If you are using explicit ranges like that then use for loops instead of while loops to make it clear that a range is being dealt with and what the range is. Consider:

#!/usr/bin/perl
use strict;
use warnings;

my $maxline = 5;
my $start = 2;
my $fileName = "testlog";

open my $info, '<', $fileName or die "Could not open '$fileName': $!";

<$info> // die "Log file '$fileName' too short\n" for 1 .. $start - 1;

for ($start .. $maxline) {
    my $line = <$info> // die "Log file '$fileName' too short\n";
    print $line;
}
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The <$info> // die ... stuff dies on end of file.