# Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities. Do not use this tag just because an inequality appears somewhere in your question.

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### Show that $|x_{(k)} - y_{(k)}| \leq \|x-y\|_2$

Given two vectors in $\mathbb R^n$, show that the absolute difference in the $k^{th}$ order statistics is bounded above by the $\ell_2$ norm. More formally, consider $x, y \in \mathbb R^n$. Show that: ...
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### Lower bound on a particular infinite sum

Consider the following sum: $$\huge\sum_{k=1}^{\infty}{ A^{-k} B^{-\frac{\phi^k-1}{\phi-1}} }$$ where $\phi>1$, $A>0$ and $B>1$. The $B$ term dominates the $A$ term for large $k$, so the sum ...
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### Asymptotics of a convoluted sum involving $r^{-\alpha}$

Let $\alpha,\beta$ be real constants satisfying $0<\alpha\leq \beta$, and let $R>1$ be an integer. Prove that \label{eq:lemma_convolution} \sum^{R-1}_{r=1}\frac{1}{r^\beta (R-r)...
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### Inequalities concerning real positive numbers [duplicate]

Prove that $\frac{a}{b+c}+\frac{b}{c+d}+\frac{c}{d+a}+\frac{d}{a+b}\geq 2$ for all positive $a,b,c,d$ $a,b,c,d,$ are any positive real numbers
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### Olympiad inequality $\frac{a}{c+5b}+\frac{b}{a+5c}+\frac{c}{b+5a}\geq \frac{1}{2}$ for all positive $a,b,c$?

How can I prove that $\frac{a}{c+5b}+\frac{b}{a+5c}+\frac{c}{b+5a}\geq \frac{1}{2}$ for all positive $a,b,c$?
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### Proving inequality using root of unity

Let $\omega$ be a complex cube root of unity. It can be shown that if $a,b,c \in \mathbb{R}$, then $$(a+b+c)(a+b\omega+c\omega^2)(a+b \omega^2 + c \omega)= a^3+b^3+c^3-3abc$$ I was wondering if this ...
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1 vote
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### Proving $x^4+x^3y-x^2y+y^2>0$ with some given constraints

I have to prove the inequality $$x^4+x^3y-x^2y+y^2>0$$ holds for all pairs $(x,y)\in\mathbb{R}^2$ such that $x^2+y^2\leq1$ and $x^2>y^2$. I would like to find a lower bound just to start, but I ...
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### Another proof of Euler's inequality via the half-angle formulas

The Euler's inequality is an immediate consequence of Euler's identity in a triangle, $$OI^2=R^2−2Rr.$$ An additional proof of the Euler's inequality is given at Elias Lampakis, Am Math Monthly, 122 ...
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### Integral involving the eror function

I would like to know if the following integral has an explicit expression or not. $$F(r)=\int_1^r e^{t^2/2} \operatorname{erf}\left(\frac{t}{\sqrt{2}}\right)dt$$ where the error function inside is ...
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### If $a,b,c,x,y,z$ are positive real numbers and $n>1$ then prove the following.

If $a,b,c,x,y,z$ are positive real numbers and $n>1$, prove that $(ax+by+cz)^n$ + $(ay+bz+cx)^n$ + $(az+bx+cy)^n$ 《 $(a+b+c)^n$($x^n$ + $y^n$ + $z^n$). I tired by Induction on n but got stuck on ...
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### Showing the maximum value using alternate approach

If $x_1,x_2...x_n$ are postive numbers satisfying $x_1\cdot x_2 \cdots x_n = 1$ , then find the maximum value of $\frac{1}{\sqrt{x_1 ^2 + 1}\,\cdot\, \sqrt{x_2^2 +1}\,\cdots\,\sqrt{x_n^2+1} }$ . ...
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### Proof inequality in $R^n$ [duplicate]

Let $x=(x_1,x_2,\dots,x_n).$ I wanted to show that $$\sum_{i=1}^{n}|x_i|-\sqrt{(\sum_{i=1}^{n}|x_i|^2)}\geq 0$$ I spent some times to show that but I am not sure if there are some inequalities that I ...
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### Find minimum of $\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2}+ab+bc+ca$

Let $a, b,c$ be the lengths of the sides of a triangle such that $a+b+c=2$, find the minimum value of $$\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2}+ab+bc+ca$$ I don't have many ideas for this problem,...
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### Converse Chebyshev inequality

The problem (and also the author´s solution): Of course I understand the derivation of the inequality etc. but I don´t see how this proves, that there is at least one entry of x with absolute value ...
1 vote
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### Vector inequality regarding p-Laplace euations

In chapter~12, (p.99) of this note it is written that "In the case $p \geq 2$, the inequality $$|b|^p \geq |a|^p + p<|a|^{p−2}a, b − a> + C(p)|b − a|^p$$ holds with a constant $C(p) > 0$....
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### Showing that $(|\left<a, b\right>| - \epsilon)^2 \leq |\left<a, Mb\right>|^2$ for self-adjoint operator $M$ such that $||(I - M)b|| < \epsilon$

Let $a, b$ be $L^2$ normalized functions and $M$ be a self-adjoint pseudodifferential operator such that $||(I - M)b|| < \epsilon$ and $\sigma(M) \leq 1$ (author of the paper I am reading has not ...
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### Why does the following hold? $u_{k+1}\le(1+AT/N)u_k+BT^pN^{-p}$ implies $u_k\le e^{AT}u_0+\frac{B}{A}\frac{T^{p-1}}{N^{p-1}}(e^{AT}-1)$

I have read a lemma in "Numerical Integration of Stochastic Differential Equations" of G.N.Milstein, which was stated without any proof as follows: Suppose that for arbitrary natural number \$...