Let $L$ be an oriented Lagrangian submanifold in an $n$-dimensional Kähler manifold $M$. Let $u:\,D\,\to \,M$ be a minimal immersion from a disk $D$ with $u(\partial D)\,\subset \,L$ such that $u(D)$ meets $L$ orthogonally along $u(\partial D)$. Then the real dimension of the space of admissible holomorphic variations is at least $n\,+\,\mu (E,\,F)$, where $\mu (E,\,F)$ is a boundary Maslov index; the minimal disk is holomorphic if there exist $n$ admissible holomorphic variations that are linearly independent over $\mathbb{R}$ at some point $p\,\in \,\partial D;$; if $M=\mathbb{C}{{P}^{n}}$ and $u$ intersects $L$ positively, then $u$ is holomorphic if it is stable, and its Morse index is at least $n\,+\,\mu (E,\,F)$ if $u$ is unstable.