# 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 $$a \geq 0, b > 0$$, $$a \sqrt{b+a^2} \leq b + a^2$$.
• $$\bigl( \sum_i a_i b_i \bigr)^2 = \sum_i \sum_j a_i b_i a_j b_j$$

## 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 $$t \in \lbrack 0,T \rbrack$$.