better quality of transparent image

Not really. Without the original to work with (photoshop/illustrator etc) you're going to struggle. It looks like the image was created as a PNG8 rather than a PNG24, so the transparency is 1 bit, and you get jagged edges.

Hopefully someone has the original and you can export a 24 bit transparent PNG out from there.

@padas - the original didn't have a white background - you've just added a white background to the loops in the letters, effectively making it less transparent!

Thanks Chris.  I just assumed and pasted rather than download and save.  

If it is already transparent, and it is being asked to make it "better quality", do you mean a higher dpi for print?  If we do that, than the resolution will be smaller.   If you want a higher dpi keeping the same resolution, that can be done from your image posted here.

@padas - Yeah - Photoshop doesn't seem to copy and paste transparent PNG very well. It doesn't keep the transparency.

I think the problem is that when you save transparent images in PNG you have 2 choices - 8bit and 24bit. The 24 bit gives you nice clean transparency (the alpha channel is 8 bit). The 8bit PNG works like the old gif - transparency is either on or off (1 bit), which is fine if your image is sitting on a plain background, but not so good if it's sat over a pattern, gradient or other image, as you see the base colour around the edges. If you download the image, open it up in photoshop and overlay it on a black background, you'll see what I mean.

Strange definition of better quality!!

The answer you've accepted is no better in quality and actually has less transparency than you started with.

Thank you for the points.  Are you accepting the part of the answer "go back to the designer"? or the image?

>The answer you've accepted is no better in quality and actually has less transparency than you started with.

It would be worse.  That's why I asked which part of the answer he is accepting.