Zalcman conjecture and Hankel determinant of order three for starlike and convex functions associated with shell-like curves

The aim of this article is to estimate an upper bound of $|H_3(1)|$, the Zalcman coefficient functional for $n=3$ and $n=4$, and also to investigate the fifth, sixth, seventh coefficients of starlike and convex functions associated with shell-like curves. Similar type of outcomes are estimated for the functions $f^{-1} $ and $\frac{z}{f\left(z\right)}$.