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Questions tagged [jensen-inequality]

this tag is meant for question related to the Jensen inquality.

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Counterexample of the converse of Jensen's inequality

Let $\phi$ be a convex function on $(-\infty, \infty)$, $f$ a Lebesgue integrable function over $[0,1]$ and $\phi\circ f$ also integrable over $[0,1]$. Then we have: $$\phi\Big(\int_{0}^{1} f(x)dx\...
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21 views

Generalize Jensen's Integral Inequality to the product of two functions

Let $E$ be a measurable set with $m(E)>0$. Let $f$, $\gamma$ be two measurable, real-valued function which are finite a.e. on $E$ with $f, \gamma$ and $f\cdot \gamma$ all integrable. Assume $\...
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2answers
45 views

Inequality using Jensen

It's a simple problem that I propose to you this is the following : Let $a,b,c,d,e$ be positive real numbers such that $abc=ab+bc+ca$ then we have : $$\frac{1}{da+eb}+\frac{1}{db+ec}+\frac{1}{dc+...
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1answer
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$\frac{a}{\sqrt{a^2 + 8bc}} +\frac{b}{\sqrt{b^2 + 8ac}} + \frac{c}{\sqrt{c^2 + 8ab}} \ge \frac{1}{\sqrt{a^3+b^3+c^3 + 24abc}}$ is true?

In one of the solutions of a problem in this site: https://artofproblemsolving.com/wiki/index.php?title=2001_IMO_Problems/Problem_2 It is used the following: If $a,b,c$ are positive real numbers ...
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Jensen's Inequality, convex function [duplicate]

f is a convex function. Prove that $\frac{f(x) + f(y) + f(z)}{3}$ + $f$($\frac{x+y+z}{3}$) $\geq$ $\frac{2}{3}$( $f$ ($\frac{x+y}{2}$) + $f$ ($\frac{x+z}{2}$) + $f$ ($\frac{y+z}{2}$) ). Starting with ...
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1answer
21 views

Is it a typo in Jensen's inequality?

I am reading a book where it is written: For any $n \geq 1$, $E(|x|^n) \geq (E(|x|))^n$ I understand that it is simple apllication of Jensen' inequality for function $f(x) = |x|^n$, but I have some ...
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1answer
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$\int_\Omega\min(f,g)\text{d}\mu\ge\frac{1}{2}\big(\int_\Omega\sqrt{f\cdot g}\ \text{d}\mu\big)^2$

Let $(\Omega,\mathcal A,\mu)$ be a $\sigma$-finite measure space, $\mathbb P,\mathbb Q$ be probability measures on $(\Omega,\mathcal A)$ with density functions $f,g: \Omega \to (0,\infty)$ concerning $...
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15 views

Jensen type inequality for a non-convex function.

I suppose that a function $f$ is $\geq 0$ on $[-1,1]$, decreasing and $f(t)(1+t)$ is concave. Moreover for every $a,b \in [-1,1]$, $a<b$, we have a (simple positive) measure $\mu_{a,b}$ such that $...
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1answer
26 views

How can I solve this inequality using convexity?

Given $a, b, c\ge 0$ and $x, y, z> 0$ and $a + b + c = x + y + z$. Show that $$a ^ 3 / x ^ 2 + b ^ 3 / y ^ 2 + c ^ 3 / z ^ 2 \ge a + b + c$$ prove inequality using convexity
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A question in random matrices

In page 78 of "An Introduction to Matrix Concentration Inequalities", it is written that if $Z$ be a random matrix, I can't understand why $$\mathbb{E}||Z||^2=\mathbb{E}\text{max}\{||ZZ^*||,||Z^*Z||\}...
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2answers
33 views

proof that involves Jensen's inequality

An exercise I have to solve indicates the following: "Show that for any discrete random variable X: e^tE[X]<= E[e^tX] where t belongs to R and is fixed. I think this is related with Jansen's ...
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3answers
84 views

Prove that if $x_1 + x_2 + … + x_n = n$, then $x_1^k + x_2^k + … + x_n^k \ge n$

$x_1, x_2, ..., x_n \in \mathbb R$ are nonegative and $k \in \mathbb R$, $k \ge 1$. Prove that if $x_1 + x_2 + ... + x_n = n$, then $x_1^k + x_2^k + ... + x_n^k \ge n$. I tried to find the smallest ...
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2answers
42 views

Can we extend convex combination to integral?

The definition of the convex combination is given as: $$x=\lambda x_1 + (1-\lambda) x_2,\forall \lambda \in [0,1]$$ It can be obviously extend to multi-terms form as: $$x=\sum_{i=1}^N \lambda_i x_i,\...
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problem regarding application of Jensen's inequality

question: For $a,b,c,d \in \mathbb{R^+}$ with $a+b+c+d = 4$, Prove $\displaystyle \sum\dfrac{a}{b(b+1)}\geq \dfrac{8}{(a+c)(b+d)}$ my attempt: $f(x)= \dfrac{1}{x^2+x}=\dfrac{1}{x(x+1)}$ is convex ...
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1answer
41 views

What Jensen inequality implies in it holds with equality

For any measurable function $f$, Jensen inequality states that $\int \phi (f) d\mu \geq \phi ( \int f d\mu)$ for convex $\phi$. Is there a proof that if the inequality holds with equality, that is $...
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1answer
124 views

Proving a convexity inequality

Given $f: \mathbb{R} \to \mathbb{R}$ convex, show that: $$ \frac{2}{3}\left(f\left(\frac{x+y}{2}\right) + f\left(\frac{z+y}{2}\right) + f\left(\frac{x+z}{2}\right)\right) \leq f\left(\frac{x+y+z}{3}\...
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1answer
59 views

Can I minimize $x^a + y^a$ by minimizing $x+y$? Is there such a mathematical identity?

I would like to minimize the following expression: $$x^a+y^a.$$ I wonder if a mathematical identity exists where a minimization of $x+y$ implies a minimization of the above. Where: a is a positive ...
4
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1answer
111 views

Prove: for all $x \in (0, 1], 2^x+2^{\frac{1}{x}} \leqslant 2^{x+\frac{1}{x}}$

This problem is from my math teacher.I tried using Calculus, the derivative function is like a black hole.Then I graphed it by Mathematica. As the following picture shows, I was strongly astonished. ...
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1answer
46 views

Bounding the variance of the maximum of convex functions of a random variable

Consider a random variable $X$ with density $f$, and a finite set of convex and increasing functions $Y_i(x)$. I am interested in bounding the variance of the max of the $Y_i$'s. I have the following ...
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Karamata + Jensen = Niculescu's inequality (version of 1991)

In fact it's my point of view but I think that this version of Niculescu's inequality is a mixture between Karamata and Jensen's inequality . So we have : Let $f(x)$ be a convex strictly ...
2
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1answer
74 views

Show that $\sum_{\text{cyc}} \frac{1}{b^2+c^2+5bc-a^2} \leq \frac{\sqrt3}{8S}$ for a triangle with sides $a$, $b$, $c$ and area $S$

Let be $a$, $b$, $c$ sides of a triangle and $S$ his area. Prove that $$\sum_{\text{cyc}} \frac{1}{b^2+c^2+5bc-a^2} \leq \frac{\sqrt3}{8S}$$ My idea: $b^2+c^2-a^2 = 2bc \cos A$, so the inequality is ...
4
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2answers
130 views

P.T. $\frac{1}{a^3(b+c)} +\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)} \ge \frac 32$

If $abc=1$ where $a,b,c$ are positive real. Prove that ,$\frac{1}{a^3(b+c)} +\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)} \ge \frac 32$. I tried to multiply the LHS by $abc$ to make the relation homogeneous ...
35
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3answers
1k views

Prove that $(1+x)^\frac{1}{x}+(1+\frac{1}{x})^x \leq 4$

Prove that $f(x)=(1+x)^\frac{1}{x}+(1+\frac{1}{x})^x \leq 4$ for all $x>0.$ We have $f(x)=f(\frac{1}{x}), f'(x)=-\frac{1}{x^2}f'(\frac{1}{x}),$ so we only need to prove $f'(x)>0$ for $0 < x <...
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0answers
51 views

How to introduce weights for Jensen's inequality into $log\sum\sum$

For log sum matrix function, I saw the example introducing the weight after sum regarding all matrices in summation for Jensen's inequality \begin{align*} f(W,H) &= \sum_{ij}[-V_{ij}log \sum_{k}...
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1answer
50 views

Generalization of $f\left(\frac{x_1 + x_2}{2}\right) < {f(x_1) + f(x_2) \over 2}$ for odd number of variables.

Let $f: \mathbb R \to \mathbb R$ satisfy the following inequality: $$ f\left(\frac{x_1 + x_2}{2}\right) < {f(x_1) + f(x_2) \over 2} $$ Show that: $$ f\left(\frac{x_1 + x_2 + x_3}{3}\...
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1answer
60 views

How to show that $f(x) = a^x$ is convex for $x \in \mathbb R$ using Jensen's inequality?

A function is called convex downwards if the following inequality is satisfied: $$ f(\lambda\cdot x_1 + (1-\lambda)\cdot x_2) \le \lambda\cdot f(x_1)+(1-\lambda)\cdot f(x_2), \; \text{where} \; \...
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2answers
58 views

For positive real numbers $x,y,z$, Prove that

For positive real numbers $x,y,z$, Prove that, $$\bigg(\frac{x^2+y^2+z^2}{x+y+z}\bigg)^{(x+y+z)}\geq x^xy^yz^z \geq \bigg(\frac{x+y+z}{3}\bigg)^{(x+y+z)}$$ I have no idea how to begin this? I ...
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0answers
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Jensen's inequality applied to Liapunov's CLT condition

The Liapunov's sufficient condition to the Central Limit Theorem says that if $\exists \delta$ for whom the following expression converges to $0$ then you can apply the CLT to the sequence's ...
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1answer
55 views

Does Jensen's Inequality hold for complex numbers?

We can use Jensen's inequality to show that if $1<p<\infty$, then there exists a constant $C>0$ such that for every $x,y \in \mathbb{R}$, we have: $$ |x+y|^p \leq C (|x|^p + |y|^p) $$ Can we ...
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1answer
59 views

Can we always find a convex combination of a given set of points to satisfy this condition?

Let $F$ be the average of $f(x)$ for $m$ given vectors $x_1,\dots,x_m$, that is,$$ F=\frac{1}{m}\sum_{i=1}^m f(x_i). $$ Here $f(x)$ is a convex function from $\mathbb{R}^n$ to $\mathbb{R}$. Can we ...
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0answers
138 views

Proof verification in a general case of an hard inequality

Related to this I have solved the general case which is the following: If $a_1,\cdots,a_n >0$, and $a_1+\cdots + a_n=n$, prove that $$\sum_{cyc}a_i^{a_i a_{i+1}} \geq (\sum_{i=1}^{n}a_ia_{i+1}...
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2answers
33 views

Denominator in Jensen's inequality

What is the intuition behind having a denominator in Jensen's inequality? For example, why do we need $n$ in a version of Jensen's inequality when all weights are equal: $$ \phi\left(\frac{\sum x_i}{n}...
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1answer
177 views

Show that $E[1/X] = 1/E[X]$ using Jensen's inequality

The function $1/x$ is convex on the interval $x > 0$ and two times differentiable. So, for $x > 0$, Jensen's inequality implies $$E(1/x) \geq 1/E(x).$$ But for $x < 0$, $1/x$ is not ...
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42 views

What inequalities can be obtained for summation of piecewise linear convex functions?

Let $f_i(x):\mathbb{R}^n\to\mathbb{R}$ be a piece-wise linear convex function with $i=1,\dots,m$. We know that $F(x)=\sum_{i=1}^m f_i(x)$ is also a piece-wise linear convex function (see this post). I ...
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1answer
36 views

An inequality on the geometric mean of sines

Let $n \in \mathbb{N}$, $n \geq 2$. Let $x_1, \ldots, x_n \in (0, \pi)$. Set $x = \frac{(x_1 + \cdots + x_n)}{n}$. Which of the following statements are true? (b) $\prod_{k=1}^n \sin x_k \leq \sin^...
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1answer
55 views

Looking for an inequality for $1 \leq p < \infty$

Let $a_1,...,a_n$ be positive real numbers and let $0 < p < 1$. Then $$(a_1 + \cdots + a_n)^p \leq a_1^p + \cdots + a_n^p. $$ Now take $ 1 \leq p < \infty$. Can we get a similar inequality, ...
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144 views

The Gudermannian function , its inverse and Jensen's inequality: in search of inequalities

In this post $\operatorname{gd}(x)$ denotes the Gudermannian function and $\operatorname{gd}^{-1}(x)$ its inverse. Look at the related Wikipedia's article to know the domain of these functions. The ...
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1answer
137 views

When do we have equality in Jensen's inequality?

I'm talking about the finite form of the inequality: $$f(q_1x_1+q_2x_2+\cdots+q_nx_n)\leq q_1f(x_1)+q_2f(x_2)+\cdots+q_nf(x_n)$$ with $$\sum_{i=1}^{n}{q_i}=1, q_i\geq 0$$ (Obviously the form ...
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1answer
66 views

Jensen's inequality for random functions in a Banach space

Suppose I have a Banach space of functions over $\mathbb{R}$ with norm $||\cdot||$. Suppose that $f$ is a random function that takes values in this space such that $||f||\leq 1$. Suppose that for all $...
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1answer
75 views

Does the following integral inequality hold?

For any square, symmetric, positive definite matrix $M\in\Re^{m\times{m}}$, a scalar $\gamma$ and a vector valued function $\omega:[0,\gamma]\to\Re^{m}$ such that the following inequality holds \begin{...
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0answers
120 views

Generalisation of Jensen's inequality for two variables?

Consider the finite form of Jensen's inequality. For the convex case ($\phi>1$): $$ \frac{\sum \delta_ix_i^\phi}{\left(\sum \delta_i x_i\right)^\phi} > 1 $$ and for the concave case ($\phi &...
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1answer
54 views

Jensen inequality for matrix exponential of positive definite matrix (laplacian)

I have a positive definite matrix $n\times n$ matrix $\mathbf{L}$. This matrix is the combinatorial Laplacian of a graph. We can compute the expected value of the Laplacian as $\mathbb{E}[\mathbf{L}]...
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2answers
86 views

Prove that $E(e^{X}) < \frac{1}{2}(e^{-K} + e^{K})$ if $E(X)=0$ and $|X|<K$ almost surely

I saw this question on a friend's problem set (I mean it, not MY homework) and kept thinking about it, although without success. Let $X$ be a a random variable such that $ -K < X < K$ for ...
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1answer
95 views

Symmetric inequality with three variables including radicals

Let $a, b, c$ be positive reals. Show that the following inequality holds: $$\frac{a}{\sqrt{a^2 + b^2}} + \frac{b}{\sqrt{b^2 + c^2}} + \frac{c}{\sqrt{c^2 + a^2}} \le \frac{3}{\sqrt2}$$ I managed to ...
0
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1answer
35 views

Expected value inequality with zero-mean variable

Let $X,Y$ be independent real valued random variables. In addition $\mathbb{E}[Y] = 0$ and $p\geq 1$. Then $$ \mathbb{E}[ |X|^p] \leq \mathbb{E}[ |X+Y|^p] $$ How can I prove this? I tried to use ...
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1answer
55 views

Optimal probability distribution minimising a function

Consider a discrete probability distribution $P_i$ over $i$ belonging to set of non-negative integers. The question is which distribution minimises the function $\sum_i P_ii^2$, such that $\sum_i ...
2
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2answers
56 views

Proving the inequality $\int_X f\ln f d\mu-\int_X f d\mu\ln \int_X f d\mu\ge \int_X f\ln g d\mu-\int_X f d\mu\ln \int_X g d\mu$

Let $f,g$ be positive integrable functions on the measure space $(X, \mathcal A, \mu )$. Such that $fln f $ and $fln g $ are also integrable $$\int_X f\ln f d\mu-\left(\int_X f d\mu\right)\ln\...
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1answer
55 views

Jensen inequality on square root

For a concave function $f$, Jensen inequality states that $$ f(ax+by) \geq af(x) + bf(y) $$ Now, $\sqrt{x}$ is a concave function, hence by Jensen inequality $$ \sqrt{x+y} \geq \sqrt{x} + \sqrt{y} $...
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1answer
113 views

Inequality proof using Cauchy-Schwarz and Jensen's inequality

I am trying to understand a proof of a simple lemma from this paper (lemma 3) which I paste here (refer to the 1st inequality, $H$ is a Hilbert space): The first step in the proof is to use ...
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1answer
42 views

When is the integrand in Jensen's formula actually integrable?

I was looking at an integrated form of Jensen's inequality for holomorphic functions: $$ \ln|f(0)|\le\frac1{\pi}\int_0^1\int_0^{2\pi}\ln|f(re^{i\theta})|r\text{d}\theta\text{d}r $$ (which can be found ...