I hope the following kind of inequality holds: let $a_i,b_i\in R$ with $b_i>0$, $\sum _{i=1}^mt_i=1$ with $t_i>0$, then
$$\frac{t_1a_1+\cdot+t_ka_k}{t_1b_1+\cdot+t_kb_k}\le\frac{a_1}{b_1}+\cdot+\frac{a_k}{b_k}.$$

I am not sure whether it is true. Does someone know some inequalities like above? Many thanks!