Sharp bounds of the norm of pre-Schwarzian of some certain starlike functions

Let $Delta$ be the open unit disc in the complex plane $mathbb{C}$, i.e. $Delta={zin mathbb{C}:|z|< 1}$ and $mathcal{H}(Delta)$ be the class of functions that are analytic in $Delta$.Also, let $mathcal{A}subset mathcal{H}(Delta)$ be the class of functions that have the following Taylor--Maclaurin series expansionbegin{equation*} f(z)=z+sum_{n=2}^{infty} a_nz^nquad(zinDelta).end{equation*}Thus, if $finmathcal{A}$, then it satisfies the following normalized conditionbegin{equation*} f(0)=0=f'(0)-1.end{equation*}The set of all univalent (one--to--one) functions $f$ in $Delta$ is denoted by $mathcal{U}$. Also, we denote by $mathcal{LU}subset mathcal{H}$ the class of all locally univalent functions in $Delta$. Let $f$ and $g$ belong to class $mathcal{H}(Delta)$. Then we say that a function $f$ is subordinate to $g$, written bybegin{equation*} f(z)prec g(z)quad{rm or}quad fprec g,end{equation*}begin{linenomath}if there exists a Schwarz function $w$ with the following propertiesbegin{equation*} w(0)=0quad{rm and}quad |w(z)|0}$, $varpi(0)=1$ and $varphi'(0)>0$.