Some properties of differential operator to the subclass of univalent functions with negative coefficients

Abstract

Various function theorists have successfully defined and investigated different kinds of analytic functions. The applications of such functions have played significant roles in geometry function theory as a field of complex analysis. In this work, therefore, a certain subclass of univalent analytic functions is defined using a generalized differential operator and we have discussed a subclass \(TS_{\sigma, \delta} ^{~ \wp} (\vartheta ,\hbar ,\ell )\) of univalent functions with negative coefficients related to differential operator in the unit disk \( \mathbb { U }=\left \{{z \in \mathbb{ C }:|z|<1}\right \}\). We obtain basic properties like coefficient inequality, distortion and covering theorem, radii of starlikeness, convexity and close-to-convexity, extreme points, Hadamard product, and closure theorems for functions belonging to our class.