> eof
Returns 1 if the next read on FILEHANDLE will return end of file or if FILEHANDLE is not open. FILEHANDLE may be an expression whose value gives the real filehandle. (Note that this function actually reads a character and then "ungetc"s it, so isn't useful in an interactive context.)

A filehandle to which nothing has been printed is like a Schrödinger's cat. You can't tell whether there will be an eof before something writes to it or closes it.

Update: I like tybalt89's answer, but I guess you wanted to be notified that the parent is waiting for the child. Use IO::Select for that:

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

use Socket;
use IO::Select;

my ($child, $parent);

socketpair $child, $parent, AF_UNIX, SOCK_STREAM, PF_UNSPEC
    or die "socketpair: $!";
$child->autoflush(1);
$parent->autoflush(1);

if (my $pid = fork) {
    close $parent  or die "close: $!\n";
    print STDOUT time % 100, ": polling child\n";
    my $select = 'IO::Select'->new;
    $select->add($child);
    my $buf = '';
  COMMUNICATION:
    while (1) {
        if ($select->can_read(0)) {
            sysread $child, $buf, 4096, length $buf
                or last COMMUNICATION;
            if ($buf =~ s/(.*)\n//) {
                print STDOUT time % 100, ": received <$1>\n";
            }
        } else {
            print STDOUT time % 100, ": waiting for child.\n";
            sleep 1;
        }
    }
    print STDOUT time % 100, ": end of input from child\n";
    wait;
} elsif (defined $pid) {
    close $child or die "close: $!\n";
    print STDOUT time % 100, ": child started\n";
    sleep 5;
    print $parent time % 100, ": child printed\n";
    sleep 2;
    print $parent time % 100, ": child printed again\n";
    close $parent
        or die "close: $!\n";
    exit
} else {
    die "cannot fork: $!"
}
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