Inequalities of Analysis - Math

USAC Colloquium

Inequalities of Analysis Andrejs Treibergs

University of Utah

Fall 2014

2. USAC Lecture: Inequalities of Analysis

The URL for these Beamer Slides: "Inequalities of Analysis"

3. References

G. Hardy, J. E. Littlewood & G. P?olya, Inequalities, 2nd ed., Cambridge university Press, Cambridge, 1991. G. P?olya & G. Szeg?o, Problems and Theorems in Analysis I, Springer Verlag, New York, 1972.

4. Outline.

Triangle and Cauchy Schwarz Inequalities Arithmetic - Geometric - Harmonic Mean Inequality

Relations among the AGH means Cauchy's proof Applications: largest triangle of given perimeter and monotonicity of the compound interest sequence Jensen's Inequality Convex functions and a proof for finitely many numbers Probabilistic interpretation H?older's, Cauchy-Schwarz's and AG Inequalities follow from Jensen's Application: largest polygons in circular arc Another proof using support functions Integral form of Jensen's, H?older's and Minkowski's Inequalities Application: least force exerted on magnetic pole at a point inside a loop wire of given length carrying a fixed current

5. Triangle Inequality.

For any two numbers x, y R we have the Triangle Inequality. |x + y | |x| + |y |.

Figure 1: Euclidean Triangle.

The name comes from the fact that the sum of lengths of two sides of a triangle exceeds the length of the third side so the lengths satisfy

C A + B.

If we have sides given as vectors x, y and x + y then the lengths satisfy

|x + y | |x| + |y |.

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