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Questions tagged [jensen-inequality]

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

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1answer
38 views

Bounding integral of square root by square root of integral

Let $f(x)\geq 0$ be a function over $[0,\infty)$. How can I lower bound $\int_{x=0}^{u}\sqrt{f(x)}dx$ by $c \sqrt{\int_{x=0}^{u}f(x)dx}$ where $\sqrt{\int_{x=0}^{u}f(x)dx}<\infty$ and $c>0$ is ...
1
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1answer
23 views

Showing $\sum_{v \in V} \deg(v)^2 \in O(m^2/n)$

Given an undirected graph $G = (V,E)$, where $|V| = n$ and $|E| = m$, I am trying to show that $\sum_{v \in V} \deg(v)^2 \in O(m^2/n)$. Using Jensen's inequality, we can show that $$\sum_{v \in V} \...
2
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0answers
29 views

Variation of Jensen's inequality

I have a more general question: can we always say that, given a convex function $g$ and some $h$ s.t. $\sum_{x\in\mathcal{X}}h(x)=1$, $A\subset\mathcal{X}$ $$\sum_{x\in A} g(x)h(x)\geq g\left(\sum_{x\...
1
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0answers
27 views

Inequality with log-sum

I'm interested by the following problem : Let $a,b,c,d>0$ such that $a+b+c+d=4$ then we have : $$ab\ln(a)+bc\ln(b)+cd\ln(c)+da\ln(d)> -1.37$$ My try for $b\geq 3,a\leq1,c\leq 1,d\leq1$: ...
1
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1answer
47 views

Reverse of Jensen's Inequality

If $E[v(x)] \geq v(E[X])$ for every random variable $X$, then $v$ is convex. I know that a function $v(x)$ is convex iff for every $x_0$ a line, we have $l_0(x) = a_0x + b_0$ exists such that $l_0(x_0)...
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0answers
19 views

Error check in book - Jensen inequality and binomial coefficient

I'm reading "Ramsey Theory" by Graham, Rothschild and Spencer, 2nd edition (see here). Page 111 & 112, they state For a given $a$, the function $f:x\mapsto \binom{x}{a}$ is concave For a given ...
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0answers
16 views

Power sum refinement inequality

We start with the following inequality : Let $a,b,c,d>0$ such that $a+b+c+d=4$ $$\sum_{cyc}a^{ab}\geq (Q)e^{\frac{P}{3(Q)}}>\pi$$ With : $$P=\sum_{cyc}a^{ab0.75}\ln(a^{ab0.75})$$ And $$Q=\...
2
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1answer
52 views

Real Analysis How to Prove Properties of Natural Logs and Integrals

Here is the question I am trying to answer: Let $f:[0,1] \to\Bbb R$ be a Riemann integrable function with $f \ge c>0$. Prove that $$\int_0^1\ln(f(x))\ dx\le \ln\left(\int_0^1 f(x)\ dx\right).$$ I ...
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3answers
84 views

Maximum value of expression $a+b+c$

If $a,b,c$ are non negative integers such that $$2(a^3+b^3+c^3)=3(a+b+c)^2.$$ Then maximum value of $a+b+c$ is ? My Try: Using Jensen Inequality Let $f(x)=x^3$. Then $f''(x)>0$ for $x>...
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1answer
44 views

Jensen's inequality with supremum

Problem In a paper I am reading now, the author claims that by Jensen's inequality, they have $$ \frac { 1 } { \lambda } \log \exp \left( \lambda \cdot \mathbb { E } _ { \epsilon } \sup _ { h \in \...
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1answer
25 views

Comparing inequalities using convexity of the function

There was a question about comparing two entropies and showing that one of them is greater than or equal to other. After getting rid of the same terms for both expressions, I am left with the ...
2
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2answers
69 views

Proving inequality using Jensen

Studying for a convex optimization exam I encountered the below question. I suspect the inequality can be proved using Jensen inequality with the function $f(x)=-\ln(1-x)$ and $x_i=a_i$ but can't work ...
2
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2answers
32 views

Prove that $ (\frac{\sum_{i=1}^n x_i}{n})^{\sum_{i=1}^n x_i} \le \prod_{i=1}^n {x_i}^{x_i}$ $, \forall x_i>0, n\ge1 $

Prove that $ (\frac{\sum_{i=1}^n x_i}{n})^{\sum_{i=1}^n x_i} \le \prod_{i=1}^n {x_i}^{x_i}$ $, \forall x_i>0, n\ge1 $ (The second sum in the left-hand side of the inequality is an exponent) I've ...
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2answers
37 views

Proving a (seemingly simple) inequality [duplicate]

I want to show that for $p \in (0,1)$,$(x +y)^p \leq x^p +y^p$. I thought of doing this: Since $p \in (0,1)$, then $\frac{1}{p} \in (1,\infty)$. I can then raise both sides of the inequality to the $\...
2
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2answers
91 views

Find maximum of $\cos(x)\cos(y)\cos(z)$ when $x + y + z = \frac{\pi}{2}$ and $x, y, z > 0$

Find maximum value of P = $\cos(x)\cos(y)\cos(z)$, given that $x + y + z = \frac{\pi}{2}$ and $x, y, z > 0$. Effort 1. I drew a quarter-circle, divided the square angle into three parts, and ...
1
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1answer
63 views

Generalization of Nesbitts's inequality

Let some (fixed) real $k >0$ and positive reals $a,b,c$. Consider the conjecture $$ \left(\frac{a}{b+c}\right)^k +\left(\frac{b}{a+c}\right)^k+\left(\frac{c}{a+b}\right)^k \geq \min \{\frac{3}{2^k} ...
4
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2answers
140 views

A conjecture about power sum : $e^{ab}+e^{bc}+e^{ca}\geq 3e^{\sqrt{abc}}$ and $a+b+c=3$

Hello I have this to propose : Let $a,b,c$ be real positive numbers such that $a+b+c=3$ then we have : $$e^{ab}+e^{bc}+e^{ca}\geq 3e^{\sqrt{abc}}$$ For a generalization I have this conjecture : ...
0
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1answer
67 views

Kovari-Sos-Turan Theorem - proof

recently I'm facing problems while trying to prove the following theorem: Theorem: Let Ka,b denote the complete bipartite graph with a and b vertices in its color-classes. Then: $$ex(n,K_{a,b}) \leq \...
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5answers
166 views

How can I prove that $\frac{e^x+e^y}{2}>e^\frac{x+y}{2}$, where $x \neq y$?

The inequality is $$\frac{e^x+e^y}{2}>e^\frac{x+y}{2}$$ where $x \neq y$. This is my first time coming across inequalities of this form thus I really don't know to approach it correctly. Here is ...
1
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0answers
43 views

A new strategy to prove a problem

It's related to this If $a+b+c=3$, then $ \frac{a}{5b+c^3}+\frac{b}{5c+a^3}+\frac{c}{5a+b^3} \geq \frac{1}{2}$ My idea to prove this : We have the following theorem : Let $x,\beta,\alpha$ be real ...
1
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1answer
36 views

Can Jensen be applied to multivariable functions?

I was wondering, suppose we have a symmetric two variable function, in my case: $f(x,y)=\frac{1}{x+y+1}$ with the restriction that $0\le x,y \le 1$, I took the second partial derivative with respect ...
1
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1answer
51 views

How can we show that $\left|a+b+c\right|^p-2\left|a\right|^p\le C\left(\left|b\right|^p+\left|c\right|^p\right)$?

Let $p\ge2$. How can we show that $$\left|a+b+c\right|^p-2\left|a\right|^p\le C\left(\left|b\right|^p+\left|c\right|^p\right)\;\;\;\text{for all }a,b,c\in\mathbb R\tag1$$ for some $C\ge0$? I'm only ...
1
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1answer
47 views

An inequality for convex function from $\mathbb R$ to $\mathbb R$

If $f(x): \mathbb R \to \mathbb R$ is a convex function, prove that $$f(x_1) + f(x_2) + f(x_3) + 3 f(\frac{x_1 + x_2 + x_3}{3}) \geq 2 f(\frac{x_1 + x_2}{2}) + 2 f(\frac{x_1 + x_3}{2}) + 2 f(\frac{x_2 ...
3
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3answers
89 views

Proof verification: For $a$, $b$, $c$ positive with $abc=1$, show $\sum_{\text{cyc}}\frac{1}{a^3(b+c)}\geq \frac32$

I would like to have my solution to IMO 95 A2 checked. All solutions I've found either used Cauchy-Schwarz, Chebyshevs inequality, the rearrangement inequality or Muirheads inequality. Me myself, I've ...
13
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3answers
265 views

Prove that $({a\over a+b})^3+({b\over b+c})^3+ ({c\over c+a})^3\geq {3\over 8}$

Let $a,b,c$ be positive real numbers. Prove that $$\Big({a\over a+b}\Big)^3+\Big({b\over b+c}\Big)^3+ \Big({c\over c+a}\Big)^3\geq {3\over 8}$$ If we put $x=b/a$, $y= c/b$ and $z=a/c$ we get $xyz=1$ ...
0
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1answer
46 views

Triple refinement of an inequality

We have the following theorem : Let $a,b,c$ be real positive numbers such that $2b\geq a+c$ then we have : $$A\geq B\geq C \geq D \geq E$$ With : $$A=3\big(\frac{\sum_{cyc}\frac{a^3}{13a^2+5b^...
1
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1answer
38 views

Jensen's type inequality for inner product

Let $\mathcal{H}$ be a real Hilbert space and $x,y,z \in \mathcal{H}$. If $x$ and $y$ are a convex combination of $z$, that is $$ z := tx + (1-t)y , \quad t \in (0,1) , $$ then we have the following ...
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0answers
40 views

Upperbounding the expected value of an L2-norm difference

I am trying to find an upperbound for $$E_I\left[\sum_{k=I}^n a_k^2 - \sum_{k=I}^n b_k^2\right],$$ where the expectation is taken over the random variable $I$. The tuples $a$ and $b$ are real and $I$ ...
1
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1answer
35 views

Lower bound of the entropy over the probability distribution

I am trying to show that: $$\inf_{z: 1^Tz=1} \sum_{i=1}^m {z_i \log z_i} = -\log m$$ I thought about Jensen inequality or induction, but none of them provided me something.
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1answer
63 views

Refinement of a strong inequality

It's related to this If $a+b+c=abc$ then $\sum\limits_{cyc}\frac{1}{7a+b}\leq\frac{\sqrt3}{8}$ . I make a little refinement wich could be usefull to prove the original one . Let $a,b,c$ be real ...
0
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1answer
47 views

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\...
0
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1answer
58 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 $\...
1
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2answers
58 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
80 views

$\frac{a}{\sqrt{a^2 + 8bc}} +\frac{b}{\sqrt{b^2 + 8ac}} + \frac{c}{\sqrt{c^2 + 8ab}} \ge \frac{1}{\sqrt{a^3+b^3+c^3 + 24abc}}$ 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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0answers
21 views

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 ...
1
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1answer
26 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 ...
2
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1answer
50 views

$\int_\Omega\min(f,g)\text{d}\mu\ge\frac{1}{2}\big(\int_\Omega\sqrt{f\cdot g}\ \text{d}\mu\big)^2$

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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0answers
25 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 $...
1
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1answer
27 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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0answers
12 views

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||\}...
0
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2answers
44 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 ...
0
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3answers
95 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 ...
0
votes
2answers
72 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,\...
0
votes
2answers
51 views

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 ...
0
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1answer
103 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 $...
6
votes
1answer
140 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}\...
1
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1answer
60 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
votes
1answer
118 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. ...
2
votes
1answer
64 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 ...
1
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0answers
67 views

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 ...