I’d play with the appearance palette in Illustrator and applying effects to your live text in the appearance palette. Here’s my live text with the appearance palette showing a “Warp Arc Lower” and a “Warp Arc” effect to it:
I used both because in your example the top was rounded as well. By starting with arc lower, I was able to bend the bottom, and then with the regular arc warp I was able to warp all of it. Here are the separate effect settings I used:
And this shows a comparison by the non warped font to the final version below it.
A final caveat is that using warping in illustrator on text tends to distort the font quite a bit. To “fix” the fact that it tends to stretch the font, I applied a 65% horizontal scale to the letters in the character palette. And To further “fix” the issue I applied a 60% horizontal scale to the center three letters “THL” since the center letters get more stretched than the outside.
Basically you may just have to tweak and play with it.