This paper studies the class consisting of univalent logharmonic mappings in the unit disk , where and are analytic in and is a normalized starlike analytic function. A representation theorem for these mappings is obtained, which yields sharp distortion estimates, and a sharp Bohr radius.

In this manuscript, we apply the simulation mappings to present and verify several original results of common and coincidence fixed point in complete S-metric spaces. Moreover, using S-metric to expand and generalized diverse results in the literature involving simulation mappings. On the other hand, we apply our major results to derive several common and coincidencefixed point theorems for right monotone simulation map in complete S-metric. As implementations, various related outcomes of fixed-point theory via specific simulation mappings are obtained in complete S-metric spaces. Additionally, illustrative examples and some applications to solve an integral equation are introduced to support our major results.

The double-controlled metric-type space $(X,\mathcal{D})$ is a metric space in which the triangle inequality has the form $\mathcal{D}(\eta ,\mu )\le {\zeta }_1(\eta ,\theta )\mathcal{D}(\eta ,\theta )+{\zeta }_2(\theta ,\mu )\mathcal{D}(\theta ,\mu )$ for all $\eta ,\theta ,\mu \in X$. The maps ${\zeta }_1,{\zeta }_2:X\times X\to [1,\infty )$ are called control functions. In this paper, we introduce a novel generalization of a metric space called a double-composed metric space, where the triangle inequality has the form $\mathcal{D}(\eta ,\mu )\le \alpha \left(\mathcal{D}(\eta ,\theta )\right)+\beta \left(\mathcal{D}(\theta ,\mu )\right)$ for all $\eta ,\theta ,\mu \in X$. In our new space, the control functions $\alpha ,\beta :[0,\infty )\to [0,\infty )$ are composed of the metric $\mathcal{D}$ in the triangle inequality, where the control functions ${\zeta }_1,{\zeta }_2:X\times X\to [1,\infty )$ in a double-controlled metric-type space are multiplied with the metric $\mathcal{D}$. We establish some fixed-point theorems along with the examples and applications.

The first objective of the present manuscript is to introduce the notion of complex‐valued ‐metric spaces as generalizing and improving the idea of ‐metric spaces by using the context of complex‐valued metric space and ‐metric spaces. The principle of contraction mappings has recently been established. Furthermore, the second objective is devoted to establishing various new fixed‐point theorems in complete complex‐valued ‐metric spaces that are improved and generalized by Azam et al. Banach’s contraction principle and various distinguished outcomes are also presented in the literatures. Additionally, some implementations of common fixed‐point theory on complete complex‐valued ‐metric spaces have been offered by applying open and closed balls. Additionally, we provide some illustrative examples as an implementation for our main results.