Let \(\mathcal A\) denote the class of all functions \(f\) analytic in the unit disk \(\mathbb{U}\), normalized by \(f(0) = f'(0) -1 = 0.\)
For given numbers \(\epsilon \in (0,1]\) and \(\alpha > 1\) the author considers the class of functions \(f \in \mathcal A\) such that \[ |(f'(z))^{\alpha} - 1 | < \epsilon. \]
In the paper an upper bound of the norm of the pre-Schwarzian derivative for these functions is computed. Moreover, the author obtains some subordination and radius results in the considered class of functions.