Prove Schwarz Inequality for x, y, z in R+

In summary, the task is to prove that for any positive real numbers x, y, and z, the expression √(x(3x+y)) + √(y(3y+z)) + √(z(3z+x)) is less than or equal to 2(x+y+z). The Schwarz inequality can be used to solve this problem by assigning components of general vectors v and w and expanding the smaller side of the equation. With this approach, the solution can be found by placing the square roots in a suitable spot in the equation.
  • #1
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Homework Statement


For x,y,z ## \in \mathbb {R^+} ##, prove that
## \sqrt {x (3 x +y) } + \sqrt {y (3y +z) } + \sqrt {z(3z +x)} \leq ~ 2(x +y+ z) ##

Homework Equations


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The Attempt at a Solution


I don't know which inequality among the above two has to be applied.
I am trying to solve it by inspection. I don't know the standard approach to solve it.
 

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  • #2
If in doubt, write the Schwarz inequality with components of general vectors v,w, expand the smaller side, then see if you can assign the components to get these square roots at a suitable spot in the equation.
 
  • #3
mfb said:
If in doubt, write the Schwarz inequality with components of general vectors v,w, expand the smaller side, then see if you can assign the components to get these square roots at a suitable spot in the equation.

I did it. Thanks a lot for guiding me.
 

1. What is the Schwarz Inequality for x, y, z in R+?

The Schwarz Inequality, also known as the Cauchy-Schwarz Inequality, is a fundamental mathematical inequality that states that for any three real numbers (x, y, z) that are all positive, the following inequality holds:
(x*y + y*z + z*x)^2 ≤ (x^2 + y^2 + z^2)(y^2 + z^2 + x^2)

2. How is the Schwarz Inequality used in mathematics?

The Schwarz Inequality is used in a variety of mathematical fields, including linear algebra, analysis, and probability theory. It is often used to prove other theorems or to establish bounds for certain mathematical objects.

3. What is the significance of the Schwarz Inequality in real analysis?

In real analysis, the Schwarz Inequality is a key tool for proving the convergence of sequences and series. It is also used to establish the existence and uniqueness of solutions to differential equations.

4. Can the Schwarz Inequality be extended to more than three variables?

Yes, the Schwarz Inequality can be extended to any number of variables. In fact, there is a general form of the inequality for any finite number of variables that are all positive. It is often referred to as the generalized Schwarz Inequality.

5. What are some applications of the Schwarz Inequality in physics and engineering?

The Schwarz Inequality is used in various applications in physics and engineering, such as in the study of electrical circuits, quantum mechanics, and signal processing. It is also used in optimization problems to find the minimum or maximum value of a function subject to certain constraints.

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