Admissible solutions to augmented nonsymmetric $k-$Hessian type equations I. The $d-$concavity of the $k-$Hessian type functions

We establish for $2\le k\le n-1$ the strict concavity of the function $f_k(\lambda )=\log ({\sigma }_k(\lambda ))$ on a subset of the positive cone $\Gamma_n={\lambda=(\lambda_{1}, \lambda_{2}, \cdots,\lambda_{n})\in $\mathbb{R}$^n; \lambda_j>0,j=1,\cdots, n}$where$\sigma_{k}(\lambda)$ is the basic symmetric polynomial of degree $k,$ $2\le k\le n\mathrm{.}$ Then we apply the result to study the so-called $d-$concavity of the $k-$Hessian type function $F_k(R)=\log (S_k(R)),$ where $S_{k}®=\sigma_{k}(\lambda®), \lambda®= \left(\lambda_{1}, \lambda_{2}, \cdots, \lambda_{n}\right) \in $\mathbb{C}$^{n}$iseigenvalue-vectorof$R \in $\mathbb{R}$^{n \times n},$ $R=\omega +\beta ,{\omega }^T=\omega ,\omega >0,{\beta }^T=-\beta \mathrm{.}$ The $d-$concavity will be used in our next paper to study the existence of admissible solutions to the Dirichlet problem for the augmented nonsymmetric $k-$Hessian type equations.