Comparing present and accumulated value of annuities with different interest rates

Assume we have k immediate (due)-annuities with different interest rates. Let {\bf i}=(i_۱,i_۲,...,i_k) and {\bf i^*}=(i^*_۱,i^*_۲,...,i^*_k) be two vectors of interest rates such that {\bf i^*} is majorized by {\bf i}. It's shown that sum of present and accumulated value of annuities-immediate with interest rate {\bf i} is grater than sum of present value of annuities-immediate with interest rate {\bf i^*}. We also prove the similar results for annuities-due.