Questions tagged [jensen-inequality]

For questions about proving and using Jensen's inequality for convex functions. To be used necessarily with the [inequality] tag.

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Relationship between the expectation of the squared L2 norm of sum of elements and the sum of expectations of squared L2 norm of elements.

I am quite confused about the relationship between the following terms, $E||\sum_{i=1}^n f(x_i) x_i||^2$ and $\sum_{i=1}^n E||f(x_i) x_i||^2$ where $x_i \in \mathbb{R}^p$ and $f(\cdot): \mathbb{R}^p \...
AJ193's user avatar
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Ratio of two Jensen's Inequality

I have these pair of numbers $ (a, b) = (\frac{4}{9}, \frac{1}{9}) $ and $(c, d) = (\frac{1}{2}, \frac{1}{6}) $. (Number mean nothing, just for illustration and simplification) Note that - (a, b) are ...
Elina Gilbert's user avatar
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The inequality of convex function including power function

I want to show the following. $$ \phi((\lambda g_1 + (1-\lambda)g_2)^k) - (\lambda g_1 + (1-\lambda) g_2)^k \nabla \phi(f^k) \le \lambda \phi(g_1^k) + (1-\lambda)\phi(g_2^k) - (\lambda g_1^k + (1-\...
user275310's user avatar
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1 answer
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Does the Jensen's inequality hold under expectation? [closed]

In other words, does $$ E\big[E[f(X)]\big] \ge E\big[f(E[X])\big] $$ holds for a convex function $f(x)$? If so why? or why not?
Alireza Azimi's user avatar
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1 answer
41 views

Is this Jensen's Inequality? (Symmetrization Lemma)

This question concerns most proofs I've seen on the so called Symmetrization Lemma. Let $\mathcal{F}$ be a class of measurable functions and $X_1,\ldots,X_n$ be independent and identically distributed ...
Paulo Mourão's user avatar
2 votes
1 answer
52 views

Jensen's inequality integral version for strongly convex functions

For a convex function $\phi$, not necessarily differentiable. If $\mu$ is a probability measure, and $f$ and $\phi(f)$ are integrable. We have: $$ \phi \left( \int f \, d\mu \right) \leq \int \phi(f) \...
wsz_fantasy's user avatar
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Jensen inequality with fraction expectation, determinant, and log function

I have a question when I try to have some analysis on the following expression. \begin{align} \mathbb{E}\left[\log_2\det\left(\mathbf{I}_L+\frac{1}{N_0+\sigma^2}\mathbf{X}\right)\right], \end{align} ...
Charlie Nie's user avatar
3 votes
1 answer
76 views

Conditional Jensen's inequality proof correctness. Queries regarding convex functions.

Let $(Ω, \mathcal{F}, P)$ be a probability space and let $\mathcal{G} ⊂ \mathcal{F}$ be a sub-$σ$-algebra. Conditional Jensen's inequality: Let $φ : R → R$ be a convex function, $X$ and $φ(X)$ be ...
Kaustav Choubey's user avatar
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Jensen's inequality with expectation of log and determinant

I have a question when I try to have some analysis on the following expression. \begin{align} \mathbb{E}\left[\log_2\det\left(\mathbf{I}_L+\mathbf{X}\right)\right], \end{align} where $\mathbb{E}$ ...
Charlie Nie's user avatar
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Single inflection point theorem

Let $f$ be a twice differentiable function on $\mathbb{R}$ with a single inflection point, let $S$ be a fixed real number and let $$g(x) = f(x) + (n-1)f(\frac{S-x}{n-1}).$$ If $x_1, x_2,...,x_n$ are ...
MNJG_1729's user avatar
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Maximizing the sum of side lengths of a circumscribed tetrahedron

The original problem posed was the following: On a sphere of radius $1$ are given four points $A,B,C,D$ such that $$AB\cdot AC\cdot AD \cdot BC \cdot BD \cdot CD = \frac{2^9}{3^3}$$ I am aware of a ...
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Let $f(x)=x^2+3x-3,x\gt0.$ If $n$ points $x_1,x_2,...,x_n$ are chosen on the x-axis, evaluate $\frac{x_1+x_2+...+x_n}n$

Question: Let $f(x)=x^2+3x-3,x\gt0.$ If $n$ points $x_1,x_2,...,x_n$ are so chosen on the x-axis such that (i) $\frac1n\sum_{i=1}^nf^{-1}(x_i)=f(\frac1n\sum_{i=1}^nx_i)$ (ii) $\sum_{i=1}^nf^{-1}(x_i)=\...
aarbee's user avatar
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Strict Jensen inequality for Ergodic Markov processes

Let $\vartheta(t)$ be a real Ergodic stationary Markov process solution to the stochastic differential equation $$ \mathrm{d}\vartheta(t)=A_1(\vartheta(t),t)\mathrm{d}t+A_2(\vartheta(t),t)\mathrm{d}W(...
Daan's user avatar
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11 votes
2 answers
346 views

Bounding "Jensen's Gap": Elementary Approaches

The point of this post is to explore some "elementary" but general ways one can quantify the "gap" in Jensen's inequality. Specifically, let $h:\mathbb R\rightarrow\mathbb R$ be a ...
Small Deviation's user avatar
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46 views

Jensen's inequality for Euclidean spaces with extended reals

Let $X$ be a convex subset of $\mathbb{R}^n$, and let $p$ be a probability density function on $X$ (i.e. $\int_X p(x) dx = 1$), let $\phi:X\to \mathbb{R}\cup\{+\infty\}$ be a convex function with ...
helloworld's user avatar
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$\sum\limits_{\mathrm{cyc}} \frac{a}{2a^2+a+1}\leq\frac{3}{4}$, with $a,b,c\in\mathbb{R}^+,\ a+b+c=3$ [duplicate]

I am trying to prove the inequality: $\sum\limits_{\mathrm{cyc}} \frac{a}{2a^2+a+1}\leq\frac{3}{4}$, with $a,b,c\in\mathbb{R}^+,\ a+b+c=3$. Now if we consider the function $f:(0,+\infty)\to\mathbb{R},\...
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Expectation of transformed mean of multiple random variables <= mean of the expectation of their transformation

I am trying to understand the following. If $M$ is the number of different random variables $X_1 ... X_M$, are there any conditions under which we can claim that: $$E\left[f(\sum_{m=1}^M w_m X_m)\...
ZXiu's user avatar
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Show that $\mathbb{E}\left(|X|^q\right) \leq \left[\mathbb{E}\left(|X|^p\right)\right]^{\frac{q}{p}}$

Let $X$ be a random variable and let $p \in (0, \infty)$ such that $\mathbb{E}\left(|X|^p\right) < \infty$. Show that for all $q \in (0, p)$, we have $\mathbb{E}\left(|X|^q\right) \leq \left[\...
Willem's user avatar
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Show inequality about riemann integrals with jensen inequality

I am having trouble proving the following inequality: Let $f,h: [0,1] \to \mathbb{R}$ continuous, $h >0$ and $\phi: \mathbb{R} \to \mathbb{R}$ continuous and convex. Show that: $$ \phi(\frac{\int^1 ...
Raoul Luqué's user avatar
-2 votes
1 answer
87 views

Popoviciu's inequality

If $f(x): \mathbb R \to \mathbb R$ is a convex function, prove that $$f(x) + f(y) + f(z) + 3 f(\frac{x + y + z}{3}) \geq 2 f(\frac{x + y}{2}) + 2 f(\frac{x + z}{2}) + 2 f(\frac{y + z}{2})$$ I proved ...
math.enthusiast9's user avatar
2 votes
1 answer
58 views

About Jensen’s inequality and Fano’s inequality [closed]

Let $X, X^′$ be independent with $X ∼ p(x)$, and $X^′ ∼ q(x)$, $x,x^′ \in \mathcal{X}$ . Then: 1.$Pr[X = X^′] ≥ 2^{−H(p)−D(p||q)}$, 2.$Pr[X = X^′] ≥ 2^{−H(q)−D(q||p)}$. I don't kown how to prove 1 and ...
Daisy u's user avatar
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Lower bounding the mean of a quadratic form with a positive semi-definite matrix

Let $\mathbf{X}\in\mathbb{R}^n$ be a vector random variable, and let $\mathbf{P}$ be an $n\times n$ positive semi-definite matrix. I'm interested in deriving lower bounds on the expected value of the ...
Matt's user avatar
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Brezis' Exercise 4.9

I'm trying to solve below exercise in Brezis' Functional Analysis, i.e., Let $(\Omega, \mathcal F, \mu)$ be a measure space with $\mu(\Omega) < \infty$. Let $J:\mathbb R \to (-\infty, +\infty]$ be ...
Akira's user avatar
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2 votes
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Convex function with a parameter H

Let us take $s_1, s_2, t_1, t_2 >0$ such that $ s_1 < t_1 < s_2 < t_2 $. Let us also assume that $H \in (0 , 1)$ and define $$ a_1 = t_2 - s_1, \quad a_2 = t_2 - t_1, \quad b_1 = s_2 - s_1,...
user60576's user avatar
3 votes
2 answers
89 views

Relation between $g(\mathbb{E}[X])$ and $\mathbb{E}[g(X)]$ when $g(X)=\frac{X}{1-X}$ is the "odds" function

Some context: Let's say we draw from an urn containing red and blue balls. We start at $n=1$, draw 1 ball, look at the color, and put it back, and repeat the process at $n=2$, $n=3$, etc., and every ...
sg1234's user avatar
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Pigeonhole coloring bound

I am stuck on this problem from a combinatorics book. I tried applying the pigeonhole principle to show that there are at least n/r numbers with the same color, and applied the formula for (x choose 2)...
theorystudent's user avatar
2 votes
0 answers
172 views

Show $2F\left(\frac{x+y}{2}\right)\geq F(x)+F(y)$

It's a follow up of Prove that $f(x,y)+f(y,z)\ge f(x,z)$ where $f(x,y)=\sqrt{x\ln x+y\ln y-(x+y)\ln(\frac{x+y}2)}$ : I don't give the details but I show using second derivative : if we have for $a,b,c,...
Erik Satie's user avatar
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2 votes
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Help me with this probability theory inequality---I have been stuck for weeks

I'm a bit stuck on a proof that seems to hold when I test in in graphing calculators, would love to hear some tips on strategies I might use. Suppose we have two finite sets $\Omega$ and $\Phi$, with ...
Spencer Dylan Pantoja's user avatar
4 votes
0 answers
108 views

A probabilistic proof of Oppenheim's inequality?

Oppenheim's inequality is a standard result about the Hadamard product of positive definite matrices. It goes as follows, let $A=(a_{ij})_{i,j\leq n},B=(b_{ij})_{i,j\leq n} \in S_n^{++}$ where $S_n^{++...
PAM's user avatar
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2 votes
2 answers
111 views

Prove that $\prod_{i=1}^n\left(\frac{1}{x_i^2}-1\right)\ge (n^2-1)^n$

Let $n$ be a positive integer and $x_1,...,x_n$ positive reals such that $x_1+\dots+x_n=1$. Prove that $$\prod_{i=1}^n\left(\frac{1}{x_i^2}-1\right)\ge (n^2-1)^n.$$ Now notice that if we apply Jensen'...
PNT's user avatar
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Show basic inequality for complex random variables

Suppose I have $n$ (not independent) complex random variables $X_i,i=1,\ldots,n$. I want to show the following \begin{equation} \mathbb{E}\left[\left|\sum_{j=1}^nX_j\right|^2\right]^{1/2}\leq \sum_{j=...
wonderer's user avatar
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1 vote
1 answer
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Reversal of inequality sign in Jensen's inequality

I am given a set of coefficients such that the affine combination $\{x_1, x_2, ..., x_n \} \notin conv(x_1, x_2, ..., x_n)$. How do I prove that under such given conditions the Jensen's inequality ...
lonewolf.py's user avatar
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1 answer
78 views

Inequality $\left(2-a-b\right)f\left(\frac{\left(2-a-b\right)^{2}-2dc}{2-a-b}\right)+af\left(a\right)+bf\left(b\right)\leq 16/25$

It's an inequality which refines the following question: Prove or disprove that the inequality is valid if $x,y,z,u$ are positive numbers and $x+y+z+u=2$. : Let $a,b,d\in[0,1]$ and $c\in[0.5,1]$ such ...
Erik Satie's user avatar
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Does the concavity of the log function justify the following inequality?

in an exercise solution I was presented with the following inequality and it's not making sense to me. $ \frac{n}{n+1}\log{(\frac{1}{n}(x_1 + x_2 +...+ x_n))} +\frac{1}{n}\log{(x_{n+1})} $ $ \leq\log{(...
Fregheit Meier's user avatar
-2 votes
3 answers
205 views

JBMO-$2014$ Inequality question [duplicate]

Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that $$ {\left(a+\frac{1}{b}\right)^2}+{\left(b+\frac{1}{c}\right)^2} +{\left(c+\frac{1}{a}\right)^2}≥3(a+b+c+1)$$ My solution: By Jensen'...
Rehman's user avatar
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1 vote
3 answers
86 views

Prove that $ (a+b)^{2n} \leq 2^{2n-1}(a^{2n}+b^{2n})$ [duplicate]

I have to prove that for all $(a;b) \in \mathbb{R}^2$, and for all $n \in \mathbb{N}$ we have: $$ (a+b)^{2n} \leq 2^{2n-1}(a^{2n}+b^{2n})$$ without using induction. I tried to use the convexity of $x^...
Sine's user avatar
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2 votes
0 answers
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Upper Bounding A Sum of Powers of Degrees in a Graph

Let $G$ be a simple, directed graph with $n$ vertices and $m$ edges. Let $V$ denote the vertex set of $G$. Given a vertex $v$ in $G$, let $\deg_{\text{out}}(v)$ denote its outdegree. Finally, let $\...
Naysh's user avatar
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1 vote
1 answer
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A 'continuous version' of Jensen's inequality

Let $X$ denote a random variable that is smoothly distributed on $[0, 1]$ with PDF $f$. Consider $$g(c) = \mathbb{P}(X< c) \phi(\mathbb{E}[X|X < c]) + \int_c^1 \phi(x)f(x) dx$$ where $\phi$ is ...
afreelunch's user avatar
4 votes
2 answers
195 views

Prove that $ \frac{x^2}{1+x} + \frac{y^2}{1+y} + \frac{z^2}{1+z} \geq \frac{1}{2} $

Assume that positive numbers a, b, c, x, y, z satisfy $cy+bz =a; az + cx = b$$bx + ay = c$. Prove that $ \frac{x^2}{1+x} + \frac{y^2}{1+y} + \frac{z^2}{1+z} \geq \frac{1}{2} $ I've tried appling ...
Leibniz-Z's user avatar
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0 votes
4 answers
120 views

Prove that $\sqrt{a+ c} + \sqrt{b + c} \le f(x) \le 2\sqrt{\frac{a+ b}{2}+c}$, $f(x)=\sqrt{a\sin^2(x) +b\cos^2(x)+c} +\sqrt{a\cos^2(x)+b\sin^2(x)+c}$ [closed]

$f(x) = \sqrt{a\sin^2(x) + b\cos^2(x) + c} + \sqrt{a\cos^2(x) + b\sin^2(x) + c}$ a, b, c are positive real numbers I turned $f(x)$ into $\sqrt{a + c + (b - a)\cos^2(x)} + \sqrt{b + c + (a - b)\cos^2(x)...
Spring's user avatar
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6 votes
1 answer
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What's the relation between the expectation of p-norm squared of a random vector before and after centering?

Let $a$ be a random vector with $\mathbb{E} a = b$ and $\mathbb{E} \|a\|_p^2 =\sigma^2$, where $1\leq p \leq \infty$. Is it true that $\mathbb{E} \|a-b\|_p^2 \leq 2\sigma^2$? It is clear to me how to ...
gladiego's user avatar
0 votes
1 answer
68 views

Application of Jensen Inequality for lower bound

In this work Jensen inequality is applied to turn Eq (11) to Eq. (12). The essential part of Equation 11: $L = \int q_{X_0}(x_{0}) log (\int q_{X_1,…,X_T | X_0}(x_1,…,x_T|x_{0})*C(x_1,…x_T) d_{x_1},…...
stat_127's user avatar
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0 answers
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Counterexample for Jensen's inequality for rank one convex function

We all know the Jensen's inequality is for all probability measures $\mu$ and all convex $h:\mathbb{R}^N\to\mathbb{R}$ it holds that \begin{equation} h\left(\int Xd\mu(X)\right)\leq \int h(X)d\mu(X). \...
Min Gao's user avatar
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1 answer
78 views

Jensen's inequality but with geometric mean

If the inequality $f({\sqrt x}{\sqrt y})⩽{\sqrt f(x)}{\sqrt f(y)}$ is satisfied for all non negative $x,y$ in the domain, what can we say about the convexity of $f$ ? Or are there any other properties ...
osdinuto's user avatar
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1 answer
54 views

$ \mathbb P[X > 0] \geq \frac{\mathbb E [X]^ 2}{ \mathbb E [X^2]}$ if the mean is positive and finite

Let $Χ$ be a random variable for witch $0 < E[X] < \infty$. Prove that: $ \mathbb P[X > 0] \geq \frac{\mathbb E [X]^ 2}{ \mathbb E [X^2]}$ Hint: Use the fact that: $(\mathbb{1}_{\{X>0\}}\...
Giorgos Mitropoulos's user avatar
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0 answers
40 views

Projection in Convex Optimization

I have fixed problem a and b, in (a) I use a counter method to prove the correctness of it, and with the conclusion if (a), (b) is easy to prove. But here comes problem c, here Δ stand for Bregman ...
Ecthelion's user avatar
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0 answers
36 views

Bounding a double expectation of the entropy function

Consider a sequence of real numbers $0 \leq a_1 \leq \cdots \leq a_n \leq 1$ with $1/n\sum_i [a_i] = \mu$. Suppose further that $H(2 \mu - \mu^2) \geq H(\mu)$ (or equivalenly $\mu \leq \frac{3 + \sqrt{...
ScoobySnacks's user avatar
0 votes
2 answers
58 views

Is $C^2$ necessary for convexity $\implies$ $f''(x) \geq 0$?

Let $f$ be a twice differential function defined on open interval $(a,b) \subseteq R$. Let $[c,d] \subseteq (a,b)$. I am able to prove that if $f''(x) \geq 0$ on $[c, d]$ then $f$ is necessarily ...
Lab's user avatar
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1 vote
1 answer
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Jensen's for continuous convex combinations

Let $d_p(u(x), v(x)) = \left( \int_0^m \left| u(x)-v(x) \right|^p dx \right)^{1/p}$ be the distance according to the $\ell_p$ norm of two functions $u,v:[0,m]\to [0,M]$. Is the following true for the ...
Jay's user avatar
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13 votes
0 answers
328 views

Any reference on Jensen inequality for measurable convex functions on a Banach space?

The only proof of Jensen inequality (and most general version) that I know is a direct consequence of the Fenchel-Moreau Theorem : If $X$ is a locally convex Hausdorff topological space, let $\mu$ be ...
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