What is the purpose of Eliot alluding to John Donne's 'The Relic' in the line 'arms that are braceleted and white and bare'?
The line in Donne's relic to which Eliot refers is "A bracelet of bright hair about the bone". This line evokes not the carnal love about which Eliot's narrator is so indecisive but rather a spiritual love that transcends mortality, and may, in fact happen only after death.
There are a few different themes evoked by the reference. The first, echoing Marvell's "To His Coy Mistress", suggests that Prufrock and the woman will both have died before he gets around to posing his "overwhelming question." In this way the bracelet serves as a memento mori.
The second is a comparison of the trivialities of the relationship in Prufrock, set against a background of London social life and gossip, against the spiritual and intellectual background of Donne. This is the theme of disassociation of the sensibility in which Donne, as part of a unified sensibility, framed emotions and spirit and intellect as part of a unified whole, but in modernity, they are separate, so the Prufrock's intellect opposes and hinders his emotional life, and his emotions only funtion when independent of ideas ("do not ask what is it")