# Mix between Slater and Jensen inequality with exponential function

We have the following conjecture :

Let $$x_i$$ be $$n$$ real numbers and $$p_i>0$$ be $$n$$ weights and $$f(x)=e^{x}$$ then we have : $$\Big(\sum_{i=1}^{n}p_i\Big)\Big(f\Big(\frac{\sum_{i=1}^{n}x_ip_i}{\sum_{i=1}^{n}p_i}\Big)f\Big(\frac{\sum_{i=1}^{n}x_if(x_i)p_i}{\sum_{i=1}^{n}p_if(x_i)}\Big)\Big)^{\frac{1}{2}}\leq \sum_{i=1}^{n}p_if(x_i)$$

This inequality is more precise than the classical Jensen's inequality with exponential .

My try :

I think we can use induction to solve this problem for $$n=2$$ we have :

$$\Big(p_1+p_2\Big)\Big(f\Big(\frac{x_1p_1+x_2p_2}{p_1+p_2}\Big)f\Big(\frac{x_1f(x_1)p_1+x_2f(x_2)p_2}{p_1f(x_1)+p_2f(x_2)}\Big)\Big)^{\frac{1}{2}}\leq p_1f(x_1)+p_2f(x_2)$$

But after this I'm really stuck...