In recent years, many authors have studied and investigated majorization results for different subclasses of analytic functions. In this paper, we give some majorization results for certain non vanishing analytic functions, whose ratios are subordinated to different domains in the open unit disk.

Inequalities are essential in solving mathematical problems in many different areas of mathematics. Among these, problems involving coefficient combinations that occurred in the Taylor–Maclaurin series of the inverse of complex-valued analytic functions are the challenging ones to solve. In the current article, our aim is to study certain coefficient-related problems that construct from coefficients of the inverse of specific analytic functions. These problems include the Zalcman and Fekete–Szegö inequalities, as well as sharp estimates of the second and third-order Hankel determinants with inverse function coefficients. Also, one of the obtained results gives an improvement of the problem that has been recently published in the journal "AIMS Mathematics".