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Inequality 10(Christos Patilas)

Problem:

If \displaystyle a,b,c\in \mathbb{R}, prove that

\displaystyle \sqrt[4]{a^4+b^4+c^4+1}\geq \sqrt[5]{a^5+b^5+c^5+1}.

Solution:

We have that

\displaystyle \sqrt[5]{\sum_{cyc}a^5+1}\leq \sqrt[5]{\sum_{cyc}\left|a\right|^5+1},

so we only need to prove the inequality for \displaystyle a,b,c\geq 0.

Let us write the given inequality into the form

\displaystyle \left(\sum_{cyc}a^4+1\right)^{\frac{5}{4}}\geq \sum_{cyc}a^5+1 and divide with \displaystyle \left(\sum_{cyc}a^4+1\right)^{\frac{5}{4}}.

Then we acquire

\displaystyle \sum_{cyc}\frac{a^5}{\left(\sum_{cyc}a^4+1\right)^{\frac{5}{4}}}+\frac{1}{\left(\sum_{cyc}a^4+1\right)^{\frac{5}{4}}}\leq 1

or

\displaystyle \sum_{cyc}\left(\frac{a^4}{\sum_{cyc}a^4+1}\right)^{\frac{5}{4}}+\frac{1}{\left(\sum_{cyc}a^4+1\right)^{\frac{5}{4}}}\leq 1.

But for all non-negative numbers holds that

\displaystyle \left(\frac{a^4}{\sum_{cyc}a^4+1}\right)^{\frac{5}{4}}\leq \frac{a^4}{\sum_{cyc}a^4+1}.

So from the above inequality we get that

\displaystyle \sum_{cyc}\left(\frac{a^4}{\sum_{cyc}a^4+1}\right)^{\frac{5}{4}}+\frac{1}{\left(\sum_{cyc}a^4+1\right)^{\frac{5}{4}}}\leq \frac{\sum_{cyc}a^4+1}{\sum_{cyc}a^4+1}=1, Q.E.D.


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