Let $a_k$ and $b_k$ be ascending positive numbers for $1\leq k \leq K+1$. If it is known that $$\frac{K\left(\exp\left(\frac{1}{K}\sum_{k=1}^K b_k\right)-1\right)}{\left(\sum_{k=1}^K \sqrt{a_k} \sqrt{\exp(b_k)-1} \right)^2}=1$$ I need to prove that $$\frac{(K+1)\left(\exp\left(\frac{1}{K+1}\sum_{k=1}^{K+1} b_k\right)-1\right)}{\left(\sum_{k=1}^{K+1} \sqrt{a_k} \sqrt{\exp(b_k)-1} \right)^2}<1.$$

From Matlab this surely holds, but I do not know why... thanks