require Net::Curl; Then, the following (assume I have use strict) fails with

Bareword "Net::Curl::Easy::new" not allowed while "strict subs" in use
+<?c>
<c>
my $c = Net::Curl::Easy::new;
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while the following does not

my $c = Net::Curl::Easy->new;
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That's because require is a runtime directive, and strict is compile time. During the parse of the script it is not distinguishable whether Net::Curl::Easy::new is a subroutine call or a bareword. Add parens to disambiguate:

my $c = Net::Curl::Easy::new();
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or wrap the require statement into a BEGIN block:

BEGIN { require Net::Curl::Easy }
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which loads, compiles and executes Net/Curl/Easy.pm during the parse, and thus makes known Net::Curl::Easy::new as being a subroutine in perl's symbol table.

But then, if Net::Curl::Easy::new is not insentitive to be called as a class method or a function, (i.e. it takes arguments and doesn't validate them proper), you may have to pass the classname, as LanX noted:

my $c = Net::Curl::Easy::new( q(Net::Curl::Easy) );
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- or not. Documentation or source code should reveal that.