  Let $S$ be a minimal surface of general type with $p_g(S)=2$ and $K^2_S=1$,
so called by a minimal $(1,2)$-surface. Then we obtain that the global log
canonical threshold of the surface $S$ via $K_S$ is greater than equal to
$\frac{1}{2}$. As an application we have \[
{\rm{vol}}(X)\ge\frac{4}{3}p_g(X)-\frac{10}{3} \] for all projective $3$-folds
$X$ of general type which answers Question 1.4 of [J. A. Chen, M. Chen, C.
Jiang, "The Noether inequality for algebraic threefolds", arXiv:1803.05553]
about Noether inequality for $X$ with $5\le p_g(X)\le 26$.
