Some Aspects on a Special Type of (α, β)-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) = beta(2)/alpha + beta + alpha alpha where alpha = root a(ij)y(i)y(j) is a Riemannian metric; beta = b(i)y(i) is a 1-form, and a is an element of(1/4,+infinity) is a real positive scalar. This kind of metric can be expressed as follows: F(alpha, beta) = alpha center dot phi(s), where phi(s) = s(2) + s + a. In this paper we will study some important results with respect to 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.