Star-likeness associated with the exponential function

Authors : Adiba NAZ, Sumit NAGPAL, V. RAVICHANDRAN

Pages : 1353-1371

View : 9 | Download : 7

Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :Given a domain $\Omega$ in the complex plane $\mathbb{C}$ and a univalent function $q$ defined in an open unit disk $\mathbb{D}$ with nice boundary behaviour, Miller and Mocanu studied the class of admissible functions $\Psiinsert ignore into journalissuearticles values(\Omega,q);$ so that the differential subordination $\psiinsert ignore into journalissuearticles values(pinsert ignore into journalissuearticles values(z);,zp`insert ignore into journalissuearticles values(z);,z^2p``insert ignore into journalissuearticles values(z);;z);\prec hinsert ignore into journalissuearticles values(z);$ implies $pinsert ignore into journalissuearticles values(z);\prec qinsert ignore into journalissuearticles values(z);$, where $p$ is an analytic function in $\mathbb{D}$ with $pinsert ignore into journalissuearticles values(0);=1$, $\psi:\mathbb{C}^3\times \mathbb{D}\to\mathbb{C}$ and $\Omega=hinsert ignore into journalissuearticles values(\mathbb{D});$. This paper investigates the properties of this class for $qinsert ignore into journalissuearticles values(z);=e^z$. As application, several sufficient conditions for normalized analytic functions $f$ to be in the subclass of star-like functions associated with the exponential function are obtained.
Keywords : Univalent functions, star like functions, differential subordination, exponential function, Janowski star like function