For \(1\le k\le n\) integer number, let \(\sigma_k\) be the \(k\)-th elementary symmetric function \[ \sigma_k(\lambda)=\sum_{1\le i_1<i_{2}<\cdots< i_k\le n}\lambda_{i_1}\lambda_{i_2}\cdots\lambda_{i_{k}}, \;\;\text{for any}\;\; \lambda=(\lambda_1,\cdots,\lambda_n)\in\mathbb{R}^n. \] Let \(\Gamma_k\) be the convex cone \[ \Gamma_k=\left\{\lambda=(\lambda_1,\cdots,\lambda_n)\in\mathbb{R}^n\; :\; \sigma_1(\lambda)>0,\dots, \sigma_k(\lambda)>0 \right\}. \] Let \((X,\alpha)\) be a compact complex manifold with Hermitian metric \(\alpha\) and non-empty boundary \(\partial X\) written in local coordinates as \[ \alpha=\sqrt{-1}\alpha_{\overline{k}j}dz^{j}\wedge d\overline{z}^{k} \] and set \(\alpha^{i\overline{k}}\) the inverse of \(\alpha_{\overline{k}j}\). Now, let \(\chi \in \Omega^{1,1}(X, \mathbb{R})\) be a differential form of type \((1,1)\), written in local coordinates as \[ \chi=\sqrt{-1}\chi_{\overline{k}j}dz^{j}\wedge d\overline{z}^{k}. \] We say \(\chi\in \Gamma_k(X, \alpha)\) if the vector of eigenvalues of the Hermitian endomorphism \(\alpha^{i\overline{k}}\chi_{\overline{k}j}\) lies in the \(\Gamma_k\) cone at each point. The authors prove that given \(\chi \in \Omega^{1,1}(X, \mathbb{R})\) differential form of type \((1,1)\), \(\psi\in C^{\infty}(X)\) satisfying \(\psi\ge c>0\), \(\varphi\in C^{\infty}(\partial X,\mathbb{R})\) if there exists a subsolution \(\underline{u}\in C^{\infty}(\overline{X}, \mathbb{R})\) of the equation \[ \sigma_k(\underline{\lambda})\ge \psi, \quad \underline{u}{|_{\partial X}}=\varphi, \] where \(\underline{\lambda}\in \Gamma_k\) are the eigenvalues of \(\chi+\sqrt{-1}\partial\overline{\partial}\underline{u}\) with respect to \(\alpha\), then there exists a unique solution \(u\in C^{\infty}(\overline{X},\mathbb{R})\) of the equation \[ \sigma_k(\lambda)= \psi, \quad u{|_{\partial X}}=\varphi, \] where \(\lambda\in \Gamma_k\) are the eigenvalues of \(\chi+\sqrt{-1}\partial\overline{\partial}u\) with respect to \(\alpha\).