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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3
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54 views

Prove that midpoint convex and bounded imply continuous

Question: Let $f$ be a function on $[a, b]$. For $\forall x\in[a, b], |f(x)|\le M$ where $M>0$, and for $\forall x, y\in[a, b], f(\frac{x+y}{2})\le\frac{f(x)+f(y)}{2}$. (1) Prove that $f$ is ...
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1answer
44 views

Expected value of absolute value of centered random variable

I am looking to prove the following: Given iid random variable's $X = X_1, X_2, \dots$, and mean $E[X] = \bar{X}$ ,show that: $$E[|X|] \geq E[|X - \bar{X}|] \tag{1}\label{1}$$ This intuitively ...
4
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1answer
71 views

Use Jensen's inequality to show $\frac{2x}{2+x} < \log(1+x) < \frac{2x+x^2}{2+2x}$ for $x>0$

Use Jensen's inequality to show $\frac{2x}{2+x} < \log(1+x) < \frac{2x+x^2}{2+2x}$ for $x>0$. I can show this without Jensen's inequality, but I'd like to see what that form of the proof ...
2
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1answer
34 views

$f(0)=f(1)=0$, $f(x)=\frac{f(x+h)+f(x-h)}{2}$ implies $f(x)=0$ for $[0, 1]$

Question: Suppose $f$ is continuous on $[0, 1]$ with $f(0)=f(1)=0$. For $\forall x\in (0, 1)$, there $\exists h>0$ with $0\le x-h<x<x+h\le1$ such that $f(x)=\frac{f(x+h)+f(x-h)}{2}$. Show ...
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28 views

sum of exponential vs exponential of sums

Let $x_1,\cdots,x_k\in (0,1]$ are $k$ reals such that $\sum_{i=1}^kx_i > 1/2$. Since $f(x)=\exp(-x)$ is convex function, from Jensen's inequality, we have that $\sum_{i=1}^k\exp(-x_i) \geq k.\exp(-\...
1
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1answer
96 views

Some doubts in a part of the proof of Backwards Martingale Convergence Theorem (Jacod-Protter)

A USEFUL RESULT (Doob's Upcrossing Inequality) Let $(X_n)_{\geq0}$ be a submartingale, let $a<b$ and let $U_n$ be the number of upcrossings of $[a,b]$ before time $n$. Then $$ \mathbb{E}\{U_n\}\...
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0answers
25 views

References for variants of Jensen inequality

I am looking for a reference for the following claim: Let $X$ be a probability space, and let $g:X \to \mathbb [0,\infty) $ be in $L^1(X)$. Let $\phi:\mathbb [0,\infty) \to [0,\infty)$ be convex and ...
4
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127 views

Conjecture about Jensen's inequality and polynomials

Hi it's related to the following conjecture An inequality for polynomials with positives coefficients : We have the first conjecture : Let $x,y>0$ then we have : $$(x+y)f\Big(\frac{x^2+...
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7 views

Direction to calculate the expected value of $\frac{1}{a\sum_{i=1}^N X_i}$ where $f_X(x)=\frac{1}{c^2 x^2} $ where $\frac{1}{c^2}<x<\infty$

The expected value of $\frac{1}{a\sum_{i=1}^N X_i}$ where $f_X(x)=\frac{1}{c^2 x^2}$ where $\frac{1}{c^2}<x<\infty $ is needed. I tried using Jensen's inequality but that did not work much as ...
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1answer
71 views

Question regarding Jensen Inequality

Following is the picture of the question regarding the application of Jensen Inequality. Following is the picture my approach to proove the inequality. Can anyone please check if my proof is ...
4
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2answers
130 views

$a+b+c+d=1, a,b,c,d ≠ 0$, then prove that $(a + \frac{1}{a})^2 + (b + \frac{1}{b})^2 + (c + \frac{1}{c})^2 + (d + \frac{1}{d})^2 \ge \frac{289}{4} $

If $a+b+c+d=1$, $a,b,c,d ≠ 0$, prove that $$\left(a + \frac{1}{a}\right)^2 + \left(b + \frac{1}{b}\right)^2 + \left(c + \frac{1}{c}\right)^2 + \left(d + \frac{1}{d}\right)^2 \ge \frac{289}{4} $$ I ...
1
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1answer
31 views

Exercise on submartingales: is $\phi(X_n)$ a submartingale, given some assumptions on $(X_n)$? Is the following solution correct?

Let $X=(X_n)_{n>0}$ be a submartingale. Show that if $\phi$ is convex and nondecreasing on $\mathbb{R}$ and if $\phi(X_n)$ is integrable for each $n$, then $Y_n=\phi(X_n)$ is also a submartingale. ...
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1answer
34 views

Can I apply Jensen Inequality here?

$X$ is a non negative random variable with decreasing density function. Let $U$ be a $Unif(0,2t)$ random variable where $t>0$. For $x>0$ define $G(X)=P(X>x)$. Then show that $$\mathbb{E}(G(U))...
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1answer
78 views

If $x+y+z=1$ prove $ \sqrt{x+\frac{(y-z)^{2}}{12}}+\sqrt{y+\frac{(z-x)^{2}}{12}}+\sqrt{z+\frac{(x-y)^{2}}{12}} \leq \sqrt{3} $

Question - Let $x, y, z$ be non-negative real numbers with sum $1 .$ Prove that $$ \sqrt{x+\frac{(y-z)^{2}}{12}}+\sqrt{y+\frac{(z-x)^{2}}{12}}+\sqrt{z+\frac{(x-y)^{2}}{12}} \leq \sqrt{3} $$ My work -...
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What is the proof of entropy property: $H\left(x,y\right)\le H\left(x\right)+H\left(y\right)$ in Shannon's paper?

In Shannon's 1948 paper titled "A Mathematical Theory of Communication", in the discussion of the entropy of the joint event, there is no proof for this inequality (or subadditivity of entropy) $$H\...
2
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1answer
114 views

Prove using Jensen's inequality that if $abcd=1$ then $\frac{1}{(1+a)^{2}}+\frac{1}{(1+b)^{2}}+\frac{1}{(1+c)^{2}}+\frac{1}{(1+d)^{2}} \geq 1$

Question - Let $a, b, c, d$ be positive real numbers such that abcd $=1 .$ Prove that $$ \begin{array}{c} \frac{1}{(1+a)^{2}}+\frac{1}{(1+b)^{2}}+\frac{1}{(1+c)^{2}}+\frac{1}{(1+d)^{2}} \geq 1 \\ \...
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4answers
61 views

MOP 2011 inequality

If $a,b,c$ are positive integers prove that $\sqrt{(a^2-ab+b^2)} +\sqrt{(b^2+c^2-bc)} +\sqrt{(a^2+c^2-ac)} +9(abc)^{1/3} \le 4(a+b+c)$ My attempt: I tried to split inequality and prove it bit by ...
1
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1answer
64 views

If $f(\frac{t_1+t_2}{2}) \leq \frac{f(t_1)+f(t_2)}{2}$, show that $f(\frac{t_1+t_2+ \cdots +t_n}{n})\leq \frac{f(t_1)+f(t_2)+\cdots f(t_n)}{n}$ [duplicate]

If $f$ is a continuous function on $[0,1]$ such that for all $t_1, t_2 \in [0,1]$, $$f\left(\frac{t_1+t_2}{2}\right) \leq \frac{f(t_1)+f(t_2)}{2}$$ Show that $$f\left(\frac{t_1+t_2+ \cdots +t_n}{n}\...
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0answers
29 views

A Turan-type inequality :$\Big(a^{(2b)^n}-b^{(2a)^n}\Big)^2\geq \Big(a^{(2b)^{n-1}}-b^{(2a)^{n-1}}\Big)\Big(a^{(2b)^{n+1}}-b^{(2a)^{n+1}}\Big)$

Hi inspired by this question Prove that if $a+b =1$, then $\forall n \in \mathbb{N}, a^{(2b)^{n}} + b^{(2a)^{n}} \leq 1$. I propose this : Let $a,b>0$ such that $a+b=1$ then we have : $$...
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35 views

Explanation for the solution associated with Jensen's inequality

While I was surfing AoPS, I found this problem and its solution: But I don't understand the application of Jensen's integral inequality (inequality on the first line of bengabriel's solution). Jensen'...
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1answer
49 views

Prove some inequality using Jensen's inequality

How do I prove this using Jensen's inequality? $$\biggl|\prod_{i=1}^n x_{i}\biggl|^p \le {n^{p-1}}\sum_{i=1}^n |x_{i}|^p$$ I've tried log on both sides but I couldn't find a common expression. on ...
1
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1answer
32 views

On Conditional Jensen Inequality Hypothesis

** Conditional Jensen Inequality Let $X$ be in $L^1$, and $\mathcal{G}$ a sigma-algebra of the space and $\phi$ a convex function. Then, $\mathbb E(\phi(X)|\mathcal{G}) \geq \phi(\mathbb E(X|\mathcal{...
1
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1answer
20 views

Consistency of Sylvester's Determinant Theorem under Applying Jensen's Inequality

Sylvester's determinant theorem states that for matrices $A\in\mathbb{R}^{n\times d}, B\in\mathbb{R}^{d\times n}$: \begin{equation} \det(I_{n}+AB)=\det(I_d+BA) \end{equation} In my case I consider $...
2
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1answer
31 views

The Jensen gap $\mathbb{E}[|\overline X|] - |\mu|$

Let $ X_1, X_2, \dots, X_n $ be a sequence of i.i.d. random variables with finite mean $ \mu $ and variance $ \sigma^2 $. Let $ \overline X = \frac{1}{n}\sum_{i=1}^n X_i $ denote the sample average. I ...
3
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1answer
67 views

Concativity of entropy without Jensen's inequality

In my information theory class I need to prove that entropy is concave (which is usually done with Jensen's inequality). But I want to use only the definition of entropy. And as the result of ...
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0answers
34 views

Another proof of Jensen’s Inequality (finite form)

I am reading through Jensen’s inequality and its proofs. I want to find an alternate proof to the tangent line proof. My attempt: If $f(x)$ is a convex function, then $\{x_{i},f(x_{i})\}\ ,\ i=1,2,.....
3
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1answer
85 views

Sum Infinite Random Variables

Let's say we generate $n$ samples independently from two independent distributions $X$ and $Y$. We know that the following is true from Jensen's Inequality: $$\ E\left[\min\left(\sum_{i=1}^{n}X_i, \...
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0answers
31 views

Jensen's inequality proof (1)

Hello I do not understand part of a very specific Jensen inequality proof. That is how exactly I prove equation (1). I do understand how to get to the left part, but not how to deal with the back ...
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2answers
28 views

Question on applying Jensen inequality on logarithm of a sum

I am confused by the meaning of the $t \in (0,1)$ parameter in Jensen's inequality $$ f( tx_1+(1-t)x_2) \le tf(x_1)+(1-t)f(x_2) $$ When I apply this to the logarithm $$ \log( tx_1+(1-t)x_2) \le t \log(...
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1answer
42 views

Construct an Unbiased Estimator

I am currently trying to prove that $\hat{\beta}$ is not an unbiased estimator of $\beta$. After proving this I need to to construct a unbiased estimator for $\beta$. I know that $$\hat{\beta} = \frac{...
3
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4answers
87 views

An inequality involving homogeneous polynomials

Let $x_1, x_2, \dots x_k \ge 0$ be non-negative real numbers. Does it follow that $$k \left( \sum_{i=1}^k x_i^3 \right)^2 \ge \left( \sum_{i=1}^k x_i^2 \right)^3 ? $$ This seems like something that ...
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28 views

If i have an objective function with a lot of constraints. How can i prove conclusively that my problem is convex/ non-convex?

How to perform convexity analysis on a difficult objective function. I know about the Hessian matrix and Jensen's inequality. Both of them are difficult to derive in my case. What other theorems in ...
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0answers
26 views

Kullback-Leibler divergence from density $f$ to density $g$.

If $f$ and $g$ are density functions that are positive over the same region, then the Kullback-Leibler divergence from density $f$ to density $g$ is defined by: $$KL(f,g) = E_f\left[\ln\left(\frac{f(...
2
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1answer
73 views

Inequality from Israel TST

Let $a, b, c, d$ be nonnegative numbers such that $a+b+c+d=18.$ Prove that: $$\sqrt{\frac{a}{b+6}}+\sqrt{\frac{b}{c+6}}+\sqrt{\frac{c}{d+6}}+\sqrt{\frac{d}{a+6}}\leq5\sqrt{\frac{2}{7}}$$ These are my ...
2
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0answers
54 views

Proof that $\tan(\sin x) > \sin(\tan x), x \in (0, \pi/2)$ [duplicate]

Assuming $x \in (0, \frac{\pi}{4})$ Write down a proof that $F(x) > 0 $ for all $x$'s where $F(x) = \tan(\sin x)-\sin(\tan x)$. All I came up with was using Jensen inequity for 1st derivative ...
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1answer
24 views

Lyapunov's CLT Limit Condition

I am trying to show that Lyapunov's condition holds in Lyapunov's CLT, and am left with the trying to show that for some $\delta >0$ $$\underset{n\rightarrow\infty}{lim} \frac{\sum_{i=1}^n w_i^{2+\...
2
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1answer
38 views

An inequality for the mgf using Jensen’s inequality

Given non-negative random variables $X_1,X_2,...$ how to show that $$\mathbb{E}\exp(t\max\limits_{1\leq i\leq n}X_i)\leq \sum\limits_{1\leq i\leq n}\mathbb{E}\exp(tX_i).$$ I think we should start ...
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0answers
41 views

Jensen's inequality and convex Lagrangian

I was reading some lecture notes, and there was a following example that I didn't quite understand. If we have a following variational problem: $ \int_{a}^{b}f(u'(x))dx$ where the Lagrangian $f$ is a ...
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1answer
59 views

Prove special case of Jensen's inequality THE OPPOSITE WAY [duplicate]

$f: \mathbb{R}^n \rightarrow \mathbb{R},f$ is continious. $$f \text{ is convex} \Leftrightarrow f\left(\dfrac{x+y}{2}\right) \le \dfrac{f(x) + f(y)}{2}\ \ \ \forall x, y \in \mathbb{R}^n$$ One side ...
2
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2answers
83 views

Find the maximum value of a sum of cosines given certain condition

In my calculus class, I've come across this problem when we were on the topic of Jensen's Inequality: \begin{multline}A=\{\cos(x_1)\cos(x_2)\dots\cos(x_n)\in\Bbb{R}:\\n\in\Bbb{N},x_1^2+...+x_n^2=1\}.\...
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39 views

In the style of the Abi-Khuzam's inequality

It's a little problem that I can solve : Let $a,b,c>0$ such that $a+b+c=\pi$ then we have : $$\Big(\sin(a)^a\sin(b)^b\sin(c)^c\Big)^{\frac{1}{\pi}}\leq \frac{3\sqrt{3}}{2\pi}\Big(a^{\...
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2answers
72 views

How to prove that $\prod_{i=1} ^ \infty x_i^{a_i} \leq \sum_{i=1} ^ \infty a_ix_i$ [duplicate]

Let $a_1,a_2,...$ be nonnegative numbers whose sum is $1$ and let $x_1,x_2,...>0$. I want to show that $\prod_{i=1} ^ \infty x_i^{a_i} \leq \sum_{i=1} ^ \infty a_ix_i$. This looks awfully similar ...
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3answers
93 views

Prove that $\frac{1}{\sqrt{a+b+2}}+\frac{1}{\sqrt{b+c+2}}+\frac{1}{\sqrt{c+d+2}}+\frac{1}{\sqrt{d+a+2}}\le 2$

Let $a,b,c,d\in \mathbb{R^+}$ such that $abcd=1$. Prove that $$\frac{1}{\sqrt{a+b+2}}+\frac{1}{\sqrt{b+c+2}}+\frac{1}{\sqrt{c+d+2}}+\frac{1}{\sqrt{d+a+2}}\le 2$$ By Cauchy-Schwarz: $$\text{LHS}^2=\...
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1answer
49 views

Jensen's inequality and LOTUS applied to entropy in probability

I am given an example and proof for entropy: (Entropy). The surprise of learning that an event with probability $p$ happened is defined as $\log_2(1/p)$, measured in a unit called bits. Low-...
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1answer
84 views

Proof of Jensen's inequality for convexity

I am studying the Jensen inequality for convexity: Let $X$ be a random variable. If $g$ is a convex function, then $E(g(X)) \ge g(E(X))$. If $g$ is a concave function, then $E(g(X)) \le g(E(X))$. ...
5
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4answers
165 views

How to prove $\frac a{\sqrt{a^2+3b^2+3c^2}}+\frac b{\sqrt{3a^2+b^2+3c^2}}+\frac{c}{\sqrt{3a^2+3b^2+c^2}}\le\frac3{\sqrt7}$ when $a,b,c>0$

I want to prove that for $a,b,c>0$ we have $$\sum_{cyc} \frac a{\sqrt{a^2+3b^2+3c^2}}= \frac a{\sqrt{a^2+3b^2+3c^2}}+\frac{b}{\sqrt{3a^2+b^2+3c^2}}+\frac{c}{\sqrt{3a^2+3b^2+c^2}}\le\frac3{\sqrt7}.$...
3
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1answer
73 views

Sum of indicators and application of Jensen's inequality

So I have stumbled upon this problem. Let $X_1, \dots, X_n \sim N(\mu, \sigma^2)$ be iid. Define: $$S = \frac{1}{n}\sum_{i=1}^n I[X_i > a]$$ $$T = I[\frac{1}{n}\sum_{i=1}^n X_i > a]$$ $a > ...
4
votes
2answers
60 views

If $f'' \ge 0$, $\int_0^2 f(x)dx \ge 2f(1)$

Question: If $f'' \ge 0$ in interval $[0, 2]$, prove that $$\int_0^2 f(x)dx \ge 2f(1)$$ The question is graphically trivial I think, but not in mathematically. I wanted to use the fact that there ...
3
votes
3answers
250 views

An exponential equation over positive real numbers $4^x+14^x+3^x=11^x+10^x$

Solve the following equation over the positive reals: $$4^x+14^x+3^x=11^x+10^x.$$ By inspecting the graph, the solutions must be $x \in \{1,2\}$ I tried using inequalities like $3^x+4^x<5^x$ for $...
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0answers
15 views

Wanted: an inequality with variable exponent

[Wanted] I want to estimate over the following quantity $ | a + b | ^ {2-2p (x)}$ Where: • $a$ and $b$ are vectors in $\mathbb{R} ^{n}$ (!?) • $p$ is a function $C^{1}$ in a limited domain in $\...

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