In resticted domain , Applying the Cauchy-Schwarz's inequality I see this problem a few days ago. a, b, c in [alpha , beta] prove that 9 <= (a+b+c)(1/a +1/b +1/c)<= ((2 alpha + beta)(alpha + 2 beta ))/( alpha beta)

Dylan Nixon 2022-10-16 Answered
In resticted domain , Applying the Cauchy-Schwarz's inequality
I see this problem a few days ago.
a , b , c [ α , β ]
prove that
9 ( a + b + c ) ( 1 a + 1 b + 1 c ) ( 2 α + β ) ( α + 2 β ) α β
Left inequality is easy by Cauchy-Schwart's inequality. But Right inequality is difficult to me. How can I approach the Right inequlity?
EDIT : Sorry, I forgot the condition : α , β > 0
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)

Dana Simmons
Answered 2022-10-17 Author has 14 answers
I hope you mean that α > 0
Let f ( a , b , c ) = ( a + b + c ) ( 1 a + 1 b + 1 c )
Thus, f is a convex function of a, of b and of c (for example, f a 2 = 2 ( b + c ) a 3 > 0), which says that
max { a , b , c } [ α , β ] f = max { a , b , c } { α , β } f = f ( α , α , β ) = ( 2 α + β ) ( α + 2 β ) α β
because f ( α , α , α ) = f ( β , β , β ) = 9 and by AM-GM
( 2 α + β ) ( α + 2 β ) α β = 2 ( α β + β α ) + 5 4 + 5 = 9.
Did you like this example?
Subscribe for all access

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 2021-11-21
What are the ways to calculate the difference between fractions where the denominators are different?
For example,
492754
asked 2022-10-11
How do I represent the following fraction in decimal?
I would like to know how do I represent the following fraction represented in base- 8 in decimal form:
( 35 25 ) 8
It seems very hard. Please somebody help me.
asked 2022-08-05
Write the ratio: 0.2 to 2 as a fraction in lowest terms.
asked 2022-07-23
Partial Fractions help!?
A + C = 0
4 A + B 8 C + D = 1
3 A + 16 C 8 D = 29
12 A + 3 B + 16 D = 5
How do I equate the coefficients? Please provide steps an an explanation.
asked 2022-06-23
Fraction Transforms
Here's a number theory problem I'm having some difficulty with:
Say we transform a fraction by the following rule: we start with some fraction m n with m > n and then convert it to f r n , where f = m / n and r = m mod n . We repeat the process until our fraction is less than 1 or an integer. Given an n is there an m so that we end up with a fraction less than 1?
For n = 2 it isn't that hard, 3, 7, 15, 31, etc. work, but what about other values of n?
asked 2022-08-07
Simplifying 2 + 6 2 + 3 ?
Is there anything else you can do to reduce it to something "nicer" other than multiplying it by 3 2 3 2 and get 6 2 + 18 12 ? The reason I think there's a nicer form is because the previous problem in the book was to simplify 3 + 2 2 3 2 2 , which simplifies nicely to ( 2 + 1 ) 2 ( 2 1 ) 2 = 2.
asked 2022-07-17
About fractions whose sum is a natural number
Some days ago I found an old problem of an olympiad that I always found interesting. It asks to replace each with the numbers 1 , 2 30 without repeating any number, such that their sum is an integer number.
+ + + + + + + + + + + + + +
I got a solution by trial-and-error and it's:
14 1 + 23 2 + 11 22 + 19 3 + 10 15 + 29 4 + 9 12 + 17 6 + 5 30 + 25 8 + 21 24 + 16 7 + 20 28 + 27 18 + 13 26 = 53.
I was wondering if there is another method different than trial-and-error. Thanks in advance for your answers.