In this article, we first define and then propose to systematically study some new subclasses of the class of analytic and bi-concave functions in the open unit disk. For this purpose, we make use of a combination of the binomial series and the confluent hypergeometric function. Among some other properties and results, we derive the estimates on the initial Taylor-Maclaurin coefficients |a2| and |a3| for functions in these analytic and bi-concave function classes, which are introduced in this paper. We also derive a number of corollaries and consequences of our main results in this paper.
