On Geometry of Some Subspaces in Hornich Space

Let $\calH$ be the collection of all locally univalent analytic functions $f$ defined on the unit disk $\mathbb{D}$ with the normalization $f(0)=f^{\prime}(0)-1=0$, and $\calS\subseteq \mathcal{H}$ be the class of all univalent functions. For $f,g\in \mathcal{H}$ and $r\in \mathbb{C}$, the Hornich operators are defined as $$r\odot f(z):=\int_0^z\{f'(\xi)\}^r\mathrm{d}\xi \quad\text{and}\quad f\oplus g(z):=\int_0^zf'(\xi)g'(\xi)\mathrm{d}\xi.$$ We study geometric properties of some subclasses of $\calS$ in the sense of the Hornich space $(\mathcal{H},\odot , \oplus )$. In fact, we prove that the classes of strongly convex functions of order $\beta$, Noshiro-Warschawski functions, and strongly Ozaki close-to-convex functions are all convex in $(\mathcal{H},\odot , \oplus )$, which generalize some known results. Meanwhile, for $M,N\in\calS$, let $T[M,N]:=\{(r,s)\in\mathbb{C}^2:r \odot f \oplus s \odot g\in N,\ for\ \forall f,g\in M \}$. We give the precise descriptions of $T[M,N]$ for some $M,N\in\calS$.