We consider three classes of functions defined using the class \({\mathcal {P}}\) of all analytic functions \(p(z)=1+cz+\cdots \) on the open unit disk having positive real part and study several radius problems for these classes. The first class consists of all normalized analytic functions f with \(f/g\in {\mathcal {P}}\) and \(g/(zp)\in {\mathcal {P}}\) for some normalized analytic function g and \(p\in {\mathcal {P}}\). The second class is defined by replacing the condition \(f/g\in {\mathcal {P}}\) by \(|(f/g)-1|<1\) while the other class consists of normalized analytic functions f with \(f/(zp)\in {\mathcal {P}}\) for some \(p\in {\mathcal {P}}\). We have determined radii so that the functions in these classes to belong to various subclasses of starlike functions. These subclasses includes the classes of starlike functions of order \(\alpha \), parabolic starlike functions, as well as the classes of starlike functions associated with lemniscate of Bernoulli, reverse lemniscate, sine function, a rational function, cardioid, lune, nephroid and modified sigmoid function.

我们考虑使用所有解析函数的类\({\mathcal {P}}\)定义的三类函数\(p(z)=1+cz+\cdots \)在具有正实部的开单位圆盘上并研究这些类的几个半径问题。第一类由所有归一化的解析函数f和\(f/g\in {\mathcal {P}}\)和\(g/(zp)\in {\mathcal {P}}\) 组成，用于一些归一化的解析函数g和\(p\in {\mathcal {P}}\)。第二类是通过更换条件定义\（F / G \在{\ mathcal {P}} \）由\（|（F / G）-1 | <1 \） ，而其他类由归一化的解析函数F与\(f/(zp)\in {\mathcal {P}}\)一些\(p\in {\mathcal {P}}\)。我们已经确定了半径，以便这些类中的函数属于星状函数的各个子类。这些子类包括\(\alpha \)阶星状函数类、抛物线类星状函数，以及与伯努利双线、反向双线、正弦函数、有理函数、心形、月形、肾形相关的类星形函数和修改的 sigmoid 函数。