The $$\mathcal {U}$$ U -Radius for Classes of Analytic Functions

Abstract

Let \(\mathcal {U}\) denote the class of normalized analytic functions \(f\) in the open unit disk \(\mathbb {D}\) satisfying $$\begin{aligned} \left| \left( \frac{z}{f(z)}\right) ^2f'(z)-1\right| < 1. \end{aligned}$$ The \(\mathcal {U}\)-radius is obtained for several classes of functions. These include the class of normalized analytic functions \(f\) satisfying the inequality \({{\mathrm{Re}}}\, f(z)/g(z)>0\) or \(\left| f(z)/g(z)-1\right| \alpha , 0 \le \alpha < 1,\) in \(\mathbb {D}.\) A recent conjecture by Obradovic and Ponnusamy concerning the radius of univalence for a product involving univalent functions is also shown to hold true.