Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course, an explicit inequality may be even more useful by itself than its application to a limit theorem, which latter is in fact a statement about the mere existence of a certain kind of inequality (recall the presence of the quantifier $\exists$ in standard definitions of the limit). 


In turn, a good inequality is oftentimes based on at least one good identity -- as something gets rewritten in some other, easier to analyze form. Identities used to prove inequalities are oftentimes simple and routine, such as $\int\_a^c=\int\_a^b+\int\_b^c$. Among well-known examples of nontrivial and useful identities in analysis and probability are Cauchy's integral theorem, Plancherel's theorem, isometric imbeddings into $L^p$, duality identities of the $\max\_x\min\_y f(x,y)=\min\_y\max\_x f(x,y)$ type (under appropriate conditions on $f$), [Spitzer's identity](http://www.ams.org/journals/tran/1956-082-02/S0002-9947-1956-0079851-X/) for random walks, and identities of [Stein's method](https://en.wikipedia.org/wiki/Stein's_method). 


There are of course a great many other known identities (some of them nontrivial), such as thousands of formulas for integrals as e.g. in the Gradshteyn and Ryzhik collection. However, it appears that not many of those formulas can lead to interesting inequalities. 


I think it would be useful for a number of people to learn about other, **not widely known identities that can be/have been used to obtain interesting inequalities**. A desirable answer to this question would state such an identity and indicate its possible/actual applications to inequalities, with appropriate references. Thank you! 



Identities and inequalities in analysis and probability
I am not sure if this one fits to this category. In this case it is a PDE (or PDI -partial differential inequality) ruling the inequality. And (I think) you can extract some identity after ``integrating'' this PDE (or PDI). 


*Let $\Omega \subset \mathbb{R}^{2}$ be a rectangular subset, and let $H(x,y) : \Omega \to \mathbb{R}$ be a bounded smooth function such that $H\_{x}, H\_{y} \neq 0$ on $\Omega$. Fix some $\lambda \in (0,1)$. Then inequality 
$$
\int\_{\mathbb{R}^{k}}\sup\_{\lambda x+(1-\lambda)y=z}H(f(x),g(y)) d\gamma\_{k}(z)\geq H\left(\int\_{\mathbb{R}^{k}}f\; d\gamma\_{k},\int\_{\mathbb{R}^{k}}g\; d\gamma\_{k} \right)
$$ 
holds for all $(f(x),g(y)):\mathbb{R}^{k}\times \mathbb{R}^{k} \to \Omega$ where $d\gamma\_{k}=e^{-x^{2}/2}\frac{1}{(2\pi)^{k/2}}dx$ is $k$ dimensional gaussian measure **if and only if** 
$$
(1-\lambda^{2}-(1-\lambda)^{2})\frac{H\_{xy}}{H\_{x}H\_{y}}+\lambda^{2}\frac{H\_{xx}}{H\_{x}^{2}}+(1-\lambda)^{2}\frac{H\_{yy}}{H\_{y}^{2}}\geq 0 \quad (1).
$$*
[Here](https://arxiv.org/abs/1605.04840/ "Boundary value Problem and the Ehrhard inequality") is the reference to the formal statement


*Applications:* 


You can try to play with some functions (or you can actually try to describe all solutions of PDE (1) -- when you have equality instead of inequality). For example, if you try $H(x,y)=\Psi(\lambda \Psi^{-1}(x)+(1-\lambda)\Psi^{-1}(y))$ then (1) takes the form 
$$
\frac{1}{\Psi'(\lambda P+(1-\lambda)Q)}\left(\frac{\Psi''(\lambda P+(1-\lambda)Q)}{\Psi'(\lambda P+(1-\lambda)Q)}-\lambda \frac{\Psi''(P)}{\Psi'(P)}-(1-\lambda)\frac{\Psi''(Q)}{\Psi'(Q)}\right),
$$
where $P=\Psi^{-1}(x), Q=\Psi^{-1}(y)$, which is nonnegative if and only if $\Psi'>0$ and $\frac{\Psi''}{\Psi'}$ is concave (or $\Psi'<0$ and $\frac{\Psi''}{\Psi'}$ is convex).


1) (Prekopa--Leindler inequality) Indeed, take $\Psi(x)=e^{x}$ in the previous example. Then you will get $H(x,y)=x^{\lambda}y^{1-\lambda}$.


2) (Ehrhard inequality). Take $\Psi(t)=\int\_{-\infty}^{t}\frac{e^{-x^{2}/2}dx}{\sqrt{2\pi}}$. This recovers Ehrhard inequality.


