This was [asked but never answered at MSE] and This question has second an open bounty,until Now no one can solve it,I think there should be a more professional here severe people may be able to solve this problem, I am looking forward to (https://math.stackexchange.com/questions/2598247/how-to-determine-the-minimal-constant-lambda-lambdan-k).

Given fixed positive integers $ n,k$ , determine the minimal constant $ \lambda = \lambda(n,k)$ for which the following inequality holds for any $ a_1,a_2,…,a_n>0$ (taking indices mod $ n$ if required): $ $ \sum_{i=1}^n\frac{a_i}{\sqrt{a_i^2+a_{i+1}^2+…+a_{i+k}^2}}\le \lambda$ $

It seem that $ \lambda=\dfrac{n}{\sqrt{k+1}}??$

I have see $ n=3,k=1$ ,it is Classical inequalities see this :https://math.stackexchange.com/questions/1481348/prove-inequality-sqrt-frac2aba-sqrt-frac2bcb-sqrt-frac2c?noredirect=1&lq=1

when $ n=4,k=2$ it is also Classical inequalities $ $ \sum_{cyc}\sqrt{\dfrac{a}{a+b+c}}\le\dfrac{4}{\sqrt{3}}$ $

But general How to solve it?Thanks