During my research I'm confronting the following inequality: $N_1(\frac{t_1}{N_1})^{1/3} +N_2(\frac{t_2}{N_2})^{1/3}<(N_1+N_2)(\frac{t_1+t_2}{N_1+N_2})^{1/3}$
With $0<t_1,t_2,N_1,N_2$ and $0<\frac{t_1}{N_1}<\frac{t_2}{N_2}<2$ Could anybody give me a hint? That'd be great ! Thanks in advance
Observe that $$\frac{t_1 + t_2}{N_1 + N_2} = \frac{t_1}{N_1} \cdot \frac{N_1}{N_1 + N_2} + \frac{t_2}{N_2} \cdot \frac{N_2}{N_1 + N_2}.$$
Divide your inequality by $N_1 + N_2$ and then apply Jensen inequality.