To prove the "##m^{\text{th}}## Powers Theorem"

  • #1
brotherbobby
702
163
Homework Statement
My textbook has listed the following theorem, calling it the "##m^{\text{th}}## Powers Theorem".

If ##a_1, a_2, \dots, a_n## be a set of positive numbers not all equal, then
1. $$\boxed{\frac{\left( \sum_{i=1}^{n} a_i^m \right)}{n} < \left(\frac{\sum_{i=1}^{n}a_i}{n}\right)^m}$$, when ##0<m<1##
2. $$\boxed{\frac{\left( \sum_{i=1}^{n} a_i^m \right)}{n} > \left(\frac{\sum_{i=1}^{n}a_i}{n}\right)^m}$$ when ##m\in \mathbb{R}-(0,1)##
Relevant Equations
I am not sure what the relevant equations to prove the above identities are
1695532956927.png
Statement : Let me copy and paste the statement as it appears in the text on the right.

Attempt : I could attempt nothing to prove the identity. The best I could do was to verify it for a given value of the ##a's, m, n##. I am not even sure what this identity is called but I will take the author's word for it - "The mth Powers Theorem".

Verify :

(1) Let some ##m=0.5 (<1)##, ##n=3## and ##a_i's = \{2,3,4\}##. Then the L.H.S. = ##\frac{\sqrt{2}+\sqrt{3}+\sqrt{4}}{3}## = 1.72. The R.H.S. = ##\left( \frac {2+3+4}{3} \right)^{0.5}## = 1.73. Hence we see that L.H.S < R.H.S.

(2) Let some ##m=2 (>1)##, ##n=3## and ##a_i's = \{2,3,4\}##. Then the L.H.S. =## \frac{2^2+3^2+4^2}{3}## = 9.67. The R.H.S. = ##\left( \frac {2+3+4}{3} \right)^2## = 9. Hence we see that L.H.S > R.H.S.

So the theorem is probably true but we can't be sure.

1695534146367.png
Moreover, let's see on the last line for the stipulation for ##m## which I copy and paste to the right.
This implies that ##m<0##, say ##m= - 0.5##. I haven't verified this case but let's assume that the theorem holds for it.

Request : A hint or help to help prove these two identities would be welcome.
 
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  • #2
What about induction on ##n##?
 
  • #3
Thank you. Let me try.
 
  • #4
The idea is that ##x^m## is convex and concave for the corresponding values of ##m##. If you draw the graph you can see why it holds. These type of inequalities go by the name Jensen's inequality. The wiki article on it is not bad.
 
  • Informative
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FAQ: To prove the "##m^{\text{th}}## Powers Theorem"

What is the ##m^{\text{th}}## Powers Theorem?

The ##m^{\text{th}}## Powers Theorem is a mathematical statement that typically involves expressing a function or a sequence in terms of its powers, often related to polynomials or algebraic expressions. It states that under certain conditions, the ##m^{\text{th}}## power of a number or a function can be expressed in a specific, often simpler, form.

Why is the ##m^{\text{th}}## Powers Theorem important?

The ##m^{\text{th}}## Powers Theorem is important because it provides a way to simplify complex expressions involving powers. This can be particularly useful in fields such as algebra, calculus, and number theory, where simplifying expressions can make solving equations and understanding functions easier.

What are the key steps to prove the ##m^{\text{th}}## Powers Theorem?

The key steps to prove the ##m^{\text{th}}## Powers Theorem generally include:1. Defining the function or sequence in question.2. Using algebraic manipulations or calculus techniques to express the ##m^{\text{th}}## power in a simpler form.3. Verifying the conditions under which the theorem holds true.4. Providing a logical and rigorous argument to show that the simplified form is equivalent to the original expression.

Can you provide an example of the ##m^{\text{th}}## Powers Theorem in action?

Sure! One common example is the Binomial Theorem, which states that for any integers \(n\) and \(x\):\[(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\]This is a specific case of the ##m^{\text{th}}## Powers Theorem where the power is distributed over a binomial expression, resulting in a sum of terms involving binomial coefficients.

What are some applications of the ##m^{\text{th}}## Powers Theorem?

The ##m^{\text{th}}## Powers Theorem has applications in various areas of mathematics and science, including:1. Solving polynomial equations.2. Analyzing sequences and series.3. Simplifying expressions in calculus.4. Cryptography, where powers of numbers play a crucial role in encryption algorithms.5. Physics, where power laws describe phenomena such as gravitational forces and electrical circuits.

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