Radius Constants for Functions with the Prescribed Coefficient Bounds

Abstract

For an analytic univalent function $f(z)=z+{\sum }_{n=2}^{\mathrm{\infty }}{a}_n{z}^n$ in the unit disk, it is well-known that $\left|a_n\right|\le n$ for $n\ge 2$. But the inequality $\left|a_n\right|\le n$ does not imply the univalence of $f$. This motivated several authors to determine various radii constants associated with the analytic functions having prescribed coefficient bounds. In this paper, a survey of the related work is presented for analytic and harmonic mappings. In addition, we establish a coefficient inequality for sense-preserving harmonic functions to compute the bounds for the radius of univalence, radius of full starlikeness/convexity of order $\alpha$ () for functions with prescribed coefficient bound on the analytic part.