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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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Integrability condition of composite function in Jensen's inequality [duplicate]

In Jensen's inequality $\phi(\mathbb{E}[X])\leq \mathbb{E}[\phi(X)]$, it is required that $\phi(X)$ is integrable. But why? If this integral is infinity, the inequality is still correct.
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Approximate Saturation of Jensen's Inequality

Let $f$ be a convex function and $X$ a random variable. Jensen's inequality states: $$\mathbb{E}[f(X)]\geq f(\mathbb{E}[X]).$$ When $f$ is an affine function, this inequality is saturated, i.e. it ...
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Inequality with two concave functions

is this inequality holds true $[\frac{\frac{f_2(x_0,y_0)}{y_0}}{\frac{f_2(x,y)}{y}}-\frac{f_1(x_0,y_0)}{f_1(x,y_0)}]\times [\frac{f_1(x,y_0)}{f_1(x_0,y_0)}-\frac{f_2(x,y)}{f_2(x_0,y_0)}]\leq 0 \; \...
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Proof of a Conditional Expectation Result Using Truncation and Conditional Jensen's Inequality

Let $X, Y$ be integrable random variables. If with probability one $E[X|Y] = Y$ and $E[Y|X] = X$, then $X = Y$ almost surely. (Hint: first assume $X, Y$ are $\mathbb{L}^2$, and then use truncation and ...
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Jensen's inequality with affine combination

From page 217 of 'Convex functions' by Arthur Wayne Roberts, Dale Varberg (exercise F) i want proof that: $f:\mathbb{R}\longrightarrow\mathbb{R}$ is affine iff $$ f\biggl(\displaystyle\sum\limits_{i=1}...
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Is the minimum of the modulus of a parametric function convex or concave?

I have a function that looks like this min_{r \in R, l \in L} |l-r| Is this function convex or concave? I’ve read somewhere that |x| and that the minimum of convex functions is also convex. But I have ...
Stella's user avatar
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5 votes
2 answers
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A certain inequality bound

Why does the following estimate hold for any $p \in [1,\infty)$, and $\delta \in (0,1)$, $$ \big ||a-b|^p - |a|^p - |b|^p \big | \leq \delta |a|^p + \frac{C_p}{\delta^p} |b|^p $$ If we use convexity ...
Document123's user avatar
2 votes
2 answers
80 views

Generalization for Jensen's inequality for expected values

I'm trying to prove this. Let $\Phi$ be a convex function and X,Y random variables on the same probability space, where $Y \geq 0$ a.s. and $E(Y) = 1$. It holds $\Phi(E(XY)) \leq E(\Phi(X)Y)$. What I ...
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Application of the Jensen's inequality

For a convex function $\phi(x)$, Jensen's inequality states: \begin{equation} E_{x}(\phi(x)) \geq \phi(E_{x}(X)) \end{equation} where $E_x$ is the expectation. When applying the Jensen's inequality ...
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Proving that $\iint f(x) g(x+y) dx dy \leq \int g^2(x) dx$

Let $D \subset \mathbb{R}^n$ be bounded, and $f,g: \mathbb{R}^n \rightarrow \mathbb{R}$. Furthermore let $f$ be nonnegative and such that $\int_D f dx= 1$. I would like to prove that $$\int_D \int_D f(...
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Power diffs: $\frac{1}{\gamma}\mathbb{E}[X^\gamma - X_*^\gamma] \leq 0 \implies \mathbb{E}[(1+ X^\gamma - X_*^\gamma)^{\frac{1}{\gamma}}] \leq 1$?

I would like to show that the following implication is true for all $\gamma<1$ and $a$ in the $\mathbb{R}^d$ simplex $\Delta^d$: $$\frac{1}{\gamma}\mathbb{E}[(a^TX)^\gamma - ({a^*}^TX)^\gamma] \leq ...
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Showing the following function is nonnegative

Let $D \subset \mathbb{R}$ and $f: D \rightarrow \mathbb{R}$ $g: \mathbb{R} \times D \rightarrow \mathbb{R}$ be nonnegative functions. Let $\mu$ be a measure on $D$ such that $\mu(D) < \infty$. I ...
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Proving a certain inequality using Jensen/Tangent line trick

This is from an Olympiad handout about inequalities. It goes as follows: Let $a\ge b \ge 1$ be two real numbers. Show that \begin{align} \frac{a}{\sqrt{a+b}}+\frac{b}{\sqrt{b+1}}+\frac{1}{\sqrt{a+1}} \...
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Under what assumptions is the Jensen's inequality sufficient for concavity (or convexity) of a function?

We know that concave (convex) functions satisfy Jensen's inequality. But could we infer the converse under some additional assumptions, i.e. under what assumptions is Jensen's inequality is an if and ...
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Prove $a^b+b^a\leq \sqrt{a}+\sqrt{b}$ [duplicate]

Prove $a^b+b^a\leq \sqrt{a}+\sqrt{b}$, if $a+b=1$ and $a>0,b>0$. It seems quite difficult for me to use the naive differentiation method (e.g. take $f(x)=a^{\frac{1}{2}-x}+b^{\frac{1}{2}+x}$ and ...
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Integral Jensen Inequality

I need to understand how does this equation derives from Jensen inequality ($\varphi(\frac{1}{b-a}\int_{a}^{b}f(x)dx) \leq \frac{1}{b-a}\int_{a}^{b}\varphi(f(x))dx$) It seems that I can show that $(\...
bob's user avatar
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Show the inequality $\sum_{\text{cyc}} \sqrt{\frac{a^3}{b^3 + (c+a)^3}} \ge 1$

For all positive reals $a, b, c$, we wish to prove the inequality $$\sum_{\text{cyc}} \sqrt{\frac{a^3}{b^3 + (c+a)^3}} \ge 1$$ My approach was Hölder: $$\left(\sum_{\text{cyc}} \sqrt{\frac{a^3}{b^3 + (...
Martin Westin's user avatar
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1 answer
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Showing inequality involving log

I want to show that for all $x_i > 0$: $$\sum_{i=1}^{n}\dfrac{x_i}{x_1 + \ldots+x_n}\;\log(x_i) \geq \log\left(\dfrac{x_1 + \ldots + x_n}{n}\right)$$ I thought of Jensen's inequality but since $\...
wheeler 's user avatar
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Prove the following from Jensen's inequality

$n \cdot \left( \frac{n+1}{2} \right)^{\left( \frac{n+1}{2} \right)} \leqslant \sum_{k=1}^{n} k^k \text{ for } n \in \mathbb{N}$ I've tried to transform the left part of inequality, but nothing worked ...
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Elementary proof that $\mathbb{E}[1/X] \geq 1/\mathbb{E}[X]$ for finitely supported positive random variables.

Let $X$ be a random variable with finite support contained in $(0,\infty)$. By Jensen's inequality we have $\mathbb{E}[1/X] \geq 1/\mathbb{E}[X]$. I am curious if there exists an elementary proof of ...
diracdeltafunk's user avatar
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1 answer
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Showing $\frac1{ab+4}+\frac1{ac+4}+\frac1{ad+4}+\frac1{bc+4}+\frac1{bd+4}+\frac1{cd+4}\geq\frac65$, for positive $a,b,c,d$ with $ab+bc+cd+da=4$

Let us take $a\geq b\geq c\geq d>0$ such that $ab+bc+cd+da=4$. Show that $$\frac{1}{ab+4}+\frac{1}{ac+4}+\frac{1}{ad+4}+\frac{1}{bc+4}+\frac{1}{bd+4}+\frac{1}{cd+4}\geq\frac{6}{5}$$ $a+b+c+d \ge 4$...
Math learner's user avatar
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Case of equality in Jensen's Inequality

I haven't really understood parts of the proof given for $\textrm{Problem 6.2}$ of Steele's Cauchy-Schwarz Master Class. Suppose that $f : [a, b] → \mathbb{R} $ is strictly convex and show that if \...
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Show $\left(x_1x_2\ldots x_n\right)^{\frac{x_1+\cdots +x_n}{n}}\le x_1^{x_1}x_2^{x_2}\ldots x_n^{x_n}$

For $x_1\ldots x_n \in \mathbb{R_{+}}$ show: $$\left(x_1x_2\ldots x_n\right)^{\frac{x_1+\cdots +x_n}{n}}\le x_1^{x_1}x_2^{x_2}\ldots x_n^{x_n}$$ Convexity and logarithmic function are the first ideas ...
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1 answer
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Prove an inequality using Jensen's inequality

Let $Q\in M_1(M_1(\Sigma)),$ whereby $M_1(\Sigma)$ denotes the space of probability measures on $\Sigma.$ We define a new measure $\mu_Q (\Gamma) = \int_{M_1(\Sigma)} \nu(\Gamma)Q(d\nu), \Gamma \in B_{...
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Bound on Jensen's gap for bivariate functions

I have recently been reading on Jensen's gap for univariate functions. Specifically, here, page 6, Theorem 2.1, an upper bound on Jensen's gap is provided that is based on absolute central moments. I ...
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Upper bound on E[ln(X)^2]

The question is to state, for positive $X$, when $\mathbb{E}[(\ln X)^2] \leq (\ln\mathbb{E}X)^2$. I think this is asking to use Jensen's inequality, but this inequality says the expected value is the ...
johnsmith's user avatar
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1 answer
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Can anyone help prove or disprove that $\frac{\Pi_i x_i ^\frac{x_i}{1+\sum_i x_i}}{1+\sum_i x_i} \geq \frac{1}{N+1}$, where $x_i>0$

I was hoping the generalize this result: How to prove $x^{x/(1+x)}/(1+x)\geq1/2$ I believe that the following inequality holds: $\frac{\Pi_i x_i ^\frac{x_i}{1+\sum_i x_i}}{1+\sum_i x_i} \geq \frac{1}{...
Vance M's user avatar
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3 answers
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How to prove $x^{x/(1+x)}/(1+x)\geq1/2$

I need to show that $\frac{x^{\frac{x}{1+x}}}{1+x} \geq \frac{1}{2}$ for all $x\geq0$. I am fairly certain this statement is true, but have no idea how to go about proving it. I know that it attains a ...
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Finding $\small{\min\limits_{a+b+c=2}\sqrt{a-2bc+3}+\sqrt{b-2ca+3}+\sqrt{c-2ab+3}.}$

Given non-negative real numbers $a,b,c$ satisfying $a+b+c=2.$ Find the minimal value of expression $$P=\sqrt{a-2bc+3}+\sqrt{b-2ca+3}+\sqrt{c-2ab+3}.$$ Source: AOPS-Jokehim. See also here. Here is ...
Dragon boy's user avatar
4 votes
3 answers
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Prove $\frac{1}{\sqrt{2a+b+c}}+\frac{1}{\sqrt{2b+a+c}}+\frac{1}{\sqrt{2c+b+a}}\le \frac{9}{2}\cdot\frac{1}{\sqrt{a}+\sqrt{b}+\sqrt{c}}.$

Let $a,b,c>0.$ Prove that $$\frac{1}{\sqrt{2a+b+c}}+\frac{1}{\sqrt{2b+a+c}}+\frac{1}{\sqrt{2c+b+a}}\le \frac{9}{2}\cdot\frac{1}{\sqrt{a}+\sqrt{b}+\sqrt{c}}.$$ WLOG, assuming that $a+b+c=1$ and we'...
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1 answer
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Does $E^2(X)/E(Y) \le E(X^2/Y)$ hold

Let $f(x),g(x)$ be $\mu$ measureable functions with $g(x) > 0$ almost surely. Does $$\frac{\left(\int f(x)\mu(x)dx\right)^2}{\int g(x)\mu(x)dx} \leq \int \frac{(f(x))^2}{g(x)}\mu(x)dx$$ hold? I ...
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Jensen's inequality for composite functions [duplicate]

Let $f(\cdot)$ be a convex function and let $g(\cdot)$ be any function. According to Jensen's inequality, we know $$f[E(X)]\leq E[f(X)].$$ However, is the following true or not? $$f[E(g(X))]\leq E[f(g(...
zijie wang's user avatar
3 votes
2 answers
178 views

How to prove $\sqrt{4a+1}+\sqrt{4b+1}+\sqrt{4c+1}\ge \frac{\sqrt{abc+4}+4\sqrt{ab+bc+ca+4}}{2}.$

Question. If $a,b,c\ge 0: a+b+c=2,$ prove that $$\color{blue}{\sqrt{4a+1}+\sqrt{4b+1}+\sqrt{4c+1}\ge \frac{\sqrt{abc+4}+4\sqrt{ab+bc+ca+4}}{2}.}$$Equality holds iff $(a,b,c)=\{(0,1,1);(0,0,2)\}.$ ...
Sickness's user avatar
2 votes
2 answers
113 views

How to prove this inequality $\sum_{r=1}^{n}a_{r}\sqrt{\frac{n-1}{1-a_{r}}}\ge\sum_{r=1}^{n}\sqrt{a_{r}}$

Let $({a_{r}})_{r=1}^{n}$ be a sequence of $n$ positive real numbers that sum to 1. Prove that for all $n>1$ :\begin{align} \sum_{r=1}^{n}a_{r}\sqrt{\frac{n-1}{1-a_{r}}}&\ge\sum_{r=1}^{n}\sqrt{...
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Bounding sum with weights changed from $2^{a_i}$ to $a_i$

I have two positive sequences $a_i$ and $b_i$ with unknown (finite) index set $I$. I also have the following: \begin{aligned} \sum_{i \in I} a_i & = \log(m) 2^n \\ \sum_{i \in I} 2^{a_i} & = m ...
Lewwwer's user avatar
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1 answer
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Entrywise matrix Jensen inequality? [closed]

Suppose we have two positive definite symmetric matrices $A,B\in \mathbb{R}^{n\times n}$ and $(X)_{ij}$ denotes the $ij$-th entry of a matrix $X$. Then do we have the following inequality? $\frac{1}{...
MikeG's user avatar
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2 votes
1 answer
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$X$ is sub-Gaussian, then $X^2$ is sub-exponential

My goal is to show the following statement without using sub-exponential or sub-Gaussian norm; Let $X$ be a zero-mean sub-Gaussian random variable with the variance proxy $\sigma^2$. Then $X^2$ is ...
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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 \...
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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
-2 votes
1 answer
75 views

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
1 vote
1 answer
173 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
143 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
194 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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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 ...
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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 ...
TG173's user avatar
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3 votes
1 answer
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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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13 votes
2 answers
1k 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
2 votes
0 answers
98 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 ...
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