Offhand, I don't know it either. But let me take a guess.
- Start with a log-normal distribution. It has the CDF $$F(x;\mu, \sigma) = \frac{1}{2} \left[1 + \operatorname{erf} \left( \frac{\ln x - \mu}{\sigma \sqrt{2}} \right) \right]$$
- Apply the truncation formula on the interval $[a,b]$ $$F(x| a < X \leq b]) = \frac{F(x) - F(a)}{F(b) - F(a)}$$ which by composition gives
$$F(x;\mu, \sigma | a < x \leq b) = \frac{\frac{1}{2} \left[1 + \operatorname{erf} \left( \frac{\ln x - \mu}{\sigma \sqrt{2}} \right) \right] - \frac{1}{2} \left[1 + \operatorname{erf} \left( \frac{\ln a - \mu}{\sigma \sqrt{2}} \right) \right]}{\frac{1}{2} \left[1 + \operatorname{erf} \left( \frac{\ln b - \mu}{\sigma \sqrt{2}} \right) \right] - \frac{1}{2} \left[1 + \operatorname{erf} \left( \frac{\ln a - \mu}{\sigma \sqrt{2}} \right) \right]}$$
$$= \frac{ \left[1 + \operatorname{erf} \left( \frac{\ln x - \mu}{\sigma \sqrt{2}} \right) \right] - \left[1 + \operatorname{erf} \left( \frac{\ln a - \mu}{\sigma \sqrt{2}} \right) \right]}{ \left[1 + \operatorname{erf} \left( \frac{\ln b - \mu}{\sigma \sqrt{2}} \right) \right] - \left[1 + \operatorname{erf} \left( \frac{\ln a - \mu}{\sigma \sqrt{2}} \right) \right]}$$
$$= \frac{\operatorname{erf} \left( \frac{\ln x - \mu}{\sigma \sqrt{2}} \right) - \operatorname{erf} \left( \frac{\ln a - \mu}{\sigma \sqrt{2}} \right) }{ \operatorname{erf} \left( \frac{\ln b - \mu}{\sigma \sqrt{2}} \right) - \operatorname{erf} \left( \frac{\ln a - \mu}{\sigma \sqrt{2}} \right) }$$