Estimates in the class of analytical functions related to the Cassini oval and some of their applications

In this article, we introduce and study a class $\mathcal{P}_n(\varphi_\lambda)$ of functions $\varphi(z) = 1 + c_n z^{n} + c_{n+1} z^{n+1} + \ldots,$ $n\geq1,$ analytic in the open unit disk $E,$ subordinate to the function $\varphi_\lambda(z)=1+{(1-\lambda)z}/{(1-\lambda z^2)},$ $0\le\lambda<1.$ From a geometric point of view, this means that the set of values of the function $\varphi(z)$ is contained within the region $\varphi_\lambda(E)$ bounded by the Cassini oval. The properties of the subordination majorant are investigated $\varphi_\lambda(z).$ Based on this, relying on the method of subordination of analytical functions, in the class $\mathcal{P}_n(\varphi_\lambda),$ precise estimates are established for $\mathrm{Re}\, \varphi(z),$ $\left|\varphi(z)\right|,$ and $\left|{z\varphi'(z)}/{\varphi(z)}\right|,$ leading to one of the classical results in a particular case. The application of these estimates to the study of extreme properties of some classes of analytical functions $f(z)$ of the form $f(z) = z + a_{n+1}z^{n+1} + a_{n+2}z^{n+2} + \ldots,$ $n\geq 1$ is considered. In particular, theorems of growth, covering, and radii of convexity are established for one class of starlike functions which is constructed by using the function $\varphi_\lambda(z)$ and generalizes the well-known subclass of starlike functions of R. Singh. Applications of the obtained results to the study of some classes of close-to-starlike and doubly close-to-starlike functions related to the function $\varphi_\lambda(z)$ are also given. In particular, in these classes, growth theorems are established and radii of starlikeness are found.
All obtained results are accurate, represent new original results as well as some generalizations of known results.