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

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}

Mathematics Asked by user818748 on December 29, 2020

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

Related Questions

Continuous $f$ has $≥2$ roots if $int_{-1}^{1} f(x)sqrt {1 – x^2} mathrm{d}x = int_{-1}^{1} xf(x) mathrm{d}x = 0$?

2  Asked on December 14, 2020

1  Asked on December 14, 2020 by quanticbolt

Finding the probability of $Y>1/4$ when conditioned on $X=x$.

2  Asked on December 14, 2020 by rexwilliamson

$arctan(z)$ and Riemann Surfaces

0  Asked on December 14, 2020 by elene

How does $A$ relate to $B$ if $A – lfloor A/B rfloor – lceil A/B rceil leq lfloor A/B rfloor times (B+1)$?

2  Asked on December 14, 2020 by nicholas

multiplicity of functions

0  Asked on December 14, 2020 by mathh

If $x_n = (prod_{k=0}^n binom{n}{k})^frac{2}{n(n+1)}$ then $lim_{n to infty} x_n = e$

2  Asked on December 13, 2020 by perlik

Find the angle that lets you make a bendy pipe

1  Asked on December 13, 2020 by john-porter

How beneficial are Gröbner bases for solving systems of equations

1  Asked on December 13, 2020 by qwaster

Evaluating $prod^{100}_{k=1}left[1+2cos frac{2pi cdot 3^k}{3^{100}+1}right]$

2  Asked on December 13, 2020 by jacky

How many nonnegative integers $x_1, x_2, x_3, x_4$ satisfy $2x_1 + x_2 + x_3 + x_4 = n$?

3  Asked on December 13, 2020 by ivar-the-boneless

Set of open sets notation

1  Asked on December 12, 2020 by jpmarulandas

Need help to understand a theorem about direct sums and regular morphism of R-modules

1  Asked on December 12, 2020 by luiz-guilherme-de-carvalho-lop

Use cylindrical coordinates to find the volume of the solid using triple integrals

2  Asked on December 12, 2020 by eric-brown

1  Asked on December 12, 2020 by jasdeep-singh

Endomorphism of $mathcal{M}_n(mathbb{R})$ such that $f({}^t M)={}^t f(M)$

2  Asked on December 12, 2020 by tuvasbien

Series Expansion by differentiation

1  Asked on December 12, 2020 by dodgevipert56

How can I prove $left|frac{e^{it_p x_j}-1}{t_p}right| leq 2|x|$?

1  Asked on December 12, 2020 by filippo-giovagnini

Are Transition Maps Implied within an Atlas?

2  Asked on December 12, 2020 by tug-witt

For primes, $p_1 + p_2 +p_3+p_4 = p_1 p_2 p_3 p_4 – 15$

4  Asked on December 12, 2020 by vlnkaowo