Univalence of Logharmonic Integrals of Cesàro Type Operator

This manuscript extends the integral transform \(\begin{aligned} C^{\alpha }_{\beta }[\varphi ](z):=\int \limits _{0}^{z} \bigg (\frac{\varphi (\zeta )}{\zeta (1-\zeta )^{\beta }}\bigg )^\alpha \, d\zeta , \quad \alpha , \beta \in \mathbb {C}, \end{aligned}\) $$\begin{array}{c}\hfill C_{\beta }^{\alpha }[\phi ](z):=\underset{0}{\overset{z}{\int }}(\frac{\phi (\zeta )}{\zeta {(1-\zeta )}^{\beta }}{)}^{\alpha }d\zeta ,\alpha ,\beta \in \mathbb{C},\end{array}$$ where \(\varphi \) $\phi$ is normalized analytic function, to the case of logharmonic mapping and study their univalence properties. This integral transform carries the well-known Alexander (i.e. \(C^{1}_0\) $C_0^1$ ) and Cesáro (i.e. \(C^{1}_1\) $C_1^1$ ) transforms. Following the idea of classical shear construction for harmonic mappings, we employ the shear construction for logharmonic mappings that is introduced by Liu and Ponnusamy in 2022 with a goal to generate logharmonic integrals of Cesáro type. In addition, intriguing relationships between harmonic and non-vanishing logharmonic integrals corresponding to the integral transforms \(C^{\alpha }_{\beta }\) $C_{\beta }^{\alpha }$ are established.