I am looking for a **reference** for the following claim:


Let $\phi:\mathbb (a,b) \to \mathbb R$ be a continuous function, and let $c \in (a,b)$ be fixed.


Suppose that "$\phi$ is convex at $c$". i.e. for any $x\_1,x\_2>0, \alpha \in [0,1]$ satisfying $\alpha x\_1 + (1- \alpha)x\_2 =c$, we have
$$
\phi(c)=\phi\left(\alpha x\_1 + (1- \alpha)x\_2 \right) \leq \alpha \phi(x\_1) + (1-\alpha)\phi(x\_2) .
$$


Then $\phi$ satisfies Jensen ineqaulity "at $c$".


**Finite form:**


Given $\lambda\_i \in [0,1],x\_i\in(0,\infty),i=1,\dots,k$ such that $\sum\_{i=1}^k \lambda\_i=1,\sum\_{i=1}^k \lambda\_ix\_i=c$, we have $$\phi(\sum\_{i=1}^k \lambda\_ix\_i) \le \sum\_{i=1}^k \lambda\_i \phi(x\_i).$$


**A more general probabilistic (measure-theoretic) form:**


Given a random variable $X \in (a,b)$ with expectation $E(X)=c$, we have
$$
\phi(c)=\phi(E(X)) \le E(\phi(X)).
$$


In addition, if $\phi$ is **strictly** convex at $c$, then equality holds if and only if $X$ is constant a.e..


Both of these forms of Jensen inequality follow from the existence of a supporting line to the graph of $\phi$ at $c$.




---


The proof of the latter fact is [not hard](https://math.stackexchange.com/a/3748714/104576), but I couldn't find a source in the literature that presents this "localized" form of Jensen inequality, under the sole assumption of "convexity **at a point**". (In fact, I couldn't even find the term "convex at a point" anywhere...).


I find it impossible to believe that this doesn't show up in existing literature. Any help would be welcomed.


*Comment:*


Convexity at $c$ does not imply that the [one-sided derivatives exist](https://math.stackexchange.com/questions/3746986/does-convexity-at-a-point-imply-existence-of-one-sided-derivatives), so the standard proof for the existence of a supporting line (subgradient) does not apply here. (When the function is convex on an interval, every number between the two-sided derivatives form a subgradient).



Convexity at a point and Jensen inequality
For any real numbers $u,v,c$ such that $u\le c\le v$, let $\mu\_{c;u,v}$ denote the unique probability distribution on the set $\{u,v\}$ with mean $c$.


Your generalization of Jensen's inequality follows immediately from the well-known fact that any probability distribution $\mu$ on $\mathbb R$ with a given mean $c\in\mathbb R$ is a mixture of probability distributions of the form $\mu\_{c;u,v}$. See e.g. [formula (2.13)](https://projecteuclid.org/euclid.ejp/1464819487).


**Details:** Indeed, that formula implies
$$Ef(X)=\int\_{S\_c} Ef(X\_{u,v})\,\nu\_X(du\times dv)$$
for some probability measure $\nu\_X$ (depending on the distribution of $X$) on the set $S\_c:=\{(u,v)\in\mathbb R^2\colon u\le c\le v\}$ and all functions $f\colon\mathbb R\to\mathbb R$ such the function $\mathbb R\ni x\mapsto f(x)-kx$ is bounded from below for some some real $k$.


Now, if $f$ is convex at $c$, then $Ef(X\_{u,v})\ge f(EX\_{u,v})=f(c)$ for all $(u,v)\in S\_c$, and hence
$Ef(X)\ge f(c)$.


**Answer to your additional question concerning the strict convexity at $c$:**
Moreover, if $f$ is strictly convex at $c$, then $Ef(X\_{u,v})> f(EX\_{u,v})=f(c)$ for all $(u,v)$ in the set, say $S\_c^\circ$, of all $(u,v)\in S\_c$ such that $u<c<v$. Hence,
$Ef(X)>f(c)$ unless $\nu(S\_c^\circ)=0$.
On the other hand, the condition $(u,v)\in S\_c\setminus S\_c^\circ$ implies that $Eg(X\_{u,v})=g(c)$ for all functions $g\colon\mathbb R\to\mathbb R$. So, the condition $Ef(X)=f(c)$ implies $\nu\_X(S\_c^\circ)=0$, which in turn implies
that $Eg(X)=\int\_{S\_c\setminus S\_c^\circ} g(c)\,\nu\_X(du\times dv)=g(c)$ for all (say) nonnegative $g\colon\mathbb R\to\mathbb R$, which means that
$P(X=c)=1$.


