Questions tagged [jensen-inequality]

For questions about proving and using Jensen's inequality for convex functions. To be used necessarily with the [inequality] tag.

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If $\sum_{cyc}\frac{a}{a+1}=1$ Show that $abc\le \frac{1}{8}$

Given positive reals $a,b$ and $c$, such that $$\sum_{cyc}\frac{a}{a+1}=1$$ Show that $$abc\le \frac{1}{8}$$ This problem is fairly easy we can just clear denominators and then use AM-GM. But since I ...
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Corrected conjecture about a possible inequality $\sum_{i=1}^{n}\sqrt{\frac{x_i+1}{4x_i^2+10x_i+4}}\leq \frac{n}{3}$ .

Hi it's a follow up of Prove $\sum_{cyc}{\sqrt{\frac{x+1}{x^2+16x+1}}}\geqslant 1$ and $ \sum_{cyc}{\sqrt{\frac{x+1}{4x^2+10x+4}}}\leqslant 1$ for $x,y,z>0,xyz=1$ : Problem : Let $x_i>0$ and $n$ ...
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Using Jensen's inequality on $\mathbb{E}[1/x]$ when x can be both positive and negative

We know the function $f(x)=\frac{1}{x}$ is convex when $x$ is positive and concave when $x$ is negative. I want to show if $\mathbb{E}[\frac{1}{x}]$ is bigger than or smaller than $\frac{1}{\mathbb{E}[...
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Simple yet hard inequality with Lipschitz functions

Let $\{w_n\}_{n=1,...N}$ be a sequence of numbers in $\mathbb R$ such that $$\sum_{i=1}^N w_n = 0\qquad \sum_{i=1}^N |w_n| \le 2.$$ Let $L_1$ denote the class of $1-$Lipschitz continuous functions ...
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Show that inequality with concave functions holds

For my calculus course, I have the following problem. I have some troubles to solve it. Suppose we have $f,g:\mathbb{R}\mapsto \mathbb{R}$ that are increasing and concave functions. Prove that for any ...
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Jensens Inequality -- Application

Let $f(g(x),x)$ be twice differentiable function in $x \in [0,1]$ with $d^2f(g(x),x)/dx^2>0$. Given probability density function $h$, let $\bar{x} \equiv \int_0^1 x h(x) dx.$ Does Jensen's ...
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prove modified jensen's inequality

My question is I'm trying to prove modified jensen's inequality so given a convex function f and $$ E= \{\sum_{0}^{n} \lambda_{i}x_{i} | \lambda_{i}>=0,\sum_{0}^{n} \lambda_{i}=1 \} $$ I want to ...
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Positive random variables with harmonic mean equal to mean have variance zero

Suppose $X>0$ is a random variable with $\mathbb{E}X=c$ and $\mathbb{E}X^{-1}=c^{-1}$. By applying Cauchy-Schwarz to $\sqrt{X}$ and $\sqrt{X^{-1}}$ and using the "equality iff linearly ...
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Jensen's Inequality Proof for Conditional Expectation (Durrett)

I have some doubts as I read over Durrett's proof on Jensen's inequality for conditional expectation. The statement is that if $\varphi$ is convex and $E|X|, E|\varphi(X)|<\infty$, then $\varphi(E(...
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$\sum_{i=1}^{n}(c_i\log_2 x_i)$ biggest when $x_i = \frac{c_i}{\sum_i c_i}$

$x_1 >0,x_2>0,...,x_n>0$,$\sum_i x_i=1$,$c_i >0$,proof $$ \sum_{i=1}^{n}(c_i\log_2 x_i) $$ max when $$x_i = \frac{c_i}{\sum_i c_i}$$ I found it similar to Jensen's inequality. but I could ...
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Is there any way to “analytically” compute the following conditional expectation?

Let me define the following SDE: $$\begin{cases} dX_{t}=X_{t}dW_{t}\\ X_{0}=1 \end{cases},$$ where $W$ is an appropriate Wiener process. My question is how we can compute $$f\left(z\right)=\mathbb{E}\...
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Computation of an integral in a consequence of the Jensen's inequality.

Proposition. Let $y_1,\dots, y_n\ge 0$, $\alpha_1,\dots, \alpha_n>0$ such that $\sum_{i=1}^n\alpha_i=1$, then $$\prod_{i=1}^ny_i^{\alpha_i}\le \sum_{i=1}^n\alpha_iy_i$$ proof. We consider $\varphi(...
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Conjecture about : $\sum_{cyc}\frac{214x^4}{133x^3 + 81y^3}\ge\frac{x+y+z}{12}+\sum_{cyc}\frac{13x^4}{8x^3+5y^3}-0.25(x-\frac{2yx}{2y+x})\ge x+y+z$

Hi it's a conjecture refinement in the simple case of Prove that $\sum_{\mathrm{cyc}} \frac{214x^4}{133x^3 + 81y^3} \ge x + y + z$ for $x, y, z > 0$ . Conjecture : Let $x,y,z>0$ such that $x\...
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Function composition with logsumexp

Given the function $g(\vec{x})=\log \sum_i \exp(x_i)$, I am curious which functions $f$ satisfy $f(g(\vec{x})) \geq g(\vec{f(x)})$? Let's let $x \in \mathbb{R}^N$ to be concrete. Useful properties of $...
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Is the expectation of minimums less than the minimum of expectations: $\mathbb{E}[\min_{k\in [N]} X_k] \leq \min_{k\in [N]} \mathbb{E}[X_k]$?

Is it true that $\mathbb{E}[\min_{k\in [N]} X_k] \leq \min_{k\in [N]} \mathbb{E}[X_k]$ for random variables $X_k$'s (or say when the $X_k$'s are non-negative)? I am tempted to say Jensen's inequality, ...
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Follow-up question on $H^s$ space forming an algebra when $2s>n$

In this post, the top answer did the following $$ \begin{split} (1+|\xi|^2)^p &\leq (1+2|\xi-\eta|^2+2|\eta|^2)^p\\ &\leq 2^p(1+|\xi-\eta|^2+1+|\eta|^2)^p\\ &\leq c(1+|\xi-\eta|^2)^p + c(...
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Find $C$ such that $\sum_{i=1}^{n}\frac{x_i^3}{\alpha x_i^{2}+\beta x_{i+1}^{2}}\geq \frac{\sum_{i=1}^{n}x_i}{C}$ is true

Problem Let $x_i>0$ and $n\geq 3$ then find the best constant which is a natural number $C=\alpha+\beta$ with $\alpha,\beta >0$ natural numbers such that $$\sum_{i=1}^{n}\frac{x_i^3}{\alpha x_i^...
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Finding the right function to use in Jensen's Inequality.

Jensen's inequality states that for a convex function $f:I \to \mathbb R$ and $x_1,...,x_n\in I$, we have $$\frac{1}{n}\sum_{i=1}^n f(x_i)\ge f\left(\frac{1}{n}\sum_{i=1}^n x_i\right)$$ But in order ...
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7 votes
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Jensen's inequality for composition of functions

I want to prove (or find a counterexample for) the following variant of Jensen's inequality. Let $f$ and $g$ be convex functions (then $f(g(x))$ and $g(f(x))$ are convex functions). From the standard ...
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Novel quirk of convex set definition

Given the definition of a convex set: A set $X\in \mathbb{R}^n$ is called convex if for all $x^1 \in X$ and $x^2 \in X$ it contains all points $$\alpha x^1 + (1-\alpha)x^2, \quad 0 < \alpha < 1....
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Expected value of $e^{tX}$ for $t > 0$

I am trying to prove the following inequality: Given a random variable $X$ with $$ 0\leq X\leq 1 \\ E[e^{tX}] \leq (1 - E[X]) + E[X]e^t, t > 0$$ I have the following steps so far: Since $$X \in [0,...
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Proving the minimum of an expected value is greater than the expected value of a minimum

I am looking to prove that $\min_{a \in \mathbb{R}} E[f(a, X)] \geq E[\min_{a \in \mathbb{R}}f(a, X)]$ where $f: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$, $f$ is guaranteed to have a ...
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2 votes
1 answer
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Counterexample to Jensen's Inequality when the convex function admits values in the extended real set

Let $(X,A,\mu)$ be a set, a $\sigma$-algebra and a measure. Suppose that $\mu(X) = 1$. Let $u : X \rightarrow {\mathbb{R}}$ and $f : \mathbb{R} \rightarrow \mathbb{R} \, \cup \, \{+\infty \} $ be an ...
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Different Versions of Jensen's Inequality?

According to wikipedia: Jensen's inequality is ${\displaystyle f(tx_{1}+(1-t)x_{2})\leq tf(x_{1})+(1-t)f(x_{2}).}$ With this, I can easily prove Inequality of arithmetic and geometric means, in ...
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Prove Jensen's inequality? [closed]

I was interested to see a proof for Jensen's inequality for the following variant: Let $X$ be a discrete random variable with finite expected value and let $h:\mathbb{R} \to \mathbb{R}$ be a convex ...
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Suppose Y is a random vairable with $E(|Y|^\alpha)<\infty$ for some $\alpha > 0$, Then $E(|Y|^\beta)<\infty$ for $0\leq \beta \leq \alpha$

During my studies for intro. to probability (undergrad) we learned about Jensen's inequality. The lecturer showed as the following theorem during one of our lectures: Suppose $Y$ is a random variable ...
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1 answer
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Deciding on the correctness of an inequality

Assume $\forall i: x_i,y_i,a_i \ge 0$, and we know that $\forall i: \frac{a_i}{1+x_i} \ge \frac{a_i}{1+y_i}$ in other words $\forall i: x_i \le y_i$. I want to know if the following inequality $$\sum_{...
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Convexity of matrix inverse of non-symmetric positive definite matrices

It is well known that if the matrices $A,B$ are symmetric positive definite then the matrix $$ rA^- + sB^- -(rA+sB)^-$$ is positive semidefinite for all positive numbers $r,s$ such that $r+s=1$. It is ...
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Lower Bound on Jensen's Inequality

Let $\mathbf{X}$ be a random vector in $\mathbb{R}^{n}$ and $f: \mathbb{R}^{n} \to \mathbb{R}$ a convex function. Jensen's inequality gives that $$f(\mathbb{E}[\mathbf{X}]) \leq \mathbb{E} f(\mathbf{X}...
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1 vote
1 answer
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How to prove this inequality involving $\tanh$?

Let $a>0, b >0$ and let $0<\alpha<\theta<\pi$. Prove that $$\frac{1}{\tanh \left(\frac{1}{\frac{\sin (\alpha )}{a \sin (\theta )}+\frac{\sin (\theta -\alpha )}{b \sin (\theta )}}\...
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3 votes
1 answer
69 views

Show inequality using the Jensen inequality

X is a non-negative random variable, with $\mathbb{E}(X) < \infty $. My goal is to show this inequality: $$\sqrt{1+(\mathbb{E}(X))^2} \leq \mathbb{E}(\sqrt{1+X^2})$$ x² is a convex function, so ...
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Application of Jensen’s inequality in graph.

This is a step in a proof that I don’t get. I’m paraphasing the argument here. Please give an explanation for the last statement. Suppose $G$ is a graph on $n$ vertices, and $G$ contains a $K_r(q)$, ...
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Jensen-Shannon divergence similarity with different probability space

I want to quantify the similarity between two probability mass functions (pmf) p and q, where q was noised with a function that changes the probability space of q. For instance if the following pmfs ...
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2 votes
1 answer
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Is my proof of the conditionnal Jensen's inequality correct?

I want to prove the conditionnal Jensen's inequality. Let $(\Omega, \mathcal H, \mathbb P)$ be a probability space, $\mathcal G \subset \mathcal H$ a sub sigma algebra, $\varphi : \mathbb R \...
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Counterexample to Jensen's inequality?

In the book `Measures, Integrals and Martingales' of René L. Schilling, pages 125-126, Jensen's inequality is stated in the following way: "Recall that a function $V:[a,b]\rightarrow\mathbb R$ on ...
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(Lower) bounds for the gap in Jensen's inequality for multi-dimensional random variables

Jensen's inequality tells us that, for a convex function $f$ on $\mathbb{R}^n$ and an $n$-dimensional random variable $X$, we have \begin{equation} f(\mathbb{E}[X]) \leq \mathbb{E}[f(X)]. \end{...
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For, $n = 1, \dots, 365$, why does: $\frac{1}{n} \sum_{k=0}^{n-1}(1 - k/365)^n \leq \left(\frac{1}{n}\int_0^n (1 - x/365)dx\right)^n$

From Jensen's I get the slightly larger bound of $$\frac{1}{n} \sum_{k=0}^{n-1}\left(1 - \frac{k}{365}\right)^n < \left(\frac{1}{n} \sum_{k=0}^{n-1}\left(1 - \frac{k}{365}\right)\right)^n.$$ But I'...
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How to apply Jensen's inequality to expected error term of an ensemble model

Consider a convex function $f(x) = x^2$ Let $E_{AV} = \frac{1}{M} \sum_{m=1}^{M} \mathbb{E}_x [(y_m(x) - f(x))^2]$ be average expected sum-of-squares error of the members of an ensemble model Let $E_{...
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$\ell_{p^\prime}(\eta)\subseteq\ell_p(\eta)$ when $p^\prime>p$

$\ell_{p^\prime}(\eta)$ and $\ell_p(\eta)$ are weighted Lebesgue sequence spaces and $\sum_{i\in\mathbb{N}}\eta_i=1$. My professor gave the hint that we should use Jensen inequality. I tried to use $\...
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p-norm and inner product inequality

I was reading Peter Lindqvist's material on the p-laplacian, there he derives a result from a convex based inequality that i have been stuck trying to show, how does one show that $$ |b|^p \geq |a|^p +...
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Exponential decay of Fisher information along the OU semigroup

I read from a paper that there is a "well-known" exponential decay of Fisher information along the OU semigroup, that is $$J(\nu^t\mid\gamma)\leq e^{-2t}J(\nu\mid\gamma),$$ where $\gamma$ is ...
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3 answers
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$\frac{1}{n}\sum_{k=1}^n E(X_k) \leqslant E(max(X_1, · · · ,X_n))$

Suppose $X_k$($1 \leqslant k\leqslant n$) be independent random variables with finite means. I know the $\frac{1}{n}\sum_{k=1}^n E(X_k)=\frac{E(X_1)+ \cdots E(X_n) }{n} \leqslant E(X_1)+ \cdots E(X_n)$...
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Induction proof of the inequality $\sum_{i=1}^na_i-\sum_{i=1}^n\frac1{a_i}\ge n(\sqrt[n]{\prod_{i=1}^na_i}-\frac1{\sqrt[n]{\prod_{i=1}^na_i}})$

Prove the following inequality: $$\sum_{i = 1}^n a_i - \sum_{i = 1}^n \frac{1}{a_i} \geq n\left(\sqrt[n]{\prod_{i = 1}^na_i}- \frac{1}{\sqrt[n]{\prod_{i = 1}^na_i}}\right)$$ Where all the variables ...
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4 answers
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Prove that $\log_27×\log_29<9$

Prove that $\log_27×\log_29<9$. I've tried couple of things like multiplying both sides by $\log_28$ Or… moving $\log_29$ to the right which would make everything look like this: $\log_27<\...
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2 votes
1 answer
159 views

simplify a function representing Jensen's gap in expectation

I have a function which I am trying to simplify given by $f(K)= \mathrm{h}_{2}\left(E\left[e^{-\frac{\tau^{2}}{K A^{2} \sigma^{2}+\sigma_{w}^{2}}}\right]\right)-\mathrm{E}\left(\mathrm{h}_{2}\left[e^{...
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4 votes
2 answers
125 views

Any bound on the Jensen's inequality with absolute value?

So we have the jensen's inequality: $$|EX| \leq E|X|$$ Any bound on the Jensen gap (upper bound or lower bound)? $$\text{gap}=E|X| - |EX|$$
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Jensen Inequality.

Sorry, I haven't touched math for some time. My math skill is very rusty. $X$ is a random variable. Can I ask when is the following equation true (or is it always true?). "Condition" here I ...
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Bartlett Statistic

Let $w = \frac{s_2}{s_1}$, where $s_1 = \bar{x}$ (empirical mean), and $s_2 = \tilde{x}$ (geometric mean). Furthermore, Jensen's inequality states that $g(EV) \leq Eg(V)$ where $g$ is a convex ...
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Jensen's inequality for Riemann integral's

Verify that if $f$ is a continuous convex function on $\mathbb{R}$ and $\varphi$ an arbitrary continuous function on $\mathbb{R}$, then Jensen's inequality $$f\left(\dfrac{1}{c}\int_{0}^{c}\varphi(t)...
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Is Jensen's inequality being used in the conclusion of the proof of the Vapnik-Chervonenkis inequality?

I am trying to resolve some compilation queries that arose in parsing the proof of the Vapnik-Chervonenkis inequality, and would appreciate some assistance on clarifying a particular step. The proof ...
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