# 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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### 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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### 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 ...
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### 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 ...
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### $f(0)=f(1)=0$, $f(x)=\frac{f(x+h)+f(x-h)}{2}$ 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 ...
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 $\sum_{i=1}^k\exp(-x_i) \geq k.\exp(-\... 1answer 96 views ### Some doubts in a part of the proof of Backwards Martingale Convergence Theorem (Jacod-Protter) A USEFUL RESULT (Doob's Upcrossing Inequality) Let$(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 $$\mathbb{E}\{U_n\}\... 0answers 25 views ### References for variants of Jensen inequality I am looking for a reference for the following claim: Let X be a probability space, and let g:X \to \mathbb [0,\infty) be in L^1(X). Let \phi:\mathbb [0,\infty) \to [0,\infty) be convex and ... 0answers 127 views ### Conjecture about Jensen's inequality and polynomials Hi it's related to the following conjecture An inequality for polynomials with positives coefficients : We have the first conjecture : Let x,y>0 then we have :$$(x+y)f\Big(\frac{x^2+... 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 $is needed. I tried using Jensen's inequality but that did not work much as ... 1answer 71 views ### Question regarding Jensen Inequality Following is the picture of the question regarding the application of Jensen Inequality. Following is the picture my approach to proove the inequality. Can anyone please check if my proof is ... 2answers 130 views ###$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 ... 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. ... 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 $$\mathbb{E}(G(U))... 1answer 78 views ### If x+y+z=1 prove \sqrt{x+\frac{(y-z)^{2}}{12}}+\sqrt{y+\frac{(z-x)^{2}}{12}}+\sqrt{z+\frac{(x-y)^{2}}{12}} \leq \sqrt{3} Question - Let x, y, z be non-negative real numbers with sum 1 . Prove that$$ \sqrt{x+\frac{(y-z)^{2}}{12}}+\sqrt{y+\frac{(z-x)^{2}}{12}}+\sqrt{z+\frac{(x-y)^{2}}{12}} \leq \sqrt{3} $$My work -... 0answers 22 views ### What is the proof of entropy property: H\left(x,y\right)\le H\left(x\right)+H\left(y\right) in Shannon's paper? In Shannon's 1948 paper titled "A Mathematical Theory of Communication", in the discussion of the entropy of the joint event, there is no proof for this inequality (or subadditivity of entropy)$$H\... 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 $$\begin{array}{c} \frac{1}{(1+a)^{2}}+\frac{1}{(1+b)^{2}}+\frac{1}{(1+c)^{2}}+\frac{1}{(1+d)^{2}} \geq 1 \\ \... 4answers 61 views ### MOP 2011 inequality If a,b,c are positive integers prove that \sqrt{(a^2-ab+b^2)} +\sqrt{(b^2+c^2-bc)} +\sqrt{(a^2+c^2-ac)} +9(abc)^{1/3} \le 4(a+b+c) My attempt: I tried to split inequality and prove it bit by ... 1answer 64 views ### If f(\frac{t_1+t_2}{2}) \leq \frac{f(t_1)+f(t_2)}{2}, show that f(\frac{t_1+t_2+ \cdots +t_n}{n})\leq \frac{f(t_1)+f(t_2)+\cdots f(t_n)}{n} [duplicate] If f is a continuous function on [0,1] such that for all t_1, t_2 \in [0,1],$$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}\... 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 : $$... 0answers 35 views ### Explanation for the solution associated with Jensen's inequality While I was surfing AoPS, I found this problem and its solution: But I don't understand the application of Jensen's integral inequality (inequality on the first line of bengabriel's solution). Jensen'... 1answer 49 views ### Prove some inequality using Jensen's inequality How do I prove this using Jensen's inequality?$$\biggl|\prod_{i=1}^n x_{i}\biggl|^p \le {n^{p-1}}\sum_{i=1}^n |x_{i}|^p$$I've tried log on both sides but I couldn't find a common expression. on ... 1answer 32 views ### On Conditional Jensen Inequality Hypothesis ** Conditional Jensen Inequality Let X be in L^1, and \mathcal{G} a sigma-algebra of the space and \phi a convex function. Then, \mathbb E(\phi(X)|\mathcal{G}) \geq \phi(\mathbb E(X|\mathcal{... 1answer 20 views ### Consistency of Sylvester's Determinant Theorem under Applying Jensen's Inequality Sylvester's determinant theorem states that for matrices A\in\mathbb{R}^{n\times d}, B\in\mathbb{R}^{d\times n}: \begin{equation} \det(I_{n}+AB)=\det(I_d+BA) \end{equation} In my case I consider ... 1answer 31 views ### The Jensen gap \mathbb{E}[|\overline X|] - |\mu| Let X_1, X_2, \dots, X_n be a sequence of i.i.d. random variables with finite mean \mu and variance \sigma^2 . Let \overline X = \frac{1}{n}\sum_{i=1}^n X_i denote the sample average. I ... 1answer 67 views ### Concativity of entropy without Jensen's inequality In my information theory class I need to prove that entropy is concave (which is usually done with Jensen's inequality). But I want to use only the definition of entropy. And as the result of ... 0answers 34 views ### Another proof of Jensen’s Inequality (finite form) I am reading through Jensen’s inequality and its proofs. I want to find an alternate proof to the tangent line proof. My attempt: If f(x) is a convex function, then \{x_{i},f(x_{i})\}\ ,\ i=1,2,..... 1answer 85 views ### Sum Infinite Random Variables Let's say we generate n samples independently from two independent distributions X and Y. We know that the following is true from Jensen's Inequality:$$\ E\left[\min\left(\sum_{i=1}^{n}X_i, \... 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 ... 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 $$\log( tx_1+(1-t)x_2) \le t \log(... 1answer 42 views ### Construct an Unbiased Estimator I am currently trying to prove that \hat{\beta} is not an unbiased estimator of \beta. After proving this I need to to construct a unbiased estimator for \beta. I know that$$\hat{\beta} = \frac{... 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 ... 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 ... 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: $$KL(f,g) = E_f\left[\ln\left(\frac{f(... 1answer 73 views ### Inequality from Israel TST Let a, b, c, d be nonnegative numbers such that a+b+c+d=18. Prove that:$$\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}}$$These are my ... 0answers 54 views ### Proof that \tan(\sin x) > \sin(\tan x), x \in (0, \pi/2) [duplicate] Assuming x \in (0, \frac{\pi}{4}) Write down a proof that F(x) > 0 for all x's where F(x) = \tan(\sin x)-\sin(\tan x). All I came up with was using Jensen inequity for 1st derivative ... 1answer 24 views ### Lyapunov's CLT Limit Condition I am trying to show that Lyapunov's condition holds in Lyapunov's CLT, and am left with the trying to show that for some \delta >0$$\underset{n\rightarrow\infty}{lim} \frac{\sum_{i=1}^n w_i^{2+\... 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 ... 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$is a ... 1answer 59 views ### Prove special case of Jensen's inequality THE OPPOSITE WAY [duplicate]$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 ... 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\}.\... 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 : $$\Big(\sin(a)^a\sin(b)^b\sin(c)^c\Big)^{\frac{1}{\pi}}\leq \frac{3\sqrt{3}}{2\pi}\Big(a^{\... 2answers 72 views ### How to prove that \prod_{i=1} ^ \infty x_i^{a_i} \leq \sum_{i=1} ^ \infty a_ix_i [duplicate] Let a_1,a_2,... be nonnegative numbers whose sum is 1 and let x_1,x_2,...>0. I want to show that \prod_{i=1} ^ \infty x_i^{a_i} \leq \sum_{i=1} ^ \infty a_ix_i. This looks awfully similar ... 3answers 93 views ### Prove that \frac{1}{\sqrt{a+b+2}}+\frac{1}{\sqrt{b+c+2}}+\frac{1}{\sqrt{c+d+2}}+\frac{1}{\sqrt{d+a+2}}\le 2 Let a,b,c,d\in \mathbb{R^+} such that abcd=1. Prove that$$\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=\... 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-... 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$E(g(X)) \le g(E(X))$. ... 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 $$\sum_{cyc} \frac a{\sqrt{a^2+3b^2+3c^2}}= \frac a{\sqrt{a^2+3b^2+3c^2}}+\frac{b}{\sqrt{3a^2+b^2+3c^2}}+\frac{c}{\sqrt{3a^2+3b^2+c^2}}\le\frac3{\sqrt7}.... 1answer 73 views ### Sum of indicators and application of Jensen's inequality So I have stumbled upon this problem. Let X_1, \dots, X_n \sim N(\mu, \sigma^2) be iid. Define:$$S = \frac{1}{n}\sum_{i=1}^n I[X_i > a]T = I[\frac{1}{n}\sum_{i=1}^n X_i > a]$$a > ... 2answers 60 views ### If f'' \ge 0, \int_0^2 f(x)dx \ge 2f(1) Question: If f'' \ge 0 in interval [0, 2], prove that$$\int_0^2 f(x)dx \ge 2f(1)$$The question is graphically trivial I think, but not in mathematically. I wanted to use the fact that there ... 3answers 250 views ### An exponential equation over positive real numbers 4^x+14^x+3^x=11^x+10^x Solve the following equation over the positive reals:$$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$...
[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 \$\...