Using the recently introduced Sălăgean integro-differential operator, three new classes of bi-univalent functions are introduced in this paper. In the study of bi-univalent functions, estimates on the first two Taylor–Maclaurin coefficients are usually given. We go further in the present paper and bounds of the first three coefficients $\left|a_2\right|$ , $\left|a_3\right|$ and $\left|a_4\right|$ of the functions in the newly defined classes are given. Obtaining Fekete–Szego inequalities for different classes of functions is a topic of interest at this time as it will be shown later by citing recent papers. So, continuing the study on the coefficients of those classes, the well-known Fekete–Szego functional is obtained for each of the three classes.

中文翻译：

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新的双单价函数类的系数相关研究

本文使用最近介绍的Sălăgean积分微分算子，介绍了三类新的双单价函数。在研究双单价函数时，通常会给出前两个Taylor-Maclaurin系数的估计值。我们在本文中进一步介绍了前三个系数的范围 $\left|{\mathrm{一种}}_2\right|$ ， $\left|{\mathrm{一种}}_3\right|$ 和 $\left|{\mathrm{一种}}_4\right|$ 给出了新定义的类中的功能。目前，对于不同类别的函数获取Fekete-Szego不等式是一个令人感兴趣的话题，稍后将通过引用最新论文加以证明。因此，继续研究这些类别的系数，对于这三个类别中的每一个，都获得了著名的Fekete-Szego函数。