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 there exists some constant C>0 sunch that $$C(\frac{a_1}{b_1}+\cdot+\frac{a_k}{b_k})\le\frac{t_1a_1+\cdot+t_ka_k}{t_1b_1+\cdot+t_kb_k}.$$

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

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!

Note: I have changed the inequality. The following $$\frac{t_1a_1+\cdot+t_ka_k}{t_1b_1+\cdot+t_kb_k}\le \max(\frac{a_1}{b_1},\cdots,\frac{a_k}{b_k}).$$ holds. The matter is $a_i (i=1,\cdots,k) $ are not necessary positive.

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!

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 there exists some constant C>0 sunch that $$C(\frac{a_1}{b_1}+\cdot+\frac{a_k}{b_k})\le\frac{t_1a_1+\cdot+t_ka_k}{t_1b_1+\cdot+t_kb_k}.$$

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