Newcastle Brown getting on my nerves

[Ofnuts]

Printable heat transfer is the completely opposite problem.

It will only work if the mug is a cone, because it needs to be a developable surface.

The shape you want it a circle arc.

Wrap the mug with paper, and draw a pair of horizontal lines and a pair or vertical ones that delimit the logo:

   

Then spread out the paper:

   

• You can see that the horizontal lines are now circle arcs, and the vertical sides are not parallel: yhey actually converge to the apex of the cone.
  
• Using the height measurements and the corresponding circumferences of the mug/glass (here, 230mm@110mm (top of label) and 178mm@0mm (base of glass), you can compute the total height of the cone (376mm below the base of the glass). The same computation gives you its summit angle (4.3°). You can also try to find it with two long rulers, but it's easier to compute it from the circumferences/height.
  
• A the top of the label the big circle has a circumference of 2*π*(110+376)=3057mm and since you want 84mm of it your label spans an arc of 84*360/3057=9.89°
  

If you start from a path, off the top of my head I can' tell if it should be done with my oft-text-along-path plugin (characters aren't distorted) or with the regular text-along-path thing. If this is a logo, you can try ofn-path-bend with an envelope that follow the top and bottom arcs. I also have an unpublished script that does the same as the text-along-path function, but  on any path.

Another way is to use Distort > Polar coordinates but this is going to be very painful.