UNIVALENCE CONDITIONS FOR A NEW INTEGRAL OPERATOR

Abstract

In the present paper, we will obtain norm estimates of the pre-Schwarzian derivatives for $F_{\lambda,\mu}(z)$, such that \[ F_{\lambda,\mu}(z) = \int_0^z \prod_{i=1}^{n} (f'_i(t))^{\lambda_i}\left( \frac{f_i(t)}{t} \right)^{\mu_i}dt \quad (z\in D),\] where $\lambda_i,\mu_i\in \mathbb{R}$, $\lambda_i=(\lambda_1,\lambda_2,\ldots,\lambda_n$, $\mu_i=(\mu_1,\mu_2,\ldots,\mu_n$ and $f_i$ belongs to the class of convex univalent functions $\mathcal{C}\subset \mathcal{S}$.