Partial sums for a generalised class of analytic functions

For a fixed natural number m, let \(\mathcal {A}_m\) ${\mathcal{A}}_m$ be the class of analytic and normalized functions \(f(z)=z+\sum _{k=m+1}^{\infty }a_kz^k\) $f(z)=z+{\sum }_{k=m+1}^{\infty }{a}_k{z}^k$ and let \(\begin{aligned} \mathcal{S}\mathcal{T}_m(\Phi ):=\left\{ f\in \mathcal {A}_m:\dfrac{zf'(z)}{f(z)}\prec \Phi (z)\right\} \end{aligned}\) $$\begin{array}{c}\hfill \mathcal{S}{\mathcal{T}}_m(\mathrm{\Phi }):=\left\{f,\in ,{\mathcal{A}}_m,:,{\displaystyle \frac{z{f}^{\prime }(z)}{f(z)}},\prec ,\mathrm{\Phi },(z)\right\}\end{array}$$ and \(\begin{aligned} \mathcal{C}\mathcal{V}_m(\Phi ):=\left\{ f\in \mathcal {A}_m:1+\dfrac{zf''(z)}{f'(z)}\prec \Phi (z)\right\} , \end{aligned}\) $$\begin{array}{c}\hfill \mathcal{C}{\mathcal{V}}_m(\mathrm{\Phi }):=\left\{f,\in ,{\mathcal{A}}_m,:,1,+,{\displaystyle \frac{z{f}^{\prime \prime }(z)}{f^{\prime }(z)}},\prec ,\mathrm{\Phi },(z)\right\},\end{array}$$ respectively, where \(\Phi \) $\mathrm{\Phi }$ is analytic univalent with \(\Phi (0)=1\) $\mathrm{\Phi }(0)=1$ . For \(f\in \mathcal{S}\mathcal{T}_m(\Phi )\) $f\in \mathcal{S}{\mathcal{T}}_m(\mathrm{\Phi })$ or \( \mathcal{C}\mathcal{V}_m(\Phi )\) $\mathcal{C}{\mathcal{V}}_m(\mathrm{\Phi })$ , we estimate the sharp lower bounds of certain ratios involving real parts of f and their \(n^{th}-\) $n^{\mathit{th}}-$ partial sum using some specific coefficient inequalities. Moreover, we consider the meromorphic functions corresponding to \(\Phi \) $\mathrm{\Phi }$ and study their geometric properties.