Some Aspects on a Special Type of $(\alpha,\beta )$-metric

The aim of this paper is twofold. Firstly, we will investigate the link between condition for functions $\phi(s)$ from $(\alpha, \beta)$-metrics Douglas type to be self-concordant and k-self concordant, other objective continue recently new introduced \beta)$-metric ([17]): $$ F(\alpha,\beta)=\frac{\beta^{2}}{\alpha}+\beta+a \alpha where $\alpha=\sqrt{a_{ij}y^{i}y^{j}}$ a Riemannian metric; $\beta=b_{i}y^{i}$ 1-form, $a\in \left(\frac{1}{4},+\infty\right)$ real positive scalar. This kind metric can expressed as follows: $F(\alpha,\beta)=\alpha\cdot \phi(s)$, $\phi(s)=s^{2}+s+a$. In study some important results in respect with above mentioned such as: Kropina change metric, Main Scalar also analyze how k-self-concordant function $\phi(s)$, linked $F$ type. functions, change, main