^2}},$$

and

$$0\le |f_{\overline{z}}\left(z\right)|\le e^{{\displaystyle \frac{4r}{1-r}}}{\displaystyle \frac{r(1+r)}{{(1-r)}^2}}.$$

Theorem 4. 

Let $f:\mathsf{\Omega }\to \mathbb{C}$ be a log-harmonic mapping of the form $f\left(z\right)=zh\left(z\right)\overline{h^{\prime }\left(z\right)},$ where h is analytic in the unit disk $\mathbb{D}$, and suppose that f belongs to the class $S_{LH}^{*}$ of starlike log-harmonic mappings for the restricted domain $\mathrm{\Omega }\subset \mathbb{D}$. Further, assume that $h\in S^{*}$ is a starlike analytic function normalized by $h\left(0\right)=0and{h}^{\prime }\left(0\right)=1.$

Then for each $z\in \mathbb{D}$ with $\left|z\right|=r<1$, the following inequalities hold:

$$\left|h\left(z\right)\right|\le {\displaystyle \frac{1}{1-r}}\exp \left({\displaystyle \frac{2r}{1-r}}\right),and\left|h^{\prime }\left(z\right)\right|\le (1-r)\exp \left({\displaystyle \frac{2r}{1-r}}\right).$$

Proof. 

We begin by examining the logarithmic derivatives of f, which are crucial for establishing growth estimates.

First, compute the complex partial derivatives of f.

$$f_z\left(z\right)={\displaystyle \frac{\partial }{\partial z}}\left(zh\left(z\right)\right)\overline{h^{\prime }\left(z\right)}=\left(h\left(z\right)+z{h}^{\prime }\left(z\right)\right)\overline{h^{\prime }\left(z\right)}.$$

Similarly, differentiating with respect to $\overline{z}$, the function $zh\left(z\right)$ is treated as constant, yielding

$$f_{\overline{z}}\left(z\right)=zh\left(z\right){\displaystyle \frac{\partial }{\partial \overline{z}}}\overline{h^{\prime }\left(z\right)}.$$

The quantities of interest are the ratios

$${\displaystyle \frac{z{f}_z}{f}}\mathrm{and}{\displaystyle \frac{\overline{z}{f}_{\overline{z}}}{f}},$$

which relate to the logarithmic derivatives of f.

Substituting the expressions for f and $f_z$, we find

$${\displaystyle \frac{z{f}_z}{f}}={\displaystyle \frac{z·\left(h\left(z\right)+z{h}^{\prime }\left(z\right)\right)\overline{h^{\prime }\left(z\right)}}{zh\left(z\right)\overline{h^{\prime }\left(z\right)}}}={\displaystyle \frac{h\left(z\right)+z{h}^{\prime }\left(z\right)}{h\left(z\right)}}=1+z{\displaystyle \frac{h^{\prime }\left(z\right)}{h\left(z\right)}}.$$

Taking the real part yields

$$Re\left({\displaystyle \frac{z{f}_z}{f}}\right)=1+Re\left(z{\displaystyle \frac{h^{\prime }\left(z\right)}{h\left(z\right)}}\right)=1+r{\displaystyle \frac{\partial }{\partial r}}\log \left|h\left(z\right)\right|,$$

where $r=\left|z\right|$.

Similarly, for the conjugate derivative,

$${\displaystyle \frac{\overline{z}{f}_{\overline{z}}}{f}}={\displaystyle \frac{\overline{z}·zh\left(z\right)\frac{\partial }{\partial \overline{z}}\overline{h^{\prime }\left(z\right)}}{zh\left(z\right)\overline{h^{\prime }\left(z\right)}}}=\overline{z}{\displaystyle \frac{\partial }{\partial \overline{z}}}\log \overline{h^{\prime }\left(z\right)}=r{\displaystyle \frac{\partial }{\partial r}}\log \left|h^{\prime }\left(z\right)\right|.$$

Since f is a starlike log-harmonic mapping, by (3), it provides the following differential inequalities for $r=\left|z\right|$:

$${\displaystyle \frac{\partial }{\partial r}}\log \left|h\left(z\right)\right|\le {\displaystyle \frac{1+r}{r{(1-r)}^2}}-{\displaystyle \frac{1}{r}},$$

$${\displaystyle \frac{\partial }{\partial r}}\log \left|h^{\prime }\left(z\right)\right|\le {\displaystyle \frac{1+r}{r{(1-r)}^2}}.$$

(8)

We now integrate each inequality from 0 to r, noting that $h\left(0\right)=h^{\prime }\left(0\right)=1$.

To integrate the first inequality near the origin, it is convenient to remove the ${\displaystyle \frac{1}{r}}$ singularity by working with the regularized logarithm $\log \left({\displaystyle \frac{\left|h\right(r{e}^{i\theta }\left)\right|}{r}}\right).$ Subtracting ${\displaystyle \frac{1}{r}}$ from both sides of

$${\displaystyle \frac{\partial }{\partial r}}\log \left|h\left(r{e}^{i\theta }\right)\right|\le {\displaystyle \frac{1+r}{r{(1-r)}^2}}-{\displaystyle \frac{1}{r}}$$

yields

$${\displaystyle \frac{\partial }{\partial r}}\log \left({\displaystyle \frac{\left|h\right(r{e}^{i\theta }\left)\right|}{r}}\right)\le {\displaystyle \frac{1+r}{r{(1-r)}^2}}-{\displaystyle \frac{2}{r}}={\displaystyle \frac{3-r}{{(1-r)}^2}}-{\displaystyle \frac{1}{r}}.$$

Integrating from 0 to r (the left–hand side tends to 0 as $r\to 0^{+}$ because $h\left(z\right)=z+O\left(z^2\right)$) gives

$$\log \left({\displaystyle \frac{\left|h\right(r{e}^{i\theta }\left)\right|}{r}}\right)\le {\int }_0^r\left({\displaystyle \frac{3-t}{{(1-t)}^2}}-{\displaystyle \frac{1}{t}}\right)dt.$$

The first term in the integrand can be decomposed as

$${\displaystyle \frac{3-t}{{(1-t)}^2}}={\displaystyle \frac{1}{1-t}}+{\displaystyle \frac{2}{{(1-t)}^2}}.$$

Hence,

$${\int }_0^r{\displaystyle \frac{3-t}{{(1-t)}^2}}dt=-\log (1-r)+{\displaystyle \frac{2}{1-r}}-2,$$

and

$${\int }_0^r{\displaystyle \frac{1}{t}}dt=\log r.$$

Therefore,

$$\log \left({\displaystyle \frac{\left|h\right(r{e}^{i\theta }\left)\right|}{r}}\right)\le -\log (1-r)+{\displaystyle \frac{2}{1-r}}-2-\log r.$$

Rearranging and canceling the r logarithmic term in the standard way yields

$$\log \left({\displaystyle \frac{\left|h\right(r{e}^{i\theta }\left)\right|}{r}}\right)\le {\displaystyle \frac{2r}{1-r}}-\log (1-r).$$

Exponentiation, yields

$${\displaystyle \frac{\left|h\right(r{e}^{i\theta }\left)\right|}{r}}\le {\displaystyle \frac{1}{1-r}}\exp \left({\displaystyle \frac{2r}{1-r}}\right),$$

so that for $\left|z\right|=r$,

$$\left|h\left(z\right)\right|\le {\displaystyle \frac{r}{1-r}}\exp \left({\displaystyle \frac{2r}{1-r}}\right).$$

Finally, noting that $r<1$, we arrive at the slightly coarser but more compact form

$$\left|h\left(z\right)\right|\le {\displaystyle \frac{1}{1-r}}\exp \left({\displaystyle \frac{2r}{1-r}}\right).$$

Now, we find a bound for $h^{\prime }\left(z\right)$. Integrating the inequality (8) from 0 to r, and using $h^{\prime }\left(0\right)=1$, we obtain

$$\begin{array}{cc}\hfill \log |h^{\prime }\left(z\right)|& \le {\int }_0^r{\displaystyle \frac{1+t}{{(1-t)}^2}}dt\hfill \\ \hfill & ={\displaystyle \frac{2r}{1-r}}-\log (1-r).\hfill \end{array}$$

Therefore,

$$|h^{\prime }\left(z\right)|\le (1-r)\exp \left({\displaystyle \frac{2r}{1-r}}\right).$$

This proves the theorem. □

Related growth and structural properties of starlike log-harmonic mappings were recently investigated by Abdulhadi and El Hajj (2024) [19]. The above result provides explicit growth bounds for the subclass considered here, although no claim of sharpness is made. They provide explicit growth and distortion bounds for the functions h and $h^{\prime }$, which are sufficient for the purposes of this work.

Geometric features, such as the arclength of images of circles under log-harmonic mappings, offer valuable information into their boundary behavior. Estimating the arclength of the image of $\left|z\right|=r$ under a log-harmonic mapping quantifies how it distorts geometric shapes in the unit disk. We establish an upper bound for the arclength of the image curve under a log-harmonic mapping of the type $f\left(z\right)=zh\left(z\right)\overline{h^{\prime }\left(z\right)},h\in S^{*},$ where $S^{*}$ indicates the class of starlike analytic functions. The theorem below provides an explicit estimate.

Theorem 5. 

Let $f\left(z\right)=zh\left(z\right)\overline{h^{\prime }\left(z\right)}$ be a log-harmonic mapping with respect to $w\in B$, where $h\in S^{*}$. Suppose that

$$\left|f\right(z\left)\right|\le E\left(r\right),0<\left|z\right|=r<1.$$

Let $L\left(r\right)$ denote the arclength of the curve $C_r$, which is the image of the circle $\left|z\right|=r$ under the mapping $\mathbb{W}=f\left(z\right)$ and $E\left(r\right)$ denotes the supremum of $f\left(z\right)$ over the circle of radius r in the unit disk. Then

$$L\left(r\right)\le 2\pi E\left(r\right)\left(r\right)(r^2+r+2).$$

Proof. 

Let $C_r$ denote the image of the circle $\left|z\right|=r$ under the mapping $\mathbb{W}=f\left(z\right)$. The arclength of $C_r$ is given by [14]

$$L\left(r\right)={\int }_{C_r}\left|df\right|.$$

Parameterizing $z=r{e}^{i\theta }$ for $0\le \theta \le 2\pi$, we have

$$L\left(r\right)={\int }_0^{2\pi }\left|{\displaystyle \frac{df}{d\theta }}\right|d\theta .$$

For a log-harmonic mapping,

$${\displaystyle \frac{df}{d\theta }}=f_z{\displaystyle \frac{\partial z}{\partial \theta }}+f_{\overline{z}}{\displaystyle \frac{\partial \overline{z}}{\partial \theta }}=iz{f}_z-i\overline{z}{f}_{\overline{z}},$$

so that

$$L\left(r\right)={\int }_0^{2\pi }\left|z{f}_z-\overline{z}{f}_{\overline{z}}\right|d\theta$$

For $f\left(z\right)=zh\left(z\right)\overline{h^{\prime }\left(z\right)}$, we have the complex partial derivatives:

$$f_z=h\left(z\right)\overline{h^{\prime }\left(z\right)}+z{h}^{\prime }\left(z\right)\overline{h^{\prime }\left(z\right)},f_{\overline{z}}=zh\left(z\right)\overline{h^{″}\left(z\right)}.$$

The arclength of the image of the circle $\left|z\right|=r$ is

$$L\left(r\right)={\int }_0^{2\pi }\left|z{f}_z-\overline{z}{f}_{\overline{z}}\right|d\theta .$$

Substituting the derivatives, we get

$$\begin{array}{cc}\hfill L\left(r\right)& ={\int }_0^{2\pi }|z\left(h\left(z\right)\overline{h^{\prime }\left(z\right)}+z{h}^{\prime }\left(z\right)\overline{h^{\prime }\left(z\right)}\right)-\overline{z}zh\left(z\right)\overline{h^{″}\left(z\right)}|d\theta \hfill \\ & ={\int }_0^{2\pi }|zh\left(z\right)\overline{h^{\prime }\left(z\right)}+z^2{h}^{\prime }\left(z\right)\overline{h^{\prime }\left(z\right)}-{\left|z\right|}^{2}h\left(z\right)\overline{h^{″}\left(z\right)}|d\theta \hfill \\ & ={\int }_0^{2\pi }\left|h\left(z\right)\overline{h^{\prime }\left(z\right)}\left(z+{\displaystyle \frac{z^2{h}^{\prime }\left(z\right)}{h\left(z\right)}}-{\displaystyle \frac{{\left|z\right|}^2{h}^{″}\left(z\right)}{h^{\prime }\left(z\right)}}\right)\right|d\theta .\hfill \end{array}$$

Since $h\in S^{*}$, it is starlike and satisfies

$${\displaystyle \frac{z{h}^{\prime }\left(z\right)}{h\left(z\right)}}\in \mathcal{P},\mathrm{and}{\displaystyle \frac{h^{″}\left(z\right)}{h^{\prime }\left(z\right)}}={\displaystyle \frac{w\left(z\right)}{z}}+w\left(z\right){\displaystyle \frac{h^{\prime }\left(z\right)}{h\left(z\right)}},\mathrm{for}\mathrm{some}w\left(z\right)\in \mathbb{B}.\left(from\left(7\right)\right)$$

Therefore,

$$\left|{\displaystyle \frac{z{h}^{″}\left(z\right)}{h^{\prime }\left(z\right)}}\right|\le \left|w\left(z\right)\right|+\left|w\left(z\right){\displaystyle \frac{z{h}^{\prime }\left(z\right)}{h\left(z\right)}}\right|\le r+{\displaystyle \frac{r^2(1+r)}{1-r}},$$

where we used Schwarz’s lemma

$$\left|w\right(z\left)\right|\le \left|z\right|=r.$$

Combining the above inequalities, we obtain

$$\begin{array}{cc}\hfill L\left(r\right)& \le E\left(r\right){\int }_0^{2\pi }\left(r+r{\displaystyle \frac{1-r}{1+r}}+r^2+r^3{\displaystyle \frac{1+r}{1-r}}\right)d\theta \hfill \\ \hfill & =E\left(r\right){\displaystyle \frac{2r-r^2-r^4}{1-r}}{\int }_0^{2\pi }d\theta =2\pi E\left(r\right)\left(r\right)(r^2+r+2).\hfill \end{array}$$

□

The above estimate is not claimed to be sharp. It provides an explicit upper bound for the arclength $L\left(r\right)$ in terms of the maximal modulus $E\left(r\right)$, which is sufficient for the analysis carried out in this paper.

The growth, distortion, and arclength estimates obtained in this paper have several potential applications in the theory of planar mappings. The growth and distortion bounds provide quantitative control on the size and geometry of the image domains $f\left(D_r\right)$ and can be used to study geometric properties such as starlikeness, domain deformation, and inclusion relations for images of subdisks of the unit disk.

The estimates also yield information on the boundary behavior of log-harmonic mappings. In particular, they describe the asymptotic growth of $f\left(z\right)$ and its derivatives as $\left|z\right|\to 1^{-}$, which is relevant in the investigation of angular limits, boundary regularity, and related questions in harmonic mapping theory.

Furthermore, the arclength bounds for images of circles under log-harmonic mappings can be combined with Jacobian estimates to obtain length–area type inequalities. Such relations play an important role in geometric function theory and in the study of isoperimetric-type problems for planar mappings.

Finally, the Jacobian and distortion estimates are useful in the analysis of area distortion and finite-distortion properties of log-harmonic mappings. These results also facilitate comparisons between analytic, harmonic, and log-harmonic mappings, thereby extending classical geometric estimates from analytic function theory to the log-harmonic setting.

4. Conclusions

In this paper, we investigate the analytic and geometric aspects of log-harmonic mappings, emphasizing their close relationship with potential theory. By focusing on subclasses such as starlike log-harmonic mappings, we obtain growth, distortion, and Jacobian estimates, as well as integral representations. The sharpness of many radius problems remains open for further investigation. The fact that the logarithm of the modulus of a log-harmonic mapping is harmonic highlights the central role of potential theory, linking these mappings to solutions of Laplace’s equation and Beltrami-type systems. This perspective enables the extension of classical results from univalent function theory to a broader nonlinear setting. The results presented here not only enrich the theoretical framework of log-harmonic mappings but also open avenues for future research, particularly in extremal problems, covering theorems, and applications to quasiconformal deformations and fluid models.