Questions tagged [jensen-inequality]

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

Jensen's inequality states that for a convex function $f$, for $\lambda\in[0,1]$, we have $$f\left[\lambda x+(1-\lambda)y\right]\leq\lambda f(x)+(1-\lambda)f(y).$$ In the context of measure-theoretic probability theory, Jensen's inequality states that given a probability space $(\Omega,\mathcal F,\mathbb P)$, given a $\mathbb P$-integrable function $f$ and convex function $\psi$, then $$\psi\left(\int_\Omega f\,\mathrm d\mathbb P\right)\leq\int_\Omega\psi( f)\mathrm d\mathbb P.$$

Jensen's inequality is sometimes written in terms of the expectation operator, i.e. if $X$ is a random variable and $\psi$ is a convex function as above, then $$\psi(\mathbb E[X])\leq\mathbb E[\psi(X)].$$ It is a broad generalization of the fact that variance is non-negative (i.e. that $\mathbb E(X^2) \le (\mathbb E X)^2$) with many consequences. For example, it gives one way to prove the AM-GM inequality.

It also has uses in combinatorics (via e.g. the discrete version: $ \sum_i\alpha_i=1,\alpha_i\ge 0$ implies $\psi(\sum_i\alpha_i x_i) \le \sum_i \alpha_i \psi(x_i)$), real analysis, harmonic analysis, and geometry.

External links: Wikipedia page on Jensen's inequality

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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 $0 < x < 1.$
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When Jensen's inequality is equality

One form of Jensen's inequality is If $X$ is a random variable and $g$ is a convex function, then $\mathbb{E}(g(X))\geq g(\mathbb{E}(X))$. Just out of curiosity, when do we have equality? If and only if $g$ is constant?
eeeeeeeeee
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Inequality : $\Big(\frac{x^n+1+(\frac{x+1}{2})^n}{x^{n-1}+1+(\frac{x+1}{2})^{n-1}}\Big)^n+\Big(\frac{x+1}{2}\Big)^n\leq x^n+1$

I have the following problem to solve : Let $x,y>0$ and $n>1$ a natural number then we have : $$\Big(\frac{x^n+y^n+(\frac{x+y}{2})^n}{x^{n-1}+y^{n-1}+(\frac{x+y}{2})^{n-1}}\Big)^n+\Big(\frac{x+y}{2}\Big)^n\leq x^n+y^n$$ The problem is equivalent…
Erik Satie
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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 $xyz=1$ and $$\Big({1\over 1+x}\Big)^3+\Big({1\over…
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If $a+b+c+d=16$, then $(a+\frac{1}{c})^2+(c+\frac{1}{a})^2 + (b+\frac{1}{d})^2 + (d+\frac{1}{b})^2 \geq \frac{289}{4}$

If $a,b,c,d$ are positive integers and $a+b+c+d=16$, prove that $$\left(a+\frac{1}{c}\right)^2+\left(c+\frac{1}{a}\right)^2+\left(b+\frac{1}{d}\right)^2+\left(d+\frac{1}{b}\right)^2 \geq \frac{289}{4}.$$ I know I have to use some inequality,…
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Proving $\int_{0}^{1}xf(x)dx \leq \frac{2}{3}\int_{0}^{1}f(x)dx$ for all concave functions $f: [0,1]\rightarrow [0,\infty)$

I'm trying to prove the inequality $$\int_{0}^{1}xf(x)dx \leq \frac{2}{3}\int_{0}^{1}f(x)dx$$ for all continuous concave functions $f: [0,1]\rightarrow [0,\infty)$. I've been working on this for a while and would just love a hint if anyone can see…
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Prove QM-AM inequality

$$\dfrac{x_1^2+ x_2^2 + \cdots + x_n^2}n \geq \left(\dfrac{x_1+x_2+\cdots+x_n}n\right)^2$$ I don't think AM, GM can be used here. And simple expansion doesn't help too. What should I do?
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Can Cauchy Schwarz inequality be proven using Jensen's inequality?

After reading a comment on If $\mathrm{E} |X|^2$ exists, then $\mathrm{E} X$ also exists, I wonder if Cauchy Schwarz inequality can be proven using Jensen's inequality?
Tim
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Proof that if $x,y>0$ and $x+y=1$, then $(2x)^{\frac 1 x}+(2y)^{\frac 1 y}\leq 2$

For positive reals $x$ and $y$ such that $x+y=1$, prove that $$(2x)^{\frac 1 x}+(2y)^{\frac 1 y}\leq 2$$ I have tried using Jensen’s inequality but it won’t cover all the possible choices for $x$ and $y$ since the concavity varies. I am trying to…
William Sun
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Prove the following inequality $\sum_{k=0}^{n}(-1)^{k}f(a_{k})\geq f ( \sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } a _ { k } )$

Suppose that a function $f$ is convex and increasing on $[0,+\infty)$ and $f(0)=0$ .Show that $$\sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } f ( a_ { k } ) \geq f \left( \sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } a _ { k } \right)$$ For any number $a _ {…
user790977
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Prove that $xy \leq\frac{x^p}{p} + \frac{y^q}{q}$

OK guys I have this problem: For $x,y,p,q>0$ and $ \frac {1} {p} + \frac {1}{q}=1 $ prove that $ xy \leq\frac{x^p}{p} + \frac{y^q}{q}$ It says I should use Jensen's inequality, but I can't figure out how to apply it in this case. Any ideas about the…
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Prove that $\sum \limits_{cyc}\frac {a}{(b+c)^2} \geq \frac {9}{4(a+b+c)}$

Given $a$, $b$ and $c$ are positive real numbers. Prove that:$$\sum \limits_{cyc}\frac {a}{(b+c)^2} \geq \frac {9}{4(a+b+c)}$$ Additional info: We can't use induction. We should mostly use Cauchy inequality. Other inequalities can be used…
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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 Jensen's inequality, we have $$ \mathbb{E_{\sim…
Jacob A
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Probability of a deviation when Jensen’s inequality is almost tight

Let $X>0$ be a random variable. Suppose that we knew that for some $\epsilon \geq 0$, \begin{eqnarray} \log(E[X]) \leq E[\log(X)] + \epsilon \tag{1} \label{eq:primary} \end{eqnarray} The question is: if $\epsilon$ is small, can we find a good bound…
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An inequality for polynomials with positives coefficients

I have found in my old paper this theorem : Let $a_i>0$ be real numbers and $x,y>0$ then we have : $$(x+y)f\Big(\frac{x^2+y^2}{x+y}\Big)(f(x)+f(y))\geq 2(xf(x)+yf(y))f\Big(\frac{x+y}{2}\Big)$$ Where :$$f(x)=\sum_{i=0}^{n}a_ix^i$$ The problem…
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