Some radius problems related to a certain subclass of analytic functions (Q477807)

For real parameters \(\alpha\) and \(\beta\) such that \(0\leq\alpha<1<\beta\), the class \(\mathcal S(\alpha,\beta)\) is defined by \[ \alpha<\mathfrak{Re}(zf'(z)/f(z))<\beta, \quad |z|<1,\quad f(z)=z+a_2z^2+\cdots. \] It is proved that if \(zf'(z)\in\mathcal S(\alpha,\beta) \), then \(f(z)\in\mathcal S(\Phi(\alpha),\Phi(\beta)) \), where \(4\Phi(x)=-1+2x+\sqrt{4x^2-4x+9}\). Also, radius problems in the class \(\mathcal S(\alpha,\beta)\) are considered.