Common Mathematical Inequalities

Mathematical inequalities are key tools in understanding relationships between numbers and functions. They play a crucial role in various fields, from geometry to statistics, helping to establish bounds and optimize solutions in Lower Division Math Foundations.

1. Triangle Inequality

   • States that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
   • Formally expressed as: ( |a + b| \leq |a| + |b| ).
   • Fundamental in geometry and analysis, establishing a basis for distance metrics.

2. Cauchy-Schwarz Inequality

   • States that for any sequences of real numbers, the square of the sum of their products is less than or equal to the product of the sums of their squares.
   • Formally expressed as: ( (a_1b_1 + a_2b_2 + ... + a_nb_n)^2 \leq (a_1^2 + a_2^2 + ... + a_n^2)(b_1^2 + b_2^2 + ... + b_n^2) ).
   • Essential in linear algebra and probability theory, often used to prove other inequalities.

3. Arithmetic Mean-Geometric Mean (AM-GM) Inequality

   • States that for any non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean.
   • Formally expressed as: ( \frac{x_1 + x_2 + ... + x_n}{n} \geq (x_1 x_2 ... x_n)^{1/n} ).
   • Useful in optimization problems and provides a foundation for various mathematical proofs.

4. Bernoulli's Inequality

   • States that for any real number ( x \geq -1 ) and integer ( n \geq 0 ), ( (1 + x)^n \geq 1 + nx ).
   • Highlights the growth of exponential functions and is often used in combinatorial proofs.
   • Important in the context of limits and approximations.

5. Young's Inequality

   • States that for any non-negative real numbers ( a ) and ( b ), and for any ( p, q > 1 ) such that ( \frac{1}{p} + \frac{1}{q} = 1 ), the inequality ( ab \leq \frac{a^p}{p} + \frac{b^q}{q} ) holds.
   • Useful in analysis, particularly in the context of integrals and function spaces.
   • Provides a method for bounding products of functions.

6. Jensen's Inequality

   • States that for a convex function ( f ), the inequality ( f\left(\frac{x_1 + x_2 + ... + x_n}{n}\right) \leq \frac{f(x_1) + f(x_2) + ... + f(x_n)}{n} ) holds.
   • Important in statistics and economics, particularly in the context of expected values.
   • Highlights the relationship between convexity and averages.

7. Hölder's Inequality

   • States that for sequences of non-negative real numbers and conjugate exponents ( p ) and ( q ), the inequality ( \sum |a_i b_i| \leq \left( \sum |a_i|^p \right)^{1/p} \left( \sum |b_i|^q \right)^{1/q} ) holds.
   • Fundamental in functional analysis and probability theory.
   • Provides a powerful tool for estimating integrals and sums.

8. Minkowski's Inequality

   • States that for any p-norms, the inequality ( ||x + y||_p \leq ||x||_p + ||y||_p ) holds.
   • Generalizes the triangle inequality to ( L^p ) spaces.
   • Important in analysis and geometry, particularly in the study of vector spaces.

9. Chebyshev's Inequality

   • States that for any random variable, the probability that it deviates from its mean by more than ( k ) standard deviations is at most ( \frac{1}{k^2} ).
   • Provides a way to estimate the spread of a distribution.
   • Useful in statistics for bounding probabilities.

10. Markov's Inequality

    • States that for any non-negative random variable ( X ) and ( a > 0 ), the probability that ( X ) is at least ( a ) is at most ( \frac{E[X]}{a} ).
    • Provides a simple way to bound probabilities without requiring knowledge of the distribution.
    • Fundamental in probability theory and statistics, often used in risk assessment.