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.

Here is an alternative formulation of the same thing:
With ascending $a_k$ and $b_k$, define
$$z_m=\frac{\prod_{k=1}^{m}(1+b_k)^{(1/m)}-1}{\frac{1}{m}\left[\sum_{k=1}^m\sqrt{a_kb_k}\right]^2}.$$

Let $m_0$ be the smallest $m$ such that $z_m<1$. Then, for $m>m_0$, the sequence $z_m$ is monotonically decaying. It is easy to find counter examples where the sequence increase for $z_m$ with $m<m_0$.


thanks