Every Inequality you will need! (Might contain mistakes)

by SomeonecoolLovesMaths, Feb 21, 2025, 1:49 PM

$\text{Useful Inequalities}$
$\text{\textbf{Cauchy–Schwarz}}$ $\left(\sum_{i=1}^n x_i y_i\right)^2 \le \left(\sum_{i=1}^n x_i^2\right) \left(\sum_{i=1}^n y_i^2\right).$
$\text{\textbf{Minkowski for }} p\ge 1$ $\left(\sum_{i=1}^n |x_i+y_i|^p\right)^{\frac{1}{p}} \le \left(\sum_{i=1}^n |x_i|^p\right)^{\frac{1}{p}} + \left(\sum_{i=1}^n |y_i|^p\right)^{\frac{1}{p}}.$
$\text{\textbf{Hölder for }} p,q>1,\; \frac{1}{p}+\frac{1}{q}=1$ $\sum_{i=1}^n |x_i y_i| \le \left(\sum_{i=1}^n |x_i|^p\right)^{\frac{1}{p}} \left(\sum_{i=1}^n |y_i|^q\right)^{\frac{1}{q}}.$
$\text{\textbf{Bernoulli}}$ $(1+x)^r \ge 1+rx,\quad x\ge -1,\; r\in\mathbb{R}\setminus(0,1).$ $\text{(Reverse for } r\in[0,1]\text{)}$
$(1+x)^r \le \frac{1}{1-rx},\quad x\in\left[-1,\frac{1}{r}\right),\; r\ge 0.$
$(1+x)^r \le 1 + \left(\frac{x}{x+1}\right)^r,\quad x\ge 0.$
$(1+x)^r \le 1 + (2r-1)x,\quad x\in[0,1],\; r\in\mathbb{R}\setminus(0,1).$
$(1+nx)^{n+1} \ge \Bigl(1+(n+1)x\Bigr)^n,\quad x\ge 0,\; n\in\mathbb{N}.$
$(a+b)^n \le a^n + nb\,(a+b)^{n-1},\quad a,b\ge 0,\; n\in\mathbb{N}.$
$\text{\textbf{Exponential Approximation}}$ $e^x \ge \left(1+\frac{x}{n}\right)^n \ge 1+x.$
$\left(1+\frac{x}{n}\right)^n \ge e^x\left(1-\frac{x^2}{n}\right),\quad n\ge 1,\; |x|\le n.$
$\frac{x^n}{n!}+1 \le e^x \le \left(1+\frac{x}{n}\right)^{n+\frac{x}{2}},\quad x,n>0.$ $e^x \ge \left(e^{\frac{x}{n}}\right)^n,\quad x,n>0.$ $\text{\textbf{Algebraic and Exponential Inequalities}}$ $xy + yx > 1, \quad xy > \frac{x}{x+y}, \quad e^x > \left(1+\frac{x}{y}\right)^y > e^{\frac{xy}{x+y}}, \quad x^y \ge e^{\frac{x-y}{x}}, \quad x,y>0.$ $\frac{1}{2-x} < x^x < x^2 - x + 1, \quad e^{2x} \le 1+x^{1-x}, \quad x\in (0,1).$ $x^{\frac{1}{r}}(x-1) \le r\,x\Bigl(x^{\frac{1}{r}}-1\Bigr), \quad x,r\ge 1, \quad 2^{-x} \le 1 - x^2, \quad x\in[0,1].$ $x\,e^x \ge x+ x^2 + \frac{x^3}{2}, \quad e^x \le x+ e^{x^2}, \quad e^x+e^{-x} \le 2e^{\frac{x^2}{2}}, \quad x\in\mathbb{R}.$ $e^{-x} \le 1 - x^2, \quad x\in [0,1.59], \quad e^x \le 1+x+x^2, \quad x<1.79.$ $\left(1+\frac{x}{p}\right)^p \ge \left(1+\frac{x}{q}\right)^q, \quad \text{for } \begin{cases} x>0,\; p>q>0, \\ -p < -q < x < 0, \\ -q > -p > x > 0. \end{cases}$ $\text{Reversed for } q<0<p, -q>x>0, \quad \text{and } q<0<p, -p<x<0.$
$\text{\textbf{Logarithmic and Harmonic Inequalities}}$ $\frac{x}{1+x} \le \ln(1+x) \le \frac{x(6+x)}{6+4x} \le x, \quad x>-1.$ $\frac{2}{2+x} \le \frac{1}{\sqrt{1+x+\frac{x^2}{12}}} \le \frac{\ln(1+x)}{x} \le \frac{1}{\sqrt{x+1}} \le \frac{2+x}{2+2x}, \quad x>-1.$ $\ln(n)+\frac{1}{n+1} < \ln(n+1) < \ln(n) + \frac{1}{n} \le \sum_{i=1}^n \frac{1}{i} \le \ln(n)+1, \quad n\ge 1.$ $|\ln x| \le \frac{1}{2}\left|\frac{x-1}{x}\right|, \quad \ln(x+y) \le \ln x + \frac{y}{x}, \quad \ln x \le y\Bigl(x^{\frac{1}{y}}-1\Bigr), \quad x,y>0.$ $\ln(1+x) \ge x-\frac{x^2}{2}, \quad x\ge 0, \quad \ln(1+x) \ge x-x^2, \quad x\ge -0.68.$
$\text{\textbf{Trigonometric and Hyperbolic Inequalities}}$ $x-\frac{x^3}{2} \le x\cos x \le x\cos\left(x^{1-\frac{x^2}{3}}\right) \le x^3\sqrt{\cos x} \le x-\frac{x^3}{6} \le x\cos x\sqrt{3} \le \sin x.$ $x\cos x \le x^3\sinh^2 x \le x\cos^2\left(\frac{x}{2}\right) \le \sin x \le \frac{x\cos x+ 2x}{3} \le x^2\sinh x.$ $\max\left\{\frac{2}{\pi},\frac{\pi^2-x^2}{\pi^2+x^2}\right\} \le \frac{\sin x}{x} \le \cos^2 x \le 1 \le 1+\frac{x^2}{3} \le \frac{\tan x}{x}, \quad x\in\left[0,\frac{\pi}{2}\right].$ $\text{\textbf{Power Mean, AM-GM, and Related Inequalities}}$ $\left( \frac{x_1^p + x_2^p + \dots + x_n^p}{n} \right)^{\frac{1}{p}} \ge \left( \frac{x_1^q + x_2^q + \dots + x_n^q}{n} \right)^{\frac{1}{q}}, \quad \text{for } p > q.$ $\frac{x_1 + x_2 + \dots + x_n}{n} \ge \sqrt[n]{x_1 x_2 \dots x_n} \ge \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \dots + \frac{1}{x_n}}.$ $\frac{x+y}{2} \ge \sqrt{xy} \ge \frac{2xy}{x+y}, \quad x, y > 0.$ $\frac{x+y+z}{3} \ge \sqrt[3]{xyz} \ge \frac{3xyz}{xy+yz+zx}.$ $\frac{x^r+y^r}{2} \ge \left( \frac{x+y}{2} \right)^r, \quad x, y > 0, \; r \ge 1.$
$\text{\textbf{Jensen’s Inequality (Convex Functions)}}$ $\varphi \left( \sum_{i=1}^{n} p_i x_i \right) \leq \sum_{i=1}^{n} p_i \varphi(x_i), \quad \text{for convex } \varphi \text{ and } p_i \geq 0, \sum p_i = 1.$
$\text{\textbf{Chebyshev’s Inequality (Monotonic Functions)}}$ $\sum_{i=1}^{n} f(x_i) g(x_i) p_i \geq \left( \sum_{i=1}^{n} f(x_i) p_i \right) \left( \sum_{i=1}^{n} g(x_i) p_i \right),$ $\text{for } x_1 \leq x_2 \leq \dots \leq x_n, \text{ with } f, g \text{ increasing}.$
$\text{\textbf{Rearrangement Inequality}}$ $\sum_{i=1}^{n} a_i b_{\sigma(i)} \geq \sum_{i=1}^{n} a_i b_{\tau(i)},$ $\text{for } a_1 \leq a_2 \leq \dots \leq a_n \text{ and } b_1 \leq b_2 \leq \dots \leq b_n,$ $\text{where } \sigma \text{ is the identity permutation and } \tau \text{ any other permutation}.$
$\text{\textbf{Majorization Inequality}}$ $\sum_{i=1}^{n} \varphi(a_i) \geq \sum_{i=1}^{n} \varphi(b_i),$ $\text{if } a \text{ majorizes } b, \text{ and } \varphi \text{ is convex}.$
$\text{\textbf{Generalized Nesbitt’s Inequality}}$ $\sum_{i=1}^{n} \frac{x_i}{x_{i+1} + x_{i+2} + \dots + x_{i+k}} \geq \frac{n}{k}, \quad x_i > 0.$
$\text{\textbf{Weighted AM-GM Inequality}}$ $w_1 x_1 + w_2 x_2 + \dots + w_n x_n \geq x_1^{w_1} x_2^{w_2} \dots x_n^{w_n},$ $\text{where } w_i \geq 0, \sum w_i = 1, \text{ and } x_i > 0.$
$\text{\textbf{Weighted Power Mean Inequality}}$ $\left( \sum w_i x_i^p \right)^{\frac{1}{p}} \geq \left( \sum w_i x_i^q \right)^{\frac{1}{q}}, \quad \text{for } p > q.$
$\text{\textbf{Karamata’s Inequality (Convex Functions and Majorization)}}$ $\sum_{i=1}^{n} \varphi(a_i) \geq \sum_{i=1}^{n} \varphi(b_i),$ $\text{if } a \text{ majorizes } b, \text{ and } \varphi \text{ is convex}.$ $\text{\textbf{Inequalities in Probability and Statistics}}$ $\Pr(|X| \geq a) \leq \frac{E[|X|]}{a}, \quad a > 0 \quad \text{(Markov's Inequality)}$ $\Pr(|X - E[X]| \geq t) \leq \frac{\operatorname{Var}(X)}{t^2}, \quad t > 0 \quad \text{(Chebyshev's Inequality)}$ $\Pr(|X - \mathbb{E}[X]| \geq \delta) \leq 2\exp\left(-\frac{2\delta^2}{\sum_{i=1}^{n} (b_i - a_i)^2}\right),$ $\text{for independent } X_i \text{ with } X_i \in [a_i, b_i] \quad \text{(Hoeffding's Inequality)}.$
$\text{\textbf{Bernstein’s Inequality}}$ $\Pr\left(S_n - E[S_n] \geq t\right) \leq \exp\left(\frac{-t^2}{2(\sigma^2 + Rt/3)}\right),$ $\text{for i.i.d. } X_i \text{ with } |X_i - E[X_i]| \leq R \text{ and variance } \sigma^2.$
$\text{\textbf{Chernoff Bound (Multiplicative Form)}}$ $\Pr(S_n \geq (1+\delta)E[S_n]) \leq \exp\left(-\frac{\delta^2}{2+\delta} E[S_n]\right),$ $\Pr(S_n \leq (1-\delta)E[S_n]) \leq \exp\left(-\frac{\delta^2}{2} E[S_n]\right),$ $\text{for independent } X_i \text{ taking values in } [0,1].$
$\text{\textbf{Azuma’s Inequality}}$ $\Pr(|S_n - E[S_n]| \geq t) \leq 2\exp\left(-\frac{t^2}{2 \sum_{i=1}^{n} c_i^2}\right),$ $\text{for a martingale } S_n \text{ with bounded differences } |S_i - S_{i-1}| \leq c_i.$
$\text{\textbf{Gibb’s Inequality (Relative Entropy)}}$ $\sum_{i=1}^{n} p_i \ln \frac{p_i}{q_i} \geq 0, \quad \text{for probability distributions } \{p_i\}, \{q_i\}.$
$\text{\textbf{Jensen’s Inequality (Convex Functions in Expectation)}}$ $\varphi(\mathbb{E}[X]) \leq \mathbb{E}[\varphi(X)],$ $\text{for convex } \varphi.$
$\text{\textbf{Fano’s Inequality (Information Theory)}}$ $H(Y|X) \leq h_b(P_e) + P_e \log(|\mathcal{Y}| - 1),$ $\text{where } P_e \text{ is the probability of classification error}.$
$\text{\textbf{Stirling’s Approximation}}$ $\sqrt{2\pi n} \left(\frac{n}{e}\right)^n \leq n! \leq e \sqrt{2\pi n} \left(\frac{n}{e}\right)^n.$ $\text{Asymptotic form: } n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n.$
$\text{\textbf{Entropy Inequalities}}$ $H(X) \leq \log|\mathcal{X}|,$ $H(X) \geq \sum_{i=1}^{n} p_i \log \frac{1}{p_i} = -\sum p_i \log p_i.$
$\text{\textbf{Log-Sum Inequality}}$ $\sum_{i=1}^{n} p_i \log \frac{p_i}{q_i} \geq \left(\sum_{i=1}^{n} p_i\right) \log \frac{\sum_{i=1}^{n} p_i}{\sum_{i=1}^{n} q_i}.$
$\text{\textbf{Generalized Mean Inequalities (Hölder’s Inequality for Sums)}}$ $\left( \sum_{i=1}^{n} p_i |x_i|^p \right)^{\frac{1}{p}} \geq \left( \sum_{i=1}^{n} p_i |x_i|^q \right)^{\frac{1}{q}}, \quad \text{for } p > q.$
$\text{\textbf{Subadditivity of Entropy}}$ $H(X, Y) \leq H(X) + H(Y).$
$\text{\textbf{Pinsker’s Inequality (Total Variation Distance and KL Divergence)}}$ $D_{\text{KL}}(P||Q) \geq \frac{1}{2} ||P - Q||_{\text{TV}}^2.$