Certain properties of an operator involving subordination

The concept of subordination can be traced back to Lindelöf since 1909, but other mathematicians like Littlewood (1925) and Rogosinski (1939) introduced the term and developed the basic theories. Subordination now plays an important role in complex analysis. The idea of univalent subordination can be stated as follows: Let $f$ and $g$ be analytic in $E$. Then $f$ is said to be subordinate to $g$, if $g$ is univalent in $E$, $f(0)=g(0)$ and $f(E)\subset g(E)$. We denote the subordination by $f\prec g$. Here, we apply a lemma of Miller and Mocanu to obtain a series of best possible subordination theorems. We also make use of an operator studied by Cho and Srivastava, and by Cho and Kim in this particular work. Thus, in this research work, we consider properties of an operator aforementioned involving subordinations with new results briefly highlighted.