description/proof of that for real number larger than \(1\), \(p\), its exponent conjugate, \(q\), and non-negative real numbers, \(r_1\) and \(r_2\), \(r_1 r_2\) is equal to or smaller than \({r_1}^p / p\) plus \({r_2}^q / q\)
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of exponent conjugate of exponent.
Target Context
- The reader will have a description and a proof of the proposition that for any real number larger than \(1\), \(p\), its exponent conjugate, \(q\), and any non-negative real numbers, \(r_1\) and \(r_2\), \(r_1 r_2\) is equal to or smaller than \({r_1}^p / p\) plus \({r_2}^q / q\).
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(p\): \(\in \mathbb{R}\), such that \(1 \lt p\)
\(q\): \(= \text{ the exponent conjugate of } p\), \(\in \mathbb{R}\)
\(r_1\): \(\in \mathbb{R}\), such that \(0 \le r_1\)
\(r_2\): \(\in \mathbb{R}\), such that \(0 \le r_2\)
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Statements:
\(r_1 r_2 \le {r_1}^p / p + {r_2}^q / q\)
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2: Proof
Whole Strategy: Step 1: deal with the \(r_1 = 0\) or \(r_2 = 0\) case; Step 2: otherwise, see that \(0 \le - t^{1 / p} + 1 / p t + 1 - 1 / p\) for \(0 \lt t\) proves the proposition; Step 3: see that \(0 \le - t^{1 / p} + 1 / p t + 1 - 1 / p\) for \(0 \lt t\).
Step 1:
Let us suppose that \(r_1 = 0\) or \(r_2 = 0\).
\(r_1 r_2 = 0\).
\(0 \le {r_1}^p / p + {r_2}^q / q\).
So, \(r_1 r_2 \le {r_1}^p / p + {r_2}^q / q\).
Step 2:
Let us suppose otherwise.
Let us see that \(0 \le - t^{1 / p} + 1 / p t + 1 - 1 / p\) for \(0 \lt t\) proves the proposition.
Let us take \(t = {r_1}^p {r_2}^{- q}\), which satisfies \(0 \lt t\).
\(0 \le - ({r_1}^p {r_2}^{- q})^{1 / p} + 1 / p ({r_1}^p {r_2}^{- q}) + 1 - 1 / p\).
\(({r_1}^p {r_2}^{- q})^{1 / p} \le 1 / p ({r_1}^p {r_2}^{- q}) + 1 / q\), but the left hand side is \(r_1 {r_2}^{- q / p}\).
\(r_1 {r_2}^{- q / p} {r_2}^q \le (1 / p ({r_1}^p {r_2}^{- q}) + 1 / q) {r_2}^{q}\), but the left hand side is \(r_1 {r_2}^{- q / p + q} = r_1 {r_2}^{(1 - 1 / p) q} = r_1 {r_2}^{q / q} = r_1 r_2\), and the right hand side is \(1 / p {r_1}^p + 1 / q {r_2}^q\).
So, \(r_1 r_2 \le {r_1}^p / p + {r_2}^q / q\).
Step 3:
Let us see that \(0 \le - t^{1 / p} + 1 / p t + 1 - 1 / p\) for \(0 \lt t\).
Let \(f: (0, \infty) \to \mathbb{R}, t \mapsto - t^{1 / p} + 1 / p t + 1 - 1 / p\).
\(d f / d t = - 1 / p t^{1 / p - 1} + 1 / p\).
\(d^2 f / d t^2 = - 1 / p (1 / p - 1) t^{1 / p - 2}\).
\(0 \lt d^2 f / d t^2\).
\(d f / d t = 0\) only when \(t = 1\).
So, \(f\) takes the minimum at \(t = 1\).
But \(f (1) = 0\).
So, \(0 \le f\).
