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

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