In this paper we introduce the concept of quadratic quantum $f$-divergence measure for a continuos function $f$ defined on the positive semi-axis of real numbers, the invertible matrix $T$ and matrix $V$ by $$\mathcal{S}_{f}\left( V,T\right) :=\mathrm{tr}\left[ \left\vert T^{\ast }\right\vert ^{2}f\left( \left\vert VT^{-1}\right\vert ^{2}\right) \right].$$ Some fundamental inequalities for this quantum $f$-divergence in the case of convex functions are established. Applications for particular quantum divergence measures of interest are also provided.

We consider an asymptotic version of $\alpha$-$\psi$ contractive mappings. We show the existence and uniqueness of fixed points. Caccioppoli's fixed point theorem is deduced from main results in this paper. Moreover, we discuss an asymptotic version of mappings related with $(c)$-comaprison functions.

Let $S$ be a $\mathbb Q$-homology projective plane, $C$ a rational unicuspidal curve on $S^0 = S - \operatorname{Sing} S$ and $C'$ the proper transform of $C$ with respect to the minimal embedded resolution of $C$. We prove that $S^0 - C$ is affine ruled if and only if $C'^2 \geq -1$ and determine the pairs $(S,C)$ when $\overline{\kappa}(S^0 -C) = -\infty$ and $C'^2 \leq -2$.

We produce the family of Calabi-Yau hypersurfaces $X_{n}$ of $(\mathbb P^{1})^{n+1}$ in higher dimension whose inertia group contains non commutative free groups. This is completely different from Takahashi's result [4] for Calabi-Yau hypersurfaces $M_{n}$ of $\mathbb P^{n+1}$.

Let $\mathcal{S}_A$ be the complex linear space of all analytic functions on the open unit disc $\mathbb D$, whose derivative can be extended to the closed unit disc $\bar{\mathbb D}$. We give the characterization of surjective, not necessarily linear, isometries on $\mathcal{S}_A$ with respect to the norm $\| f \| _{\sigma} = |f(0)| + \sup \{|f'(z)| : z \in \mathbb D \}$ for $f \in \mathcal{S}_A$.