The objective of the present article is to introduce new subclasses of bi-Bazilevič functions, bi-quasi-convex functions and $\alpha$-exponentially bi-convex functions involving functions with bounded boundary rotation of order $\nu$. For the above-said newly defined classes, we obtain first two initial coefficient bounds. In addition, the familiar Fekete–Szegö coefficient inequality is too found for the newly introduced subclasses of bi-univalent functions. Apart from the new findings that are obtained, it also improves the prior estimates that are presented already in the literature.

By using the Mittag–Leffler function associated with functions of bounded boundary rotation, the authors introduce a few new subclasses of bi-univalent functions involving the Mittag–Leffler function with bounded boundary rotation in the open unit disk $\mathbb{D}.$ For these new classes, the authors establish initial coefficient bounds of $\left|a_2\right|$ and $\left|a_3\right|$. Furthermore, the famous Fekete–Szegö coefficient inequality is also obtained for these new classes of functions.

In the present article, the authors introduce the concept of m-fold symmetric bi-univalent functions with bounded boundary rotation and introduce three new subclasses of m-fold symmetric bi-univalent functions. For the newly defined classes, the authors obtain first two initial coefficient bounds. Furthermore, the Fekete-Szegö bounds are also obtained for these three new classes of functions. Apart from many new results and new corollaries, the results in this article also improves earlier existing bounds available in the literature.