Related to this I have solved the general case which is the following:

If $a_1,\cdots,a_n >0$, and $a_1+\cdots + a_n=n$, prove that $$\sum_{cyc}a_i^{a_i a_{i+1}} \geq (\sum_{i=1}^{n}a_ia_{i+1})\ln\left(\frac{\sum_{i=1}^{n}a_ia_{i+1}}{n}\right)+n> n(1-\frac{1}{e})$$

Where $a_{n+1}=a_1$

My proof:

We begin with a substitution we put:


After this we take the theorem 1.5 of this paper and put:





We get:

$$\frac{n}{\sum_{i=1}^{n}f(x_i)}\leq 1+\frac{\sum_{i=1}^{n} -x_i}{\sum_{i=1}^{n}f(x_i)}$$


$$n+\sum_{i=1}^{n}x_i\leq \sum_{i=1}^{n}f(x_i)$$

Which is equivalent to the inequality:

$$n+\sum_{cyc}\ln(a_i^{a_i a_{i+1}})\leq \sum_{cyc}a_i^{a_i a_{i+1}}$$

Now we use Jensen inequality to give:

$$\sum_{cyc}\ln(a_i^{a_i a_{i+1}})\geq (\sum_{i=1}^{n}a_ia_{i+1})\ln\left(\frac{\sum_{i=1}^{n}a_ia_{i+1}}{n}\right)$$

So we get:

$$\sum_{cyc}a_i^{a_i a_{i+1}} \geq (\sum_{i=1}^{n}a_ia_{i+1})\ln\left(\frac{\sum_{i=1}^{n}a_ia_{i+1}}{n}\right)+n>n(1-\frac{1}{e})$$

My question is:

Can you check my proof and tell me if there exist a better constant for the original inequality?


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