Convergence rate in the law of logarithm for negatively dependent random variables under sub-linear expectations

Let $\{X,X_n,n\ge 1\}$ be a sequence of identically distributed, negatively dependent (NA) random variables under sub-linear expectations, and denote $S_n={\sum }_{i=1}^n{X}_i$, $n\ge 1$. Assume that $h(\cdot )$ is a positive non-decreasing function on $(0,\mathrm{\infty })$ fulfulling ${\int }_1^{\mathrm{\infty }}(th(t){)}^{-1}\text{dif}t=\mathrm{\infty }$. Write $Lt=\ln \max \{\text{me},t\}$, $\psi (t)={\int }_1^t(sh(s){)}^{-1}\text{dif}s$, $t\ge 1$. In this sequel, we establish that ${\sum }_{n=1}^{\mathrm{\infty }}(nh(n){)}^{-1}\text{vv}\{\mathrm{\mid }{S}_n\mathrm{\mid }\ge (1+\epsilon )\sigma \sqrt{2nL\psi (n)}\}<\mathrm{\infty }$, $\mathrm{\forall }\epsilon >0$ if $\text{ee}(X)=\text{ee}(-X)=0$ and $\text{ee}(X^2)={\sigma }^2\in (0,\mathrm{\infty })$. The result generalizes that of NA random variables in probability space.