A study of generalized distribution series and their mapping properties in univalent function theory

This paper deals with the study of some classes of analytic functions in the open unit disk defined by using subordinations and connected with the distribution series, and these new classes reduces to some inequalities involving the first and second-order derivatives of the functions. For a particular choice of parameters, these new classes reduce to many others studied by different authors, and give an extension of many of these previously investigated classes. First, we determine several simple sufficient conditions for generalized distribution series that belong to the new defined classes $S(D,E;\delta ,\rho )$ and $K(D,E;\delta ,\rho )$, and we establish certain inclusion properties between the subclasses $\mathcal{H}(\lambda ,\eta )$ and $K(D,E;\delta ,\rho )$. In addition, we obtain sufficient conditions to show that some related series belong to certain subclasses of analytic functions. Each of the results are followed by some special cases that were recently obtained in different articles. Additionally, specific instances of our primary outcomes are briefly mentioned. The main tools for obtaining our new results consist of a few simple inequalities presented in the second section, while the scope of the paper is to extend and continue the studies connecting specific problems of analytic functions with generalized distribution series.