You're interpolating $input with a value such as '../log.dummy-10-01-06-05.gz' into the pattern m#PROCESSING ../$input#. After the interpolation, that's m#PROCESSING ../../log.dummy-10-01-06-05.gz#. That's different from the working pattern you had before (m#PROCESSING ../log.dummy-10-01-06-05.gz#).

One thing you'll want to do, since you're matching literal strings, is to use quotemeta or \Q in the pattern. As it is, when you interpolate $input, the initial dots in '../log.dummy-10-01-06-05.gz' could match any character (as they normally would in a regexp).

In short, I think you want this:

m#PROCESSING \Q$input#
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If I may make a few more suggestions...

• Please don't use # as delimiters. It really plays havoc with my context-sensitive editor. I personally like m{}.
• Since you're looking for literal strings, index would be faster (but maybe harder to read).
• Use three argument open: open my $zcat_fh, '-|', "zcat $filename" or die ...
• Have a look at Writeup Formatting Tips. Specifically, you need <code> tags around your code to make it look right.
• If you don't already, use strict and warnings.