Most proofs in optimization consist in using inequalities for a particular function class in some creative way. This is a cheatsheet with inequalities that I use most often. It considers class of functions that are convex, strongly convex and $L$-smooth.

Setting. $f$ is a function $\mathbb{R}^p \to \mathbb{R}$. Below are a set of inequalities verified when $f$ belongs to a particular class of functions and $x, y, z \in \mathbb{R}^p$ are arbitrary elements in its domain. For simplicity I'm assuming that functions are differentiable, but most of these are also true replacing the gradient with a sub-gradient.

$f$ is $L$-smooth. This is the class of functions that are differentiable and its gradient is Lipschitz continuous.

• $\|\nabla f(y) - \nabla f(x) \| \leq L \|x - y\|$
• $|f(y) - f(x) - \langle \nabla f(x), y - x\rangle| \leq \frac{L}{2}\|y - x\|^2$
• $\nabla^2 f(x) \preceq L\qquad \text{ (assuming $f$ is twice differentiable)} $

$f$ is convex:

• $f(\lambda x + (1 - \lambda)y) \leq \lambda f(x) + (1 - \lambda)f(y)$ for all $\lambda \in [0, 1]$.
• $f(x) \leq f(y) + \langle \nabla f(x), x - y\rangle$
• $0 \leq \langle \nabla f(x) - \nabla f(y), x - y\rangle$
• $f(\mathbb{E}X) \leq \mathbb{E}[f(X)]$ where $X$ is a random variable (Jensen's inequality).
• $x = \text{prox}_{\gamma f}(x) + \gamma \text{prox}_{f^*/\gamma}(x/\gamma)$, where $f^*$ is the Fenchel conjugate (Moreau's decomposition, )

$f$ is both $L$-smooth and convex:

• $\frac{1}{L}\|\nabla f(x) - \nabla f(y)\|^2 \leq \langle \nabla f(x) - \nabla f(y), x - y\rangle$
• $0 \leq f(y) - f(x) - \langle \nabla f(x), y - x\rangle \leq \frac{L}{2}\|x - y\|^2$
• $f(x) \leq f(y) + \langle \nabla f(x), x - y\rangle - \frac{1}{2 L}\|\nabla f(x) - \nabla f(y)\|^2$
• $f(x) \leq f(y) + \langle \nabla f(z), x - y \rangle + \frac{L}{2}\|x - z\|^2$ (three points descent lemma)

$f$ is $\mu$-strongly convex. Set of functions $f$ such that $f - \frac{\mu}{2}\|\cdot\|^2$ is convex. It includes the set of convex functions with $\mu=0$. Here $x^*$ denotes the minimizer of $f$.

• $f(x) \leq f(y) + \langle \nabla f(x), x - y \rangle - \frac{\mu}{2}\|x - y\|^2$
• $f(x) \leq f(y) + \langle \nabla f(y), x - y\rangle + \frac{1}{2\mu}\|\nabla f(x) - \nabla f(y)\|^2$
• $\mu\|x - y\|^2 \leq \langle \nabla f(x) - \nabla f(y), x - y\rangle$
• $\frac{\mu}{2}\|x-x^*\|^2\leq f(x) - f(x^*)$
• $f(\alpha x + (1 - \alpha)y) \leq \alpha f(x) + (1 - \alpha)f(y) - \alpha(1 - \alpha)\frac{\mu}{2}\|x - y\|^2$

$f$ is both $L$-smooth and $\mu$-strongly convex.

• $\frac{\mu L}{\mu + L}\|x - y\|^2 + \frac{1}{\mu + L}\|\nabla f(x) - \nabla f(y)\|^2 \leq \langle \nabla f(x) - \nabla f(y), x - y\rangle$
• $\mu \preceq \nabla^2 f(x) \preceq L \qquad \text{ (assuming $f$ is twice differentiable)}$

References

Most of these inequalities appear in the Book: "Introductory lectures on convex optimization: A basic course" by Nesterov (2013, Springer Science & Business Media). Another good (and free) resource is the book "Convex Optimization" by Stephen Boyd and Lieven Vandenberghe.