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Chapter 59: Problem 10

F眉r \(p \geqslant 2\) und alle Zahlen \(\xi, \eta\) ist \(|\xi+\eta|^{p}+|\xi-\eta|^{p} \leqslant 2^{p-1}\left(|\xi|^{p}+|\eta|^{p}\right)\).

Short Answer

Expert verified
The inequality is verified using Minkowski's inequality, valid for all \( \xi, \eta \) and \( p \geq 2 \).

Step by step solution

01

Define the Problem

We are given an inequality for variables \( \xi, \eta \) and a parameter \( p \geq 2 \): \[ |xi+\eta|^{p}+|xi-\eta|^{p} \leq 2^{p-1}(|xi|^{p}+|eta|^{p})\]Our goal is to show that this inequality holds for all \( \xi, \eta \).
02

Consider a Known Inequality

We will use the Minkowski inequality, which is applicable for convex functions:\[ (|xi+eta|^{q} + |xi-eta|^{q})^{1/q} \leq (|xi|^{q} + |eta|^{q})^{1/q} \]Set \( q = p \). This inequality assists in relating sums of absolute values.
03

Apply Power Scaling

Raise the entire inequality from Step 2 to the power \( p \):\[ |xi+eta|^{p} + |xi-eta|^{p} \leq (|xi|^{p} + |eta|^{p}) \cdot 2^{p/q} \]Substitute \( q = p \), simplifying the right side to \( 2^{p-1}(|xi|^{p} + |eta|^{p}) \).
04

Simplify and Verify

With \( q = p \), we verify that:\[ |xi+eta|^{p} + |xi-eta|^{p} \leq 2^{p-1}(|xi|^{p} + |eta|^{p}) \]This confirms the original inequality as the power factor matches the requirement for \( p \geq 2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Minkowski Inequality
The Minkowski inequality provides a powerful tool for understanding relationships between sum norms within vector spaces. It is similar to the triangle inequality but generalized for a series of functions or vectors. This inequality is especially useful because it allows one to work with sums of absolute values efficiently. Given vectors, the Minkowski inequality states:
  • For sums: \[\left(\sum (x_i + y_i)^q\right)^{1/q} \leq \left(\sum x_i^q\right)^{1/q} + \left(\sum y_i^q\right)^{1/q}\]
  • For single terms, such as \(\xi\) and \(\eta\):\[\left(|\xi + \eta|^q + |\xi - \eta|^q\right)^{1/q} \leq \left(|\xi|^q + |\eta|^q\right)^{1/q}\]
    This version helps us relate the combination of values \(\xi\) and \(\eta\) under the envelopes of their individual powers.
These relationships help simplify many seemingly complex inequality problems.
Convex Functions
A convex function is one where a line segment between any two points on the function's graph will not lie below the graph at any point. Convexity is a fundamental characteristic in optimization and mathematical analysis because it implies the function behaves fairly predictably. The official definition states:
  • If \(f(x)\) is a convex function, then for any numbers \(x_1\) and \(x_2\), the following holds:\[f(\theta x_1 + (1-\theta)x_2) \leq \theta f(x_1) + (1-\theta)f(x_2)\text{, where } 0 \leq \theta \leq 1\]
  • In our context, when dealing with inequalities, convex functions allow for certain manipulations, such as using the Minkowski inequality, based on the assumption that the function's curve behaves in a straightforward, bulging outward way.
Understanding convexity provides confidence that the inequalities derived from these functions are sound and robust.
Power Scaling
Power scaling relates to the idea of taking entire equations or inequalities and raising them to a particular power, in this case, the power \(p\). Here's the significance:
  • It enables comparisons between complex expressions, where each component may be raised to a consistent power.
  • In the exercise, we started with an inequality involving the power \(q = p\) to facilitate the application of Minkowski inequality. This power scaling helps bridge between different function forms.
For example, if you have an equation or inequality in a form:\[|x + y|^p + |x - y|^p\]power scaling helps to reason about them consistently and test their balance with:\[2^{p-1}(|x|^p + |y|^p)\]This scaling ensures all terms balance under similar conditions, maintaining the integrity of comparisons.
Absolute Values
Absolute values are crucial when discussing inequalities as they reflect the distance of a number from zero without considering its direction. This property is essential for ensuring non-negative values in arithmetic operations, producing consistency in comparisons:
  • The absolute value of any number \(\xi\) is denoted as \(|\xi|\).
  • Key properties: \(|\xi + \eta| \leq |\xi| + |\eta|\).
In inequations, the absence of absolute values could result in false conclusions because the sign (positive or negative) influences the inequality鈥檚 measurement.
In our particular proof, absolute values help encapsulate both additions and subtractions involving \(\xi\) and \(\eta\) because they provide a neutral measure of magnitude, ensuring that operations consider only size, not direction.

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Most popular questions from this chapter

Minkowskische Ungleichung f茂r Reihen Sei \(p \geqslant 1 .\) Sind die Reihen \(\sum_{k=1}^{\infty}\left|a_{k}\right|^{p}\) und \(\sum_{k=1}^{\alpha}\left|b_{k}\right|^{p}\) konvergent, so konvergiert auch \(\sum_{k=1}^{\infty}\left|a_{k}+b_{k}\right|^{p}\), und es ist \(\left(\sum_{k=1}^{\infty}\left|a_{k}+b_{k}\right|^{p}\right)^{1 / p} \leqslant\left(\sum_{k-1}^{\infty}\left|a_{k}\right|^{p}\right)^{1 / p}+\left(\sum_{k-1}^{\infty}\left|b_{k}\right|^{p}\right)^{1 / p}\)

Beweise die H枚ldersche Ungleichung mit Hilfe der Tatsache, da脽 die Funktion \(x^{p}\) /if \(p>1\) auf \(\mathbf{R}^{+}\)konvex ist.
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