Inequality with Differentiation

Click For Summary
SUMMARY

The discussion focuses on proving the inequality \(xy \leq \frac{x^p}{p} + \frac{y^q}{q}\) for \(p > 1\) and \(q = \frac{p}{p-1}\). The participants emphasize the role of differentiation in establishing this inequality, particularly through the function \(f(x,y) = \frac{x^p}{p} + \frac{y^q}{q} - xy\). The equality condition occurs when \(x\) and \(y\) are proportional, specifically when \(x^p = y^q\).

PREREQUISITES
  • Understanding of the Hölder's inequality
  • Familiarity with differentiation techniques
  • Knowledge of the concepts of convex functions
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study Hölder's inequality in detail
  • Learn about the properties of convex functions
  • Explore differentiation techniques in multivariable calculus
  • Investigate the conditions for equality in inequalities
USEFUL FOR

Students and educators in mathematics, particularly those studying analysis, optimization, and inequality proofs.

steelphantom
Messages
158
Reaction score
0

Homework Statement


Let p > 1, and put q = p/(p-1), so 1/p + 1/q = 1. Show that for any x > 0, y > 0, we have

xy <= xp/p + yq/q, and find the case where equality holds.

Homework Equations



The Attempt at a Solution


This is in the differentiation chapter of my analysis book (Browder), so I'm going to go out on a limb here and assume that some aspect of differentiation comes into play here. Unfortunately, I don't really know how to start. Could someone get me started here? Thanks!
 
Physics news on Phys.org
Maybe try to find the minimum possible value of f(x,y)=x^p/p + y^q/q -xy
 

Similar threads

Verify Green's Formula for a Simple DE
  • · Replies 1 ·
Replies
1
Views
1K
Computing conditional extinction probabilities for pollen cells
  • · Replies 1 ·
Replies
1
Views
1K
Given unit impulse response, obtain differential equation
  • · Replies 7 ·
Replies
7
Views
2K
Finding the rate of change of x in the equation
  • · Replies 8 ·
Replies
8
Views
2K
On weakly singular equations and Frobenius' method
  • · Replies 7 ·
Replies
7
Views
2K
Non-Differentiable Function proof
  • · Replies 8 ·
Replies
8
Views
1K
Is Every Differential 1-Form on a Line the Differential of Some Function?
  • · Replies 3 ·
Replies
3
Views
2K
f(x) = 2x+1, proving that it is continuous when p = 1 with 𝛿 and ε
  • · Replies 7 ·
Replies
7
Views
2K
Prove Sturm-Liouville differential operator is self adjoint.
  • · Replies 4 ·
Replies
4
Views
2K
True or false questions about Divergence and Curl
  • · Replies 1 ·
Replies
1
Views
1K