English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $f=h+\overline{g}$ be a normalized harmonic mapping in the unit disk$\ID$. In this paper, we obtain the sharp radius of univalence, fullystarlikeness and fully convexity of the harmonic linear differential operators$D_f^{\epsilon}=zf_{z}-\epsilon\overline{z}f_{\overline{z}}~(|\epsilon|=1)$ and$F_{\lambda}(z)=(1-\lambda)f+\lambda D_f^{\epsilon}~(0\leq\lambda\leq 1)$ whenthe coefficients of $h$ and $g$ satisfy harmonic Bieberbach coefficientsconjecture conditions. Similar problems are also solved when the coefficientsof $h$ and $g$ satisfy the corresponding necessary conditions of the harmonicconvex function $f=h+\overline{g}$. All results are sharp. Some of the resultsare motivated by the work of Kalaj et al. \cite{Kalaj2014} (Complex Var.Elliptic Equ. 59(4) (2014), 539--552).

Radius of fully starlikeness and fully convexity of harmonic linear differential operator