I am currently writting a paper and ended up with an expression that looks like the following:
$$\frac{\sum_{i=1}C_{i}}{\sum_{i=1} {N_i}^\gamma {C_{i}}^{1-\gamma}}$$
for $i=1...K$, $C_i>0$, $N_i \in [0,1]$ such that $\sum_iN_i=1$, and $\gamma \in [0,1]$I want to know if the following inequality holds:
$$\frac{\sum_{i=1}C_{i}}{\sum_{i=1} {N_i}^\gamma {C_{i}}^{1-\gamma}}\leq \frac{K\max_kC_{k}}{\sum_{i=1} {N_i}^\gamma \max_k\{{C_{k}}\}^{1-\gamma}} $$ $$=\frac{\max_k \{C_k\} K}{\max_k \{C_k\}^{1-\gamma}\sum_{i=1} {N_i}^\gamma}$$ $$=\frac{\max_k \{C_k\}^\gamma K}{\sum_{i=1} {N_i}^\gamma}$$
Note that I am basically trying to apply max inside both sums. My intuition is that this inequality holds. However, I haven´t been able to prove it, so now I have my doubts. What do you think?
The inequality is false. Take $K=2$, $\gamma=1/2$ and $$ C_1=100,\quad C_2=1,\quad N_1=1,\quad N_2=100. $$ Then $$ \frac{\sum_{i=1}C_{i}}{\sum_{i=1} {N_i}^\gamma {C_{i}}^{1-\gamma}}=\frac{101}{20}>5 $$ and $$ \frac{K\max_iC_{i}}{\sum_{i=1} {N_i}^\gamma \max_k\{{C_{k}}\}^{1-\gamma}}=\frac{200}{10\cdot(10+1)}<2 $$
PS. A commentator correctly remarked that the hypothesis was $N_i\in[0,1]$ and $N_1+N_2=1$, so the example must be rescaled (both members are $\gamma$-homogeneous in $N_i$). I leave the example as it is, because it is easier to evaluate the square roots. However the idea is simple: you have problems when in the sum $\sum_i N_i^\gamma C_i^{1-\gamma}$ very small $N_i$ meet the big $C_i$ and viceversa.