On certain subclasses of analytic and harmonic mappings

Let $\mathcal{H}$ be the class of harmonic functions $f=h+\overline{g}$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$, where $h$ and $g$ are analytic in $\mathbb{D}$ with the normalization $h(0)=g(0)=h'(0)-1=0$. Let $\mathcal{D}_{\mathcal{H}}^0(\alpha, M)$ denote the class of functions $f=h+ \overline{g}\in\mathcal{H}$ satisfying the conditions $\left|(1-\alpha)h'(z)+\alpha zh''(z)-1+\alpha\right|\leq M+\left|(1-\alpha)g'(z)+\alpha zg''(z)\right|$ with $g'(0)=0$ for $z\in\mathbb{D}$, $M>0$ and $\alpha\in(0,1]$. In this paper, we investigate fundamental properties for functions in the class $\mathcal{D}_{\mathcal{H}}^0(\alpha, M)$, such as the coefficient bounds, growth estimates, starlikeness and some other properties. Furthermore, we obtain the sharp bound of the second Hankel determinant of inverse logarithmic coefficients for normalized analytic univalent functions $f\in\mathcal{P}(M)$ in $\mathbb{D}$ satisfying the condition $\text{Re}\left(zf''(z)\right)>-M$ for $0<M\leq 1/\log4$ and $z\in\mathbb{D}$.