Questions tagged [jensen-inequality]

For questions about proving and manipulating the AM-GM inequality. To be used necessarily with the [inequality] tag.

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2
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34 views

Contraction of Bellman Operator under general $L_p$ norms

We know that the Bellman Operator $$ TV(s) = \max_a r(s,a) + \sum_{s' \in S}p(s'|s,a)V(s') $$ is a contraction under $L_\infty$ norm.For reference one can see the following link Proof that Bellman ...
2
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1answer
34 views

Equality in Problem $2.12$, Rudin's RCA - Show that $h = \int_{\Omega} h\, d\mu$ a.e.

Suppose $\mu(\Omega) = 1$, and $h: \Omega\to [0,\infty]$ is measurable. If $A = \int_\Omega h\, d\mu$, prove that $$\sqrt{1+A^2} \le \int_\Omega \sqrt{1+h^2}\, d\mu$$ and that equality holds iff $h = ...
2
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1answer
111 views

Showing the data processing inequality

Let $X \sim P_\theta$ for some distribution $P_\theta$ parametrized by $\theta \in \Theta \subset \mathbb R$ and $Y \sim Q(\cdot | X)$ for some distribution $Q$. Assume that $P_\theta$ has a density $...
2
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1answer
33 views

Is there a “reverse” Jensen's inequality up to a constant?

Specifically I was wondering if for fixed $0<p<1$ there exists a constant $C$ such that $(E|X|)^p\leq C_pE(|X|^p)$ for any random variable $X$, giving an inverse of Jensen's inequality. From ...
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0answers
62 views

Relating Fatou's Lemma and Jensen's Inequality

If there is a sense in which $\liminf$ is a "concave" function, then we would expect Fatou's Lemma as a consequence of Jensen's inequality: Is there any way to make this precise? For a small ...
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1answer
35 views

Proof involving Jensen's inequality on random variables [closed]

Is it possible to use the Jensen's inequality to state the following: $$E\left(\frac{1}{aX+b}\right)> \frac{1}{E(aX+b)}$$ where $X$ is a random variable and $a$ and $b$ are constants? If not, is ...
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2answers
25 views

Is $E(E(Y|Z)|X)^2 \le E(E(Y|Z)^2|X))$ correct? (Jensen'sinequality)

I know that Jensen's inequality states that: $\varphi(E(V)) \le E(\varphi(V))$, where $E$ stand for expected value, $V$ for a random variable and $\varphi$ must be a convex function. Let $X$, $Y$ and $...
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2answers
49 views

Strict (or lack therefore) concavity of square root function

For the square root function $g(x) = \sqrt{x}$, we have \begin{align} g'(x) = 0.5x^{-0.5} \\ g''(x) = -0.25x^{-1.5} \\ \end{align} So the second derivative is negative everywhere except if $x =...
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1answer
43 views

Multivariate Jensen's inequality in a probabilistic setting: When does equality hold?

I am having trouble understanding the concept of Jensen's inequality in a probabilistic setting with multiple variables. Specifically, I am interested in cases where equality holds. I understand that ...
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1answer
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Jensen Inequality and Expected value

I am trying to proof that: Let f $\in$ $C^2(R,R)$,$f''>0$, and $E[f(X)]=f(E[X])$. I am trying to proof that X should be a constant. I know that $f(E[X])\leq E[f(X)]$, this holds because of the ...
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52 views

proof verification: Jensen's inequality

Following is the Jensen's inequality in Rudin's "Real and Complex Analysis".(Thm 3.3) Let $\mu$ be a positive measure on a $\sigma-$algebra $\mathfrak M$ in a set $\Omega$, where $\mu(\...
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0answers
76 views

Showing a convolution is Lipschitz. In measure.

Let $\Lambda:\mathbb{R}^d\to \mathbb{R}^d$, be Lipschitz. Is the convolution of $\Lambda$ with a measure Lipshchitz? That is do we have $$ \int_{\mathbb{R}^d} \| \Lambda*\rho(x) - \Lambda*\mu(x)\|^2 ~ ...
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Jensen inequality cross entropy

I don't remember how to do math Prove that $\sum_{j=1}^{k} r_{j}{\log p_{j}} -\sum_{j=1}^{k} r_{j}{\log r_{j}} <=0$ Can I just put all expressions in one sum symbol, use log probability $\log(p_{j}...
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1answer
52 views

Show that $E_f(\log f)\ge E_f (\log g)$

Let $f(x), g(x)$ be probability densities defined on $R^n$ with $f(x), g(x)>0$ for all $x$. Show that $E_f(\log f(x))\ge E_f (\log g(x))$ with $E_f(h(x))=\int_{-\infty}^\infty h(x) f(x) dx$ is the ...
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1answer
29 views

Jensen inequality and mixed risk portfolio

Let $X$ be a random variable indicating the gains in some investment, let $\mu = E[X]$ be the expected value, and $U(X)$ is the utility obtained from that gain. If $u$ is convex, then $U^{\prime \...
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2answers
38 views

Is the integral over a convex mapping of the solution to the heat equation monotonic?

I have the following problem: Suppose $u = u(t,x)$ solves the Cauchy-Problem for the (one-dimensional) heat equation: $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$$ $$u(0,x) = g(...
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1answer
56 views

Use Jensen's inequality to show that $e^{\frac{\sum^3_{i=1} X_i}{3}}$ is not an unbiased estimator of $e^{\theta}$

My problem: Let $X_{1}, X_{2},X_{3}$ be I.I.D. Poisson variables with common mean $\theta > 0$. Use Jensen's inequality to show that $e^{\frac{\sum^3_{i=1}X_i}{3}}$ is not an unbiased estimator of $...
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1answer
19 views

An inequality in expected values of Random Variable with different PDFs

Prove that $E \ [log \ f(X)] \ge E \ [log \ q(X)]$, if X is a random variable with PDF $f(x)$ and $q(x)$ is any valid PDF. Hint provided: Use Jensen's Inequality. This means: $E\ [log \ f(X)] = \int f(...
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2answers
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Proving Jensen's inequality for the general case starting from the finite case

For a finite convex combination, Jensen’s inequality is given by: $$f\left(\sum_i^n a_i x_i\right) \leq \sum_i^n a_i f(x_i)$$ for a convex $f$. Proving this is not so bad starting from the definition ...
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0answers
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Prove that $c_V(X) \geq c_V(f(X))$ for $X>0$ and $f$ concave, positive, increasing,

I have a square-integrable random variable $X > 0$. Function $f : \mathbb{R}^+ \rightarrow \mathbb{R}^+ $ is concave, monotonically increasing and $f(0) = 0$. I would like to prove that (and make ...
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2answers
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How to remember which direction Jensen's inequality is in?

For a real valued random variable and a convex function $f$, Jensen's inequality is given by: $$f(\mathbb E X) \leq \mathbb E f(X)$$ How do you remember which direction the inequality is in? I always ...
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1answer
60 views

On the requirements of the Jensen inequality for conditional expectation

I'm often reading this version of the Jensen inequality for conditional expectation: Let $(\Omega,\mathcal A, P)$ a probability space and $X$ a integrable random variable. Then for any convex ...
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Proof that function larger than its bounds

Setup. Let $f_k(x) = 1-\exp\left(-\frac{xk+(1-x)(k+1)}r\right)$ and $g_k(x) = xf_k(x)^k + (1-x)f_k(x)^{k+1}$. The domains are $x \in (0, 1)$, $k \in \mathbb{N}$ and $r \in \mathbb{R}$. An intuitive ...
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2answers
118 views

Alternative approaches to prove the following inequality

For $a,b,c \in \mathbb{R^+},$ prove that $$\left(\dfrac{2a}{b+c} \right)^{\frac{2}{3}} + \left(\dfrac{2b}{c+a}\right)^{\frac{2}{3}} + \left(\dfrac{2c}{a+b}\right)^{\frac{2}{3}} \geq 3.$$ I managed to ...
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0answers
56 views

Using Cauchy-Schwarz and Jensen's inequality

I need a certain estimation from a paper in which it has been described to use the Cauchy-Schwarz inequality and Jensen's inequality. I did some calculations and had a few tries but I don't seem to ...
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1answer
40 views

Applying Jensen's inequality to a convex tarsnformation of a non-convex simple random variable

My objective is to (numerically) maximize an expectation of the form of $$ E[\exp(g(\mathbf{X}))] $$ where $\mathbf{X}$ is random variable and $g$ is a non-convex function on $\mathbf{X}$. If I ...
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1answer
97 views

Equality in Jensen's inequality for non convex functions

Jensen's inequality, $\mathbb{E}[\phi(X)]\geq\phi(\mathbb{E}[X])$, holds for any convex function $\phi$ and random variable $X$. Given that $\mathbb{E}[\phi(X)]=\phi(\mathbb{E}[X])$ for some non-...
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1answer
134 views

Double refinement of Nesbitt's inequality

It's a double refinement of Nesbitt's inequality : Claim :Prove or disprove this statement. Let $a,b,c>0$ such that $1\leq c\leq b\leq 2$ then we have : $$\frac{a}{\sqrt[3]{4(b^3+c^3)}}+\frac{b}{a+...
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1answer
154 views

A Polynomial-Inequality problem from Vietnam National Olympiad 2021

Abstract I have just completed 2 days of our National Olympiad. This year's problems are not difficult, yet new and strange to many of the students. Mentioned below will be problem 5 out of 7, the ...
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1answer
87 views

Jensen Inequality Euclidean Norm

I'm trying to prove $$\mathbb{E}\|X\|\geq \|\mathbb EX\| $$ for any random matrix $X$ and the spectral norm $\|\cdot \|$. To finish my proof all I need is the same statement for the euclidean norm $|\...
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1answer
40 views

Dominated convergence theorem by Jensen inequality

Define functional $\psi(x_n):=\limsup_n x_n$ on a sequence of positive numbers $x_1,x_2...$. We can check that $\psi$ is convex. Then if $f_n \to f$ pointwise, by Jensen's inequality, $$\limsup_n \int ...
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0answers
59 views

Expected value of integral with random variable on bound

I'm trying to write the following as a single integral, however I did not get the decired results yet, and tried searching this site without succes. What I'm trying to rewrite is: $$\mathbb{E}\int^...
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0answers
162 views

Refinement of $\left(\frac{a+1}{a+b} \right)^{\frac25}+\left(\frac{b+1}{b+c} \right)^{\frac25}+\left(\frac{c+1}{c+a} \right)^{\frac25} \geqslant 3$

It's a refinement of Prove $\left(\frac{a+1}{a+b} \right)^{\frac25}+\left(\frac{b+1}{b+c} \right)^{\frac25}+\left(\frac{c+1}{c+a} \right)^{\frac25} \geqslant 3$ . It's a attempt to find a simple proof ...
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1answer
127 views

Show the inequality : $a^{(2(1-a))}+b^{(2(1-b))}+c^{(2(1-c))}+c\leq 1$

Claim : Let $0.5\geq a \geq b \geq 0.25\geq c\geq 0$ such that $a+b+c=1$ then we have : $$a^{(2(1-a))}+b^{(2(1-b))}+c^{(2(1-c))}+c\leq 1$$ To prove it I have tried Bernoulli's inequality . For $0\leq ...
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0answers
30 views

Is there a simpler way to prove that a fractional function with three variables is convex?

How to prove that f(x) is convex? For all $$x_i>0$$ we have $$ f(x) = \frac{1}{x_1 - \frac{1}{ x_2 - \frac{1}{x_3} }} $$ I tried to take the second derivative (hessian matrix) but it was too long ...
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1answer
35 views

Some converse of Jensen inequality for some extreme random variable

This problem arises in my research in a more general form. It concerns information theory and statistical bounds using information theory. I tagged this question with Jensen inequality since the ...
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127 views

Existence of an asymptote for $g(x)=\frac{f(x)f'(x)+f(1)f'(1)}{f'(x)+f'(1)}-f\left(\frac{xf'(x)+f'(1)}{f'(x)+f'(1)}\right)$

Working with the Slater's inequality (companion of Jensen's inequality) I find this statement : Let $f(x)$ be a continuous,$n$ times differentiable ,convex and non constant on $(0,\infty)$ and ...
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1answer
33 views

Verify that $M_{X}(t) \ge e^{t\mu}$ using Jensen's identity

Suppose that a real valued random variable with a probability density of $$f(x) = \begin{cases} \frac{1}{8}(x+1) &:-1 <x<5\\ 0 &: \text{else} \end{cases}$$. I need to verify that $M_{X}(...
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1answer
209 views

Probability of a deviation when Jensen’s inequality is almost tight

Let $X>0$ be a random variable. Suppose that we knew that for some $\epsilon \geq 0$, \begin{eqnarray} \log(E[X]) \leq E[\log(X)] + \epsilon \tag{1} \label{eq:primary} \end{eqnarray} The question ...
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1answer
154 views

Prove this refinement of Nesbitt's inequality based on another

Let $a,b,c\in[1,2]$ such that $a,b$ are constants then prove : $$f(c)=\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{b+a}\geq h(c)=(c-1)\frac{g(2)-g(1)}{2-1}+g(1)\geq g(c)=\sqrt{\frac{9}{4}+\frac{3}{2}\frac{(a-...
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2answers
85 views

Stronger than Nesbitt's inequality using convexity and functions

Hi it's a refinement of Nesbitt's inequality and for that, we introduce the function : $$f(x)=\frac{x}{a+b}+\frac{b}{x+a}+\frac{a}{b+x}$$ With $a,b,x>0$ Due to homogeneity we assume $a+b=1$ and we ...
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0answers
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prove for a maximal inequality using Jensen's inequality

Problem Let $\{Z_i\}_{i \in \{1,\dots, n\}}$ be a family of independent random variables and $v>0$. Given that $$ \forall i \in \{1,\dots,n\}, \lambda > 0: \mathbb{E}[e^{\lambda Z_i}] \leq e^{\...
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1answer
35 views

Convexity in NMF proof

I am currently studying a proof for the NMF multiplicative update rule based on generalized Kullback-Leibler divergence in the paper Algorithms for Non-negative Matrix Factorization, and for some ...
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1answer
57 views

Inequality between geometric mean and harmonic mean [closed]

Suppose that $x_1,\dots,x_n>0$. Prove that $$\sqrt[n]{x_1x_2\dots x_n}\geq \dfrac{n}{\frac{1}{x_1}+\dots+\frac{1}{x_n}}$$ by using Jensen's inequality for some suitable function $f$.
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34 views

Ask for a proof that expectation falls in convex domain

I am stuck at the first sentence of the following proof of Jensen’s inequality, appearing in Erhan Çınlar's text "Probability and Stochastics", P70-71: I am pretty frustrated not only by my ...
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1answer
14 views

Upperbound of Kullback-Leibler between marginals

I want to know if for every pair of $p(x, y)$ and $q(x, y)$ the following inequality $$ \int_X p(x) \log \frac{p(x)}{q(x)} dx \leq \int_X\int_Y p(x, y) \log \frac{p(x, y)}{q(x, y)}dxdy $$ is valid. I ...
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1answer
65 views

Applying Jensen's inequality to find a bound

I'm not sure how to begin with this question, apart from writing some basic relationships based on Jensen's inequality. Any suggestions would be appreciated. For arbitrary α and β satisfying 0 < α ...
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1answer
115 views

Jensen's Inequality

For arbitrary α and β satisfying 0 < α ≤ β, use Jensen’s inequality to find a bound of the form E[|X|$^α$] ≤ f(E[|X|$^β$]) for some function f. Doesn't this bound follow naturally from the fact ...
2
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1answer
40 views

Undertanding Jensen's inequality proof on countable basis Hilbert with Radon measure

Given the following proposition : $\textbf{Proposition :}$ $X$ Hilbert space with countable basis, $\mu$ Radon measure on $X$, $\phi \in L_{\mu}^{1}(X)$ and $f \in \left\lbrace f : X \longmapsto \...
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0answers
117 views

Finding bounds for the absolute value of function raised to positive constant

I am trying to show the below inequality $$|x|^{a_1} + |y|^{a_2} \leq C(|x-y|^{a_1} + |y|^{\max(a_1,a_2)}) $$ for constants ($a_1>0,a_2$>0) and $C$ is some other constant. What is the quickest ...

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