inequality ( 4 var

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SunnyEvan

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SunnyEvan

#1 h Apr 4, 2025, 5:19 AM

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Let $a,b,c,d \in R$ , such that $a+b+c+d=4 .$ Prove that :
$a^4+b^4+c^4+d^4+3 \geq \frac{7}{4}(a^3+b^3+c^3+d^3)$ $a^4+b^4+c^4+d^4+ \frac{76}{25} \geq \frac{44}{25}(a^3+b^3+c^3+d^3)$

This post has been edited 6 times. Last edited by SunnyEvan, Apr 8, 2025, 10:45 AM

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SunnyEvan

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SunnyEvan

#5 h Apr 4, 2025, 9:55 AM

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no one ?

This post has been edited 1 time. Last edited by SunnyEvan, Apr 4, 2025, 9:55 AM

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ektorasmiliotis

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ektorasmiliotis

#6 h Apr 4, 2025, 9:59 AM

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i havent try it because i am not home rn,but i think lagrange multipliers work

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arqady

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arqady

#9 h Apr 5, 2025, 2:47 PM

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ektorasmiliotis wrote:

i havent try it because i am not home rn,but i think lagrange multipliers work

For the first LM does not help, I think, but I am wrong, of course. BW helps. Also, the Vasc's RCF Theorem helps.

This post has been edited 2 times. Last edited by arqady, Apr 7, 2025, 6:36 PM

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SunnyEvan

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SunnyEvan

#10 h Apr 6, 2025, 11:00 AM

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What are "LM" and "BW"?
@arqady.

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sqing

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sqing

#11 h Apr 6, 2025, 12:53 PM • 1 Y

Y by SunnyEvan

SunnyEvan wrote:

What are "LM" and "BW"?
@arqady.

LM
BW

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SunnyEvan

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SunnyEvan

#12 h Apr 7, 2025, 9:42 AM • 1 Y

Y by teomihai

Thank you very much @ Sqing :-D

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teomihai

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teomihai

#14 h Apr 7, 2025, 10:05 AM

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it is good second?

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arqady

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arqady

#15 h Apr 7, 2025, 6:40 PM • 1 Y

Y by teomihai

SunnyEvan wrote:

Let $a,b,c,d \in R$ , such that $a+b+c+d=4 .$ Prove that :

$a^4+b^4+c^4+d^4+ \frac{252}{25} \geq \frac{88}{25}(a^3+b^3+c^3+d^3)$ equality cases : ?

It's wrong. Try $a=b=c=1.1$ .

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SunnyEvan

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SunnyEvan

#16 h Apr 8, 2025, 9:57 AM

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Let $a,b,c,d \in R$ , such that $a+b+c+d=4 .$ Prove that :
$a^4+b^4+c^4+d^4+ 2\sqrt{273} \geq \frac{\sqrt{273}-13}{2}(a^3+b^3+c^3+d^3)+30$

This post has been edited 1 time. Last edited by SunnyEvan, Apr 8, 2025, 10:44 AM

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GeoMorocco

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GeoMorocco

#18 h Apr 8, 2025, 12:04 PM • 1 Y

Y by SunnyEvan

SunnyEvan wrote:

Let $a,b,c,d \in R$ , such that $a+b+c+d=4 .$ Prove that :
$a^4+b^4+c^4+d^4+3 \geq \frac{7}{4}(a^3+b^3+c^3+d^3)$

Let $f(x) = 4x^4+3-7x^3$ . $f$ has two inflexion points and is right convex for $x \geq 1$ . Therefore it is enough to check the inquality for $a \leq 1 \leq b=c=d$ . In this case, we get:
$\frac{1}{27}(a-1)^2(27a^2+(a-1)^2) \geq 0$ Equality for $a=b=c=d=1$ .

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SunnyEvan

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SunnyEvan

#20 h Yesterday at 2:07 PM

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GeoMorocco wrote:

SunnyEvan wrote:

Let $a,b,c,d \in R$ , such that $a+b+c+d=4 .$ Prove that :
$a^4+b^4+c^4+d^4+3 \geq \frac{7}{4}(a^3+b^3+c^3+d^3)$

Let $f(x) = 4x^4+3-7x^3$ . $f$ has two inflexion points and is right convex for $x \geq 1$ . Therefore it is enough to check the inquality for $a \leq 1 \leq b=c=d$ . In this case, we get:
$\frac{1}{27}(a-1)^2(27a^2+(a-1)^2) \geq 0$ Equality for $a=b=c=d=1$ .

Thanks for your help.

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