Coefficient characterizations and sections for some univalent functions

Let $\mathscr G(\alpha)$ denote the class of locally univalent normalized analytic functions $f$ in the unit disk $|z|<1$ satisfying the condition
$$ \mathrm{Re}\left(1+\frac{zf''(z)}{f'(z)}\right)<1+\frac\alpha2\qquad\text{for}\quad|z|<1 $$
and for some $0<\alpha\le1$. We firstly prove sharp coefficient bounds for the moduli of the Taylor coefficients $a_n$ of $f\in\mathscr G(\alpha)$. Secondly, we determine the sharp bound for the Fekete–Szegö functional for functions in $\mathscr G(\alpha)$ with complex parameter $\lambda$. Thirdly, we present a convolution characterization for functions $f$ belonging to $\mathscr G(\alpha)$ and as a consequence we obtain a number of sufficient coefficient conditions for $f$ to belong to $\mathscr G(\alpha)$. Finally, we discuss the close-to-convexity and starlikeness of partial sums of $f\in\mathscr G(\alpha)$. In particular, each partial sum $s_n(z)$ of $f\in\mathscr G(1)$ is starlike in the disk $|z|\le1/2$ for $n\ge11$. Moreover, for $f\in\mathscr G(1)$, we also have $\mathrm{Re}(s'_n(z))>0$ in $|z|\le1/2$ for $n\ge11$.