Open this image in Mozilla or Chrome browser, So you show the display some white pixels in the outer side of gif. I didn't apply any effect, just used two flat colors and round or text in that.
How can i fix this issue?
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You cannot use antialias in transparent GIF images since it's only 8-bit, but you can fake it. When exporting using "Save for Web" use the option Matte to match the background color of where the animation will be placed. It adds a few extra colored pixels at the edges of the image.
The downside is that you can only match one color, so it won't work well on top of gradients or a background with a different color.