Even if that's a 7 years old question, people new to R should consider using the data.table, package.
A data.table is a data.frame so all you can do for/to a data.frame you can also do. But many think are ORDERS of magnitude faster with data.table.
vec <- 1:10
library(data.table)
DT <- data.table(start=c(1,3,5,7), end=c(2,6,7,9))
DT[,new:=apply(DT,1,function(row) mean(vec[ row[1] : row[2] ] ))]