Entire functions on banach spaces with the U$U$-property

Let E$E$ be a Banach space without a copy of l1$l_{1}$ and with the U$U$-property. We show that every entire function on E$E$ which is weakly continuous on bounded sets is bounded on bounded sets of E$E$. We answer this way, in the affirmative, to a problem raised by Aron, Herves, and Valdivia in 1983, for these spaces. In particular, this is true for every Banach space which is an M$M$-ideal in its bidual.