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

Abstract

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 $k,$ $2\le k\le n.$ Then we apply the result to study the so-called $d-$concavity of the $k-$Hessian type function $F_k(R)=\log \left(S_k(R)\right),$ where $S_k(R)={\sigma }_k(\lambda (R)),\lambda (R)=\left({\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_n\right)\in {\mathbb{C}}^n$ is eigenvalue-vector of $R\in {\mathbb{R}}^{n\times n},$ $R=\omega +\beta ,{\omega }^T=\omega ,\omega >0,{\beta }^T=-\beta .$ 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.