Given the style and position of the signature it appears that the portrait was photographed by an Eileen Fitzmaurice.
The Limerick Leader on 9th July 1938 published news about a court case (which was dismissed) involving a Miss Eileen Fitzmaurice for "overholding" of rent at her premises at 20 O'Connell Street which was used as a photographic studio. It implies that it had been in use as her studio since 1930. If she was withholding rent then business may not have been good and she may have closed soon afterwards so the photos may have been taken earlier than first thought.
(FWIW, Ferguson's Chemist used to be at 20 O'Connell Street.)