# 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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1 vote
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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 ...
1 vote
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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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### 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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### $\ell_{p^\prime}(\eta)\subseteq\ell_p(\eta)$ 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 \$\...