Police were called to Barnsley Hospital on May 16 following reports that a baby girl had suffered catastrophic injuries.
The baby died on Monday (September 14) at 2.50pm after life support was withdrawn.
A post mortem examination was carried out yesterday (September 15) and a cause of death was unascertained, pending further medical enquiries.
A 31-year-old woman and a 38-year-old man, both from Athersley, have been arrested on suspicion of murder and are currently in police custody.
Detective chief inspector Phil Etheridge, the senior investigating officer, said: "The injuries suffered by this little girl when she was just two days old were so catastrophic that she could not recover and her life support was withdrawn after four months.
“Our thoughts are with those affected and our specialist officers are supporting them throughout this incredibly difficult time.
"We have a dedicated team of officers assigned to this investigation. Two people are in custody and we are working hard to establish the exact circumstances leading up to this child's death.
"I know that the local community will be shocked and deeply saddened to hear of her death and it's important that people feel able to talk about their concerns but I urge everyone to be mindful of what they post on social media, and not to speculate on what may have occurred.
“If you have information that may assist, it's important that this is passed to the police to form part of the investigation."
Anyone with concerns or information is asked to call police on 101 quoting incident number 207 of September 16.