Let \((X,\mathcal M,\mu)\) be a measure space, and let \(p,q\in[1,\infty]\) be conjugate exponents, \(\frac1p+\frac1q=1\), unless stated otherwise.
Between integrals
Hölder. \(\norm{fg}_1 \leq \norm f_p\norm g_q\) for measurable \(f,g\). For \(1<p<\infty\) and \(\norm f_p,\norm g_q<\infty\), equality holds if and only if \(\alpha\abs f^p = \beta\abs g^q\) almost everywhere for some constants \(\alpha,\beta\geq0\), not both zero.
Minkowski. \(\norm{f+g}_p \leq \norm f_p + \norm g_p\) for \(1\leq p\leq\infty\). For \(0<p<1\) the inequality fails: on \([0,2]\), \(\norm{\chi_{[0,1]}+\chi_{[1,2]}}_p = 2^{1/p} > 2 = \norm{\chi_{[0,1]}}_p+\norm{\chi_{[1,2]}}_p\).
Minkowski’s integral inequality. For \(\sigma\)-finite measure spaces, measurable \(f\) on \(X\times Y\), and \(1\leq p<\infty\), \(\norm{\int_Y f(\cdot, y)\dy}_p \leq \int_Y \norm{f(\cdot,y)}_p \dy\).
Jensen. If \(\mu(X)=1\), \(f\in L^1(\mu)\) is real-valued, and \(\varphi\colon\RR\to\RR\) is convex, then \(\varphi\qty(\int f\,d\mu) \leq \int\varphi\circ f\,d\mu\). The inequality fails when \(\mu(X)\neq1\): for \(\mu\) twice Lebesgue measure on \([0,1]\), \(f\equiv1\), and \(\varphi(t)=t^2\), the left side is \(4\) and the right side is \(2\).
Young. For \(f\in L^p(\RR^n)\), \(g\in L^q(\RR^n)\) with \(1\leq p,q,r\leq\infty\) and \(\frac1r = \frac1p+\frac1q-1\), \(\norm{f*g}_r \leq \norm f_p\norm g_q\).
Between a norm and a measure
Chebyshev. For \(0<p<\infty\) and \(t>0\), \(\mu(\theset{\abs f > t}) \leq t^{-p}\norm f_p^p\).
Borel–Cantelli. If \(\sum_n \mu(E_n) <\infty\), then almost every point lies in only finitely many \(E_n\). Combined with Chebyshev’s inequality: if \(\sum_n\norm{f_n}_p^p<\infty\), then \(\mu(\theset{\abs{f_n}>t})\) is summable for each \(t>0\), so \(f_n\to0\) almost everywhere.
Elementary inequalities
Young’s inequality for products. For \(a,b\geq0\) and \(1<p<\infty\), \(ab \leq \frac{a^p}p + \frac{b^q}q\); Hölder’s inequality follows by applying it to \(\abs f/\norm f_p\) and \(\abs g/\norm g_q\) and integrating.
Convexity of \(t\mapsto t^p\). For \(p\geq1\), \(\abs{a+b}^p \leq 2^{p-1}\qty(\abs a^p + \abs b^p)\).
Reverse triangle inequality. \(\abs{\norm a - \norm b} \leq \norm{a - b}\) in any normed space.
Cauchy–Schwarz. \(\abs{\inner fg}\leq\norm f\norm g\) in an inner product space; in \(L^2\) it is Hölder’s inequality with \(p=q=2\).
Bernoulli. \((1+x)^n \geq 1 + nx\) for \(x\geq-1\) and \(n\in\NN\).
Exponential bound. \(1 + x\leq e^x\) for \(x\in\RR\), so \(\prod_{i}(1-a_i)\leq \exp\qty{-\sum_i a_i}\) for \(a_i\in[0,1]\).
The full statements are on Appendix: common inequalities.