Geometric properties on (j,k)-symmetric functions related to starlike and convex function

Commun. Korean Math. Soc. Published online July 27, 2021

Priyabrat Gochhayat and Anuja Prajapati
Sambalpur University; Sambalpur University

Abstract : For $j=0,1,2,\cdots,k-1;~~k\geq 2;~\text{and}~-1\leq B<A\leq1$, we have introduced the functions classes denoted by $\mathcal{ST}_{[j,k]}(A,B)$ and $\mathcal{K}_{[j,k]}(A,B)$ respectively called the generalized $(j,k)$-symmetric starlike and convex functions. We first proved the sharp bounds on $\mid f(z)\mid$ and $\mid f^{\prime}(z)\mid$. Various radii related problems, such as radius of $(j,k)$-symmetric starlikeness, convexity, strongly starlikeness and parabolic starlikeness are determined. The quantity $\mid a_{3}^{2}-a_{5}\mid$, which provide the initial bound on Zalcman functional is obtained for the functions in the family $\mathcal{ST}_{[j,k]}.$ Furthermore, the sharp Pre-Schwarzian norm is also established for the case when $f$ is a member of $\mathcal{K}_{[j,k]}(\alpha)$ for all $0\leq \alpha <1.$

Keywords : Analytic and univalent function, Subordination, (j, k)-symmetric starlike and convex functions, distortion, growth, radius of starlikeness, radius of convexity, radius of strong starlikeness, radius of parabolic starlikeness, Zalcman functional, Pre-