Strictly Nef Divisors

Let $X$ be a projective manifold of dimension $n$ and $L$ a strictly nef line bundle on $X$. Then $K_X+tL$ is ample if $t>n+1$ in the following cases. 1.) $\text{dim}X=3$ unless (possibly) $X$ is a Calabi-Yau with $c_2 \cdot L=0$; 2.) $\kappa (X)\ge n-2$; 3.) $\text{dim}\alpha (X)\ge n-2$, with $\alpha :X\to A$ the Albanese map.