Dimensions of certain sets of continued fractions with non-decreasing partial quotients

Let $[a_1(x),a_2(x),a_3(x),\cdots ]$ be the continued fraction expansion of $x\in (0,1)$. This paper is concerned with certain sets of continued fractions with non-decreasing partial quotients. As a main result, we obtain the Hausdorff dimension of the set $$\{x\in (0,1):a_1(x)\le a_2(x)\le \cdots ,\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{\log a_n(x)}{\psi (n)}=1\}$$ for any $\psi :\mathbb{N}\to {\mathbb{R}}^{+}$ satisfying $\psi (n)\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$.