My Favorite (In)Equalities

Here, I will slowly be populating a list of some of my favorite (in)equalities in mathematics. These turned out to be quite useful at times.

Algebraic

  • For all \(a, b \in \mathbb{R}, m \in \mathbb{N}_+\), it holds that \(a^n – b^n = (a-b) ( a^{n-1} + a^{n-2} b + \cdots + b^{n-1} )\).
  • For \(a \geq 0, b > 0\), \(a \sqrt{b+a^2} \leq b + a^2\).

Analysis

  • For \(n \in \mathbb{N}_+\), \(\sum_{t=1}^n 1/t \leq 1 + \ln{n}\)

Probability Theory

  • Markov’s Inequality: If \(X\) is a nonnegative random variable and \(a>0\), then \(\mathbb{P}( X \geq a ) \leq \mathbb{E}(X) / a \).

Differential Equations

  • Gronwell’s Inequality: Let \(I\) denote an interval of the real line of the form \(\lbrack a,\infty )\), \(\lbrack a,b \rbrack, \) or \(\lbrack 0,b)\) with \(a<b\). Let \(\alpha, \beta, u\) be realvalued functions on \(I\). Assume that \(\beta, u\) are continuous, and that the negative part of $\alpha$ is integrable on every closed and bounded subinterval of \(I\).
    If \(\beta\) is nonnegative and if \(u\) satisfies $$u(t) \leq \alpha(t) + \int_a^t \beta(s) u(s) ds$$ for all \(t \in I\), then $$u(t) \leq \alpha(t) + \int_a^t \alpha(s) \beta(s) \exp{ \bigl( \int_s^t \beta(r) dr \bigr) } ds$$ for all \(t \in I\).
    If in addition \(\alpha\) is nondecreasing, then $$u(t) \leq \alpha(t) \exp{ \bigl( \int_a^t \beta(r) ds \bigr) }$$ for all \(t \in I\).
  • Bihari-LaSalle’s Inequality: Let \(u,f\) be nonnegative continuous functions defined on \(\lbrack 0,\infty \rbrack\), and let \(w\) be a continuous nondecreasing function defined on \(\lbrack 0,\infty )\) and \(w(u) > 0\) on \((0,\infty)\). If \(u\) satisfies the following integral inequality, $$u(t) \leq \alpha + \int_0^t f(s) w(u(s)) ds$$ for \(t \in \lbrack 0,\infty )\), with \(\alpha\) a nonnegative constant, then $$u(t) \leq G^{\gets}\Bigl( G(\alpha) + \int_0^t f(s) ds \Bigr)$$ for \(t \in \lbrack 0,T \rbrack\). Here, \(G\) is defined by $$G(x) = \int_{x_0}^x \frac{dy}{w(y)}$$ for \(x \geq 0, x_0 >0\), and \(G^\gets\) is the inverse function of \(G\); and \(T\) is chosen so that $$G(\alpha) + \int_0^t f(s) ds \in Dom(G^\gets)$$ for all \(t \in \lbrack 0,T \rbrack\).