# Given four real numbers $a,b,c,d$ so that $1leq aleq bleq cleq dleq 3$. Prove that $a^2+b^2+c^2+d^2leq ab+ac+ad+bc+bd+cd.$

Mathematics Asked by user818748 on December 29, 2020

Given four real numbers $$a, b, c, d$$ so that $$1leq aleq bleq cleq dleq 3$$. Prove that
$$a^{2}+ b^{2}+ c^{2}+ d^{2}leq ab+ ac+ ad+ bc+ bd+ cd$$

My solution
$$3a- dgeq 0$$
begin{align}Rightarrow dleft ( a+ b+ c right )- d^{2}= dleft ( a+ b+ c- d right ) & = dleft ( 3a- d right )+ dleft ( left ( b- a right )+ left ( c- a right ) right )\ & geq bleft ( b- a right )+ cleft ( c- a right ) \ & geq left ( b- a right )^{2}+ left ( c- a right )^{2} \ & geq frac{1}{2}left ( left ( b- a right )^{2}+ left ( c- a right )^{2}+ left ( c- a right )^{2} right )\ & geq frac{1}{2}left ( left ( b- a right )^{2}+ left ( c- b right )^{2}+ left ( c- a right )^{2} right )\ & = a^{2}+ b^{2}+ c^{2}- ab- bc- ca end{align}

It's wrong.

Try $$(a,b,c,d)=(1,1,1,4).$$ For these values we need to prove that $$19leq15,$$ which is not so true.

The following inequality is true already.

let $${a,b,c,d}subset[1,3].$$ Prove that: $$a^2+b^2+c^2+d^2leq ab+ac+bc+ad+bd+cd.$$

We can prove this inequality by the Convexity.

Indeed, let $$f(a)=ab+ac+bc+ad+bd+cd-a^2-b^2-c^2-d^2$$.

Thus, $$f$$ is a concave function, which says that $$f$$ gets a minimal value for an extreme value of $$a$$,

id est, for $$ain{1,3}$$.

Similarly, for $$b$$, $$c$$ and $$d$$.

Thus, it's enough to check our inequality for $${a,b,c,d}subset{1,3}$$, which gives that our inequality is true.

Correct answer by Michael Rozenberg on December 29, 2020