Question about an inequality which seems right but not easy to prove The origin problem is as follo

misurrosne 2022-06-24 Answered
Question about an inequality which seems right but not easy to prove
The origin problem is as follows:
let a , b , c , d are positive real numbers,and a + b + c + d = 4 prove:
a 2 b + b 2 c + c 2 d + d 2 a 4 + 1 4 [ ( a b ) 2 + ( b c ) 2 + ( c d ) 2 + ( d a ) 2 ]
The solution is easy enough, which is to plug 4 = a + b + c + d into to R H S and then move it to L H S, notice:
[ ( a b ) 2 b + ( b c ) 2 c + ( c d ) 2 d + ( d a ) 2 a ] ( a + b + c + d ) [ ( a b ) 2 + ( b c ) 2 + ( c d ) 2 + ( d a ) 2 ]
which would finally lead to the proof of the problem.
However, when I tried to solve the problem, I applied the Lagrange identical equation to the R H S of the inequality, then I left out some quadratic term which finally leads to the inequality:
a 2 b + b 2 c + c 2 d + d 2 a a 2 + b 2 + c 2 + d 2
Which seems quite right, but I'm not really sure if it is. However, numerical tests imply that it is right. But I'm still not sure.
That stuff is kind of like Chebyshev inequality but it can't be directly used here.
From another point of view, my question is whether the following strengthening of the origin inequality is right or not:
let a , b , c , d are positive real numbers,and a + b + c + d = 4 then:
a 2 b + b 2 c + c 2 d + d 2 a 4 + 1 4 [ ( a b ) 2 + ( b c ) 2 + ( c d ) 2 + ( d a ) 2 + ( a c ) 2 + ( b d ) 2 ]
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

hopeloothab9m
Answered 2022-06-25 Author has 25 answers
c y c a 2 b 4 = c y c ( a 2 b 2 a + b ) = c y c ( a b ) 2 b ( c y c | a b | ) 2 4 1 4 c y c ( a b ) 2
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-06-16
Why does x / y = x / y / y
Sorry for the awkward title, hard to to sum a mathematical problem with words alone.
Having said that, I recently learned that the root of any value, x, and then that over value y, is identical to the root of x / y / y , as in x / y / y
For e.g, 35 / 7 = 5 / 7 and I cannot logically deduce why. So far, I know that:
35 = 7 × 5 35 7 = 7 × 5 7
And somehow this being equal to:
√(5/7)
So it seems that we divided the numerator, √35 by √5 to give √7. However, 7/√7 is not √7! Yet, apparently it is?
So, a fraction is of course equal to itself if scaled up or down by the same amount, in that 10/2 is the same as 5/1, or 5. So scaling down √35 by √5 makes sense, as it only leaves √7, but it seems that the same cannot be said for the denominator, 7. 7/√7 = √7!? Yet, the two expressions really are the same, √35/7 and √(5/7).
Thanks for any help in advance.
P.S This works for any values, so it's no coincidence.
asked 2022-08-19
How does this separation of the variable from the fraction result in
3 1 x & not 3 x
I wish to write the following expression as the product of a whole number or a fraction and variable expression. The answer given in my textbook is as follows:
3 x = 3 1 1 x = 3 1 1 x = 3 1 x
However, I am not sure how to get to 3 1 x . When I complete the multiplication I get to 3 1 1 x = 3 x , which of course loops me back round!
asked 2022-06-06
rational numbers as upper limit of a summation?
a quick question: Is it a legit way to use a fraction as the upper limit of a summation? Given is a frequency f and a sample rate f s . I want to use a sum like this:
k = 1 f s 2 f sin ( 2 π k )
Is this a legit way to write it?
asked 2022-05-01
Linear Equation Problem
Evaluating the expression below:
2(6x)3=9(x+5)6+13
The answer is 2313 but to obtain this answer what specific method do you use?
asked 2021-11-19
How many separate fractions can this be broken down to before simplifying?
4a3b6ab+2ab22ab
asked 2022-06-15
Proof of an inequality involving different exponents
I was asked the following interesting question but I can not come up with a proof.
Let a , b , c be positive real numbers. Prove that
( ) a 3 b 3 + c 3 + b 3 a 3 + c 3 + c 3 a 3 + b 3 a 2 b 2 + c 2 + b 2 a 2 + c 2 + c 2 a 2 + b 2
One way to prove ( ) may be to consider the following function
f ( x ) = a x b x + c x + b x a x + c x + c x a x + b x
and prove that f ( x ) is a non-decreasing function on [ 2 , 3 ]. When we take its derivative, the formula looks quite tedious and we just can not continue on. So is there any better solution?
asked 2021-12-17
How do you write 0.09 as a fraction?