Questions tagged [jensen-inequality]
For questions about proving and manipulating the AM-GM inequality. To be used necessarily with the [inequality] tag.
287
questions
3
votes
0answers
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 ...
2
votes
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
votes
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
votes
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 ...
1
vote
0answers
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
vote
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
$$
\mathbb{E}\{U_n\}\...
1
vote
0answers
25 views
References for variants of Jensen inequality
I am looking for a reference for the following claim:
Let $X$ be a probability space, and let $g:X \to \mathbb [0,\infty) $ be in $L^1(X)$.
Let $\phi:\mathbb [0,\infty) \to [0,\infty)$ be convex and ...
4
votes
0answers
127 views
Conjecture about Jensen's inequality and polynomials
Hi it's related to the following conjecture An inequality for polynomials with positives coefficients :
We have the first conjecture :
Let $x,y>0$ then we have :
$$(x+y)f\Big(\frac{x^2+...
0
votes
0answers
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 ...
0
votes
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
votes
2answers
130 views
$a+b+c+d=1, a,b,c,d ā 0$, 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 $a+b+c+d=1$, $a,b,c,d ā 0$, 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
vote
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.
...
0
votes
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 $$\mathbb{E}(G(U))...
2
votes
1answer
78 views
If $x+y+z=1$ prove $ \sqrt{x+\frac{(y-z)^{2}}{12}}+\sqrt{y+\frac{(z-x)^{2}}{12}}+\sqrt{z+\frac{(x-y)^{2}}{12}} \leq \sqrt{3} $
Question -
Let $x, y, z$ be non-negative real numbers with sum $1 .$ Prove that
$$
\sqrt{x+\frac{(y-z)^{2}}{12}}+\sqrt{y+\frac{(z-x)^{2}}{12}}+\sqrt{z+\frac{(x-y)^{2}}{12}} \leq \sqrt{3}
$$
My work -...
1
vote
0answers
22 views
What is the proof of entropy property: $H\left(x,y\right)\le H\left(x\right)+H\left(y\right)$ in Shannon's paper?
In Shannon's 1948 paper titled "A Mathematical Theory of Communication", in the discussion of the entropy of the joint event, there is no proof for this inequality (or subadditivity of entropy)
$$H\...
2
votes
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
$$
\begin{array}{c}
\frac{1}{(1+a)^{2}}+\frac{1}{(1+b)^{2}}+\frac{1}{(1+c)^{2}}+\frac{1}{(1+d)^{2}} \geq 1 \\
\...
0
votes
4answers
61 views
MOP 2011 inequality
If $a,b,c$ are positive integers prove that
$\sqrt{(a^2-ab+b^2)} +\sqrt{(b^2+c^2-bc)} +\sqrt{(a^2+c^2-ac)} +9(abc)^{1/3} \le 4(a+b+c)$
My attempt:
I tried to split inequality and prove it bit by ...
1
vote
1answer
64 views
If $f(\frac{t_1+t_2}{2}) \leq \frac{f(t_1)+f(t_2)}{2}$, show that $f(\frac{t_1+t_2+ \cdots +t_n}{n})\leq \frac{f(t_1)+f(t_2)+\cdots f(t_n)}{n}$ [duplicate]
If $f$ is a continuous function on $[0,1]$ such that for all $t_1, t_2 \in [0,1]$,
$$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}\...
0
votes
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 :
$$...
1
vote
0answers
35 views
Explanation for the solution associated with Jensen's inequality
While I was surfing AoPS, I found this problem and its solution:
But I don't understand the application of Jensen's integral inequality (inequality on the first line of bengabriel's solution). Jensen'...
0
votes
1answer
49 views
Prove some inequality using Jensen's inequality
How do I prove this using Jensen's inequality?
$$\biggl|\prod_{i=1}^n x_{i}\biggl|^p \le {n^{p-1}}\sum_{i=1}^n |x_{i}|^p$$
I've tried log on both sides but I couldn't find a common expression.
on ...
1
vote
1answer
32 views
On Conditional Jensen Inequality Hypothesis
**
Conditional Jensen Inequality
Let $X$ be in $L^1$, and $\mathcal{G}$ a sigma-algebra of the space and $\phi$ a convex function. Then, $\mathbb E(\phi(X)|\mathcal{G}) \geq \phi(\mathbb E(X|\mathcal{...
1
vote
1answer
20 views
Consistency of Sylvester's Determinant Theorem under Applying Jensen's Inequality
Sylvester's determinant theorem states that for matrices $A\in\mathbb{R}^{n\times d}, B\in\mathbb{R}^{d\times n}$:
\begin{equation}
\det(I_{n}+AB)=\det(I_d+BA)
\end{equation}
In my case I consider $...
2
votes
1answer
31 views
The Jensen gap $\mathbb{E}[|\overline X|] - |\mu|$
Let $ X_1, X_2, \dots, X_n $ be a sequence of i.i.d. random variables with finite mean $ \mu $ and variance $ \sigma^2 $. Let $ \overline X = \frac{1}{n}\sum_{i=1}^n X_i $ denote the sample average. I ...
3
votes
1answer
67 views
Concativity of entropy without Jensen's inequality
In my information theory class I need to prove that entropy is concave (which is usually done with Jensen's inequality). But I want to use only the definition of entropy. And as the result of ...
0
votes
0answers
34 views
Another proof of Jensenās Inequality (finite form)
I am reading through Jensenās inequality and its proofs. I want to find an alternate proof to the tangent line proof.
My attempt:
If $f(x)$ is a convex function, then $\{x_{i},f(x_{i})\}\ ,\ i=1,2,.....
3
votes
1answer
85 views
Sum Infinite Random Variables
Let's say we generate $n$ samples independently from two independent distributions $X$ and $Y$. We know that the following is true from Jensen's Inequality:
$$\
E\left[\min\left(\sum_{i=1}^{n}X_i, \...
0
votes
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 ...
0
votes
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 $$ \log( tx_1+(1-t)x_2) \le t \log(...
0
votes
1answer
42 views
Construct an Unbiased Estimator
I am currently trying to prove that $\hat{\beta}$ is not an unbiased estimator of $\beta$. After proving this I need to to construct a unbiased estimator for $\beta$. I know that $$\hat{\beta} = \frac{...
3
votes
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 ...
0
votes
0answers
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 ...
0
votes
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:
$$KL(f,g) = E_f\left[\ln\left(\frac{f(...
2
votes
1answer
73 views
Inequality from Israel TST
Let $a, b, c, d$ be nonnegative numbers such that $a+b+c+d=18.$
Prove that:
$$\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}}$$
These are my ...
2
votes
0answers
54 views
Proof that $\tan(\sin x) > \sin(\tan x), x \in (0, \pi/2)$ [duplicate]
Assuming $x \in (0, \frac{\pi}{4})$ Write down a proof that $F(x) > 0 $ for all $x$'s where $F(x) = \tan(\sin x)-\sin(\tan x)$.
All I came up with was using Jensen inequity for 1st derivative ...
0
votes
1answer
24 views
Lyapunov's CLT Limit Condition
I am trying to show that Lyapunov's condition holds in Lyapunov's CLT, and am left with the trying to show that for some $\delta >0$
$$\underset{n\rightarrow\infty}{lim} \frac{\sum_{i=1}^n w_i^{2+\...
2
votes
1answer
38 views
An inequality for the mgf using Jensenās 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 ...
0
votes
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 ...
-1
votes
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
votes
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\}.\...
0
votes
0answers
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 :
$$\Big(\sin(a)^a\sin(b)^b\sin(c)^c\Big)^{\frac{1}{\pi}}\leq \frac{3\sqrt{3}}{2\pi}\Big(a^{\...
0
votes
2answers
72 views
How to prove that $\prod_{i=1} ^ \infty x_i^{a_i} \leq \sum_{i=1} ^ \infty a_ix_i$ [duplicate]
Let $a_1,a_2,...$ be nonnegative numbers whose sum is $1$ and let $x_1,x_2,...>0$. I want to show that $\prod_{i=1} ^ \infty x_i^{a_i} \leq \sum_{i=1} ^ \infty a_ix_i$.
This looks awfully similar ...
1
vote
3answers
93 views
Prove that $\frac{1}{\sqrt{a+b+2}}+\frac{1}{\sqrt{b+c+2}}+\frac{1}{\sqrt{c+d+2}}+\frac{1}{\sqrt{d+a+2}}\le 2$
Let $a,b,c,d\in \mathbb{R^+}$ such that $abcd=1$. Prove that $$\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=\...
0
votes
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-...
1
vote
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
votes
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
$$\sum_{cyc} \frac a{\sqrt{a^2+3b^2+3c^2}}=
\frac a{\sqrt{a^2+3b^2+3c^2}}+\frac{b}{\sqrt{3a^2+b^2+3c^2}}+\frac{c}{\sqrt{3a^2+3b^2+c^2}}\le\frac3{\sqrt7}.$...
3
votes
1answer
73 views
Sum of indicators and application of Jensen's inequality
So I have stumbled upon this problem.
Let $X_1, \dots, X_n \sim N(\mu, \sigma^2)$ be iid. Define:
$$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 $...
0
votes
0answers
15 views
Wanted: an inequality with variable exponent
[Wanted] I want to estimate over the following quantity
$ | a + b | ^ {2-2p (x)}$
Where:
ā¢ $a$ and $b$ are vectors in $\mathbb{R} ^{n}$ (!?)
ā¢ $p$ is a function $C^{1}$ in a limited domain in $\...
