"rām dūt atulit bal dhāmā" is Chaupai 2 of the Hanuman Chalisa, which Tulsidas wrote in Awadhi. It means: You are Rama's messenger, the home of strength beyond measure. You are known as Anjani's son and as the son of the Wind.
राम दूत अतुलित बल धामा ।अंजनि-पुत्र पवनसुत नामा ॥rām dūt atulit bal dhāmāañjani-putra pavansut nāmā
You are Rama's messenger, the home of strength beyond measure. You are known as Anjani's son and as the son of the Wind.
In the Valmiki Ramayana, Jambavan reminds Hanuman of his birth: his mother Anjana was the wife of Kesari, and the Wind god was his father. The Manas Sundar Kand calls him the home of matchless strength.
अप्सराऽप्सरसां श्रेष्ठा विख्याता पुञ्जिकस्थला। अञ्जनेति परिख्याता पत्नी केसरिणो हरेः॥
Awadhi text: Goswami Tulsidas, Gita Press reading, checked against three Gita Press printings and Rambhadracharya's Mahaviri. English meaning and word glosses: Wisdom's own, following the traditional reading, edited by Ankur Shukla.