A new subclass of Ma-Minda starlike functions associated with a heart-shaped curve

In this paper, we extend the q-derivative operator, which plays an essential role in quantum calculus. Indeed, by using the Hadamard product and generalized Koebe function we define the following (\alpha,\beta,\gamma)-derivative operator\begin{equation*}    {\rm d}_{\alpha,\beta,\gamma} f(z)=\frac{۱}{z}\left\{f(z)*\mathfrak{L}_{\alpha,\beta,\gamma}(z)\right\},\end{equation*}where\begin{equation*}    \mathfrak{L}_{\alpha,\beta,\gamma}(z)=\frac{۲(۱-\gamma)z}{(۱-\alpha z)(۱-\beta z)},\end{equation*}and \alpha\in[-۱,۱], \beta\in[-۱,۱], \alpha\beta\neq \pm۱ and \gamma\in[۰,۱). Then by subordination relation, the operator {\rm d}_{\alpha,\beta,\gamma} f(z), and a special function \phi_\delta(z)=۱+\delta z/\exp(\delta z) (۰<\delta\leq۱), we define a new particular Ma-Minda class. We investigate some properties of this class, such as, radius problem and coefficient estimate.

کلیدواژه ها:

Unit disk ، Analytic functions ، starlike function ، subordination ، Radius problems ، Coefficients problems