From Longman Dictionary of Contemporary Englishnot to be outdonenot to be outdoneCOMPETE WITH/TRY TO BEATin order not to let someone else do better or seem better than you Not to be outdone, other computer manufacturers are also donating machines to schools. → outdo
Examples from the Corpus
not to be outdone• Bob Dole, the front-runner, was not to be outdone.• Not to be outdone by the girls, the boys' team also won its second team title.• The Tramroad Company, not to be outdone, illuminated some of their own cars both inside and out.• Although only fragments remain, Dunseverick manages not to be outdone in the legend stakes.• King Charles, not to be outdone, then made the abandoned wife a duchess, the title to die with her. 