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Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

17
votes
1answer
518 views

If $\prod x_k\neq0$ and $\sum\frac{x_k}{x_{k+1}}=0$, then $|\sum x_kx_{k+1}|\le (\max|x_k|- \min|x_k|)\sum x_k$

Question: Suppose that $x_{1},x_{2},\cdots,x_{n}$ are real numbers, such that $$x_{1}x_{2}\cdots x_{n}\neq 0$$ and $$\dfrac{x_{1}}{x_{2}}+\dfrac{x_{2}}{x_{3}}+\cdots+\dfrac{x_{n}}{x_{1}}=0$$ ...
17
votes
3answers
1k views

Trace inequality for real matrices

Is there any general result characterizing real matrices $A$ such that $$[\mathrm{tr}(A)]^2\leq n\mathrm{tr}(A^2)?$$ I can see that the inequality holds if: all eigenvalues of $A$ are real (by the ...
17
votes
1answer
360 views

Prove inequality$\sum\limits_{n|i+j+k}x_{i}y_{j}z_{k}\le n^2$

Being given an integer $n\ge 2$, and $x_{i},y_{i},z_{i}\in \mathbb{R}$ ($i=1,2,\cdots,n$) such that $$\sum_{i=1}^{n}(x^3_{i}+y^3_{i}+z^3_{i})=3n$$ show that $$\sum_{i+j+k=n}x_{i}y_{j}z_{k}\le n^2.$$ ...
17
votes
2answers
15k views

Show that $\frac 1 {\sqrt{x+y}}+\frac 1 {\sqrt{y+z}}+\frac 1 {\sqrt{z+x}}\geq2+\frac 1 {\sqrt2}$.

Given $x,y,z\geq0$ and $xy+yz+zx=1$. Show that $\displaystyle\frac 1 {\sqrt{x+y}}+\frac 1 {\sqrt{y+z}}+\frac 1 {\sqrt{z+x}}\geq2+\frac 1 {\sqrt2}$. I've tried many things but all failed. The only ...
17
votes
1answer
406 views

$x^3-3x-3=0$, prove that $10^x<127$

$x$ is the real root of the equation $$3x^3-5x+8=0,\tag 1$$ prove that $$e^x>\frac{40}{237}.$$ I find this inequality in a very accidental way,I think it's very difficult,because the actual value ...
17
votes
3answers
571 views

show this inequality $ab+bc+ac+\sin{(a-1)}+\sin{(b-1)}+\sin{(c-1)}\ge 3$

let $a,b,c>0$ and such $a+b+c=3abc$, show that $$ab+bc+ac+\sin{(a-1)}+\sin{(b-1)}+\sin{(c-1)}\ge 3$$ Proposed by wang yong xi since $$\dfrac{1}{ab}+\dfrac{1}{bc}+\dfrac{...
17
votes
1answer
696 views

Geometric inequality $\frac{R_a}{2a+b}+\frac{R_b}{2b+c}+\frac{R_c}{2c+a}\geq\frac{1}{\sqrt3}$

Let $P$ be a point inside $\triangle{ABC}$. Let $AP=R_a$, $BP=R_b$, $CP=R_c$, $AB=c$, $BC=a$ and $CA=b$. Prove that: $$\frac{R_a}{2a+b}+\frac{R_b}{2b+c}+\frac{R_c}{2c+a}\geq\frac{1}{\sqrt3}$$ I ...
16
votes
12answers
16k views

Prove that $ n < 2^{n}$ for all natural numbers $n$. [duplicate]

Prove that $ n < 2^{n} $ for all natural numbers $n$. I tried this with induction: Inequality clearly holds when $n=1$. Supposing that when $n=k$, $k<2^{k}$. Considering $k+1 <2^{k}+1$, ...
16
votes
4answers
5k views

How to find out which number is larger without a calculator?

So I have a question which is: Which is larger? $$2.2^{3.3} \text{ or } 3.3^{2.2} $$ Now I need to find out with using a calculator but the answer is $3.3^{2.2}$. The only thing I could think of ...
16
votes
11answers
3k views

How to prove that adding $n$ to the numerator and denominator will move the resultant fraction close to $1$?

Given a fraction: $$\frac{a}{b}$$ I now add a number $n$ to both numerator and denominator in the following fashion: $$\frac{a+n}{b+n}$$ The basic property is that the second fraction is suppose ...
16
votes
7answers
480 views

How do we know that $x^2 + \frac{1}{x^2}$ is greater or equal to $2$?

For one problem, we were supposed to know that: $$x^2 + \frac{1}{x^2}\geq 2.$$ How do you deduce this instantly when looking at the expression above?
16
votes
7answers
3k views

Smallest Possible Power

When working on improving my skills with indices, I came across the following question: Find the smallest positive integers $m$ and $n$ for which: $12<2^{m/n}<13$ On my first attempt, I ...
16
votes
7answers
25k views

How to prove $\log n < n$?

Sorry if this is a silly question but most books claim $\log n < n$ for $n \geq 1$ without providing any proof, saying it's too obvious. Could someone give me a rigorous proof? Is there some trick ...
16
votes
7answers
961 views

Prove that $\left (\frac{1}{a}+1 \right)\left (\frac{1}{b}+1 \right)\left (\frac{1}{c}+1 \right) \geq 64.$

Let $a,b,$ and $c$ be positive numbers with $a+b+c = 1$. Prove that $$\left (\dfrac{1}{a}+1 \right)\left (\dfrac{1}{b}+1 \right)\left (\dfrac{1}{c}+1 \right) \geq 64.$$ Attempt Expanding the LHS we ...
16
votes
4answers
8k views

Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$

I have a small question that I think is very basic but I am unsure how to tackle since my background in computing inequalities is embarrasingly weak - I would like to show that, for a real number $p ...
16
votes
4answers
434 views

Proving that $\sin x \ge \frac{x}{x+1}$

Prove that $$ \sin x \ge \frac{x}{x+1}, \space \space\forall x \in \left[0, \frac{\pi}{2}\right]$$
16
votes
3answers
434 views

Conjecture: $\pi(x)\ge \pi\circ\pi(x)+\pi\circ\pi\circ\pi(x)+\cdots$

$x\ge 13\implies\pi(x)\ge \pi\circ\pi(x)+\pi\circ\pi\circ\pi(x)+\cdots$ Can this be proved?
16
votes
6answers
589 views

$\log_9 71$ or $\log_8 61$

I am trying to know which one is bigger :$$\log_9 71$$ or $$\log_8 61$$ how can i know without using a calculator ?
16
votes
3answers
2k views

Reverse Cauchy Schwarz for integrals

Let $f,g$ be two continuous positive functions over $[a,b]$ Let $m_1$ and $M_1$ be the minimum and maximum of $f$ Let $m_2$ and $M_2$ be the minimum and maximum of $g$ Prove that $$\...
16
votes
4answers
349 views

CSB inquality: is $\|x\|^2\|y\|^2 - \langle x,y \rangle^2$ a square in any obvious way?

Suppose $x=(x_1,x_2),y = (y_1,y_2) \in \mathbb{R}^2$. I noticed that \begin{align*} \|x\|^2 \|y\|^2 - \langle x,y \rangle^2 &= x_1^2y_1^2 + x_1^2 y_2^2 + x_2^2 y_1^2 + x_2^2 y_2 ^2 - (x_1^2 y_1^2 +...
16
votes
7answers
6k views

Proof for $\sin(x) > x - \frac{x^3}{3!}$

They are asking me to prove $$\sin(x) > x - \frac{x^3}{3!},\; \text{for} \, x \, \in \, \mathbb{R}_{+}^{*}.$$ I didn't understand how to approach this kind of problem so here is how I tried: $\sin(...
16
votes
4answers
368 views

An inequality on sequences with each term dividing sum of two neighbouring terms

Positive integers $x_1, x_2, \dots, x_n$ ($n \ge 4$) are arranged in a circle such that each $x_i$ divides the sum of the neighbors; that is $$\frac{x_{i-1}+x_{i+1}}{x_i} = k_i $$ is an integer for ...
16
votes
3answers
857 views

How to prove this inequality $\frac{x^y}{y^x}+\frac{y^z}{z^y}+\frac{z^x}{x^z}\ge 3$

let $x,y,z$ be positive numbers, and such $x+y+z=1$ show that $$\dfrac{x^y}{y^x}+\dfrac{y^z}{z^y}+\dfrac{z^x}{x^z}\ge 3$$ My try: let $$a=\ln{\dfrac{x^y}{y^x}},b=\ln{\dfrac{y^z}{z^y}},c=\ln{\...
16
votes
3answers
308 views

A curious triangle inequality

Let $ABC$ be a triangle. Pick a point $P$ inside the triangle. How would you show that \begin{equation} |PA|+|PB|+|PC|+\min\{|PA|,|PB|,|PC|\}\leq |AB|+|BC|+|CA|. \end{equation}
16
votes
2answers
15k views

Properties of $\liminf$ and $\limsup$ of sum of sequences: $\limsup s_n + \liminf t_n \leq \limsup (s_n + t_n) \leq \limsup s_n + \limsup t_n$

Let $\{s_n\}$ and $\{t_n\}$ be sequences. I've noticed this inequality in a few analysis textbooks that I have come across, so I've started to think this can't be a typo: $\limsup\limits_{n \...
16
votes
1answer
272 views

Prove that $E\left(\frac{XY}{X^2+Y^2}\right) \geqslant 0$ for i.i.d. $X$ and $Y$

Let $X$ and $Y$ be two independent identically distributed random variables. Prove that $$E\left(\frac{XY}{X^2+Y^2}\right) \geqslant 0.$$ I tried to manipulate with the expression $\dfrac{XY}{X^2+...
16
votes
2answers
350 views

How to prove this inequality about $xyz=1$

Let $x,y,z>0$,and such $xyz=1$, show that $$\dfrac{x+y}{x^3+x}+\dfrac{y+z}{y^3+y}+\dfrac{z+x}{z^3+z}\ge 3$$ I tried use $AM-GM$ inequality $$\dfrac{x+y}{x^3+x}+\dfrac{y+z}{y^3+y}+\dfrac{z+x}{z^3+...
16
votes
2answers
483 views

How to prove this inequality $\big|x\sin{\frac{1}{x}}-y\sin{\frac{1}{y}}\big|<2\sqrt{|x-y|}$?

For any real numbers $x,y\neq 0$,show that $$\Big|x\sin{\dfrac{1}{x}}-y\sin{\dfrac{1}{y}}\Big|<2\sqrt{|x-y|}$$ I found this problem when I dealt with this problem. But I can't prove it. Maybe the ...
16
votes
2answers
660 views

An integral inequality (one variable)

Anyone has an idea to prove the following inequality? Let $g:\left(0,1\right)\rightarrow\mathbb{R}$ be twice differentiable and $r\in\left(0,1\right)$ such that $$ r\left(g"\left(x\right)+\dfrac{g'\...
16
votes
4answers
720 views

$a,b,c >0$, prove $\sqrt[2]{\frac{a}{b+c}}+\sqrt[3]{\frac{b}{c+a}}+\sqrt[4]{\frac{c}{a+b}} \geqslant \frac{7}{12} \cdot2^{\frac67} \cdot 3^{\frac47}$

$a,b,c >0$, prove $$\sqrt[2]{\frac{a}{b+c}}+\sqrt[3]{\frac{b}{c+a}}+\sqrt[4]{\frac{c}{a+b}} \geqslant \frac{7}{12} \cdot2^{\frac67} \cdot 3^{\frac47}$$ What I tried: 1) It seems like a Nesbitt ...
16
votes
2answers
1k views

$\forall x,y>0, x^x+y^y \geq x^y + y^x$

Prove that $\forall x,y>0, x^x+y^y \geq x^y + y^x$ A friend of mine told me none of the teachers in my school have succeeded in proving this seemingly simple inequality (it was asked at an oral ...
16
votes
4answers
236 views

Proving $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)<3$ for all positive integers $n$

Prove that $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)<3$ for all positive integers $n$ This problem is copied from Math Olympiad Treasures by ...
16
votes
2answers
7k views

Various proofs of Hardy's inequality

For any $p > 1$ and for any sequence $\{a_j\}_{j=1}^\infty$ of nonnegative numbers, a classical inequality of Hardy states that $$ \sum\limits_{k=1}^n\left(\frac{\sum_{i=1}^ka_i}{k}\right)^p\le \...
16
votes
1answer
413 views

Prove that there are four distinct real number $x,y,z,w$,such $|xz+yw|\ge |\sqrt{5}(xw-yz)|$

For any nine distinct real numbers,there exsit four distinct real number $x,y,z,w$,such $$|xz+yw|\ge |\sqrt{5}(xw-yz)|$$ I think can use pigeonhole principle to solve it?Thanks
16
votes
1answer
422 views

How prove this inequality $\tan{(\sin{x})}>\sin{(\tan{x})}$

How prove this inequality $$\tan{(\sin{x})}>\sin{(\tan{x})},0<x<\dfrac{\pi}{2}$$ this PDF give a ugly methods : http://wenku.baidu.com/link?url=CHnWPdmjsqSmNAQhL4bOmDfUVc0Tc5nWCBQWNB1lweG-...
16
votes
4answers
836 views

Prove $(x+y)(y^2+z^2)(z^3+x^3) < \frac92$ for $x+y+z=2$

$x,y,z \geqslant 0$ and $x+y+z=2$, Prove $$(x+y)(y^2+z^2)(z^3+x^3) < \frac92$$ While numerical method can solve this problem, I am more interested in classical solutions. I tried this problem for ...
16
votes
3answers
640 views

How to find this minimum of the value

Let $x_{i}$, where $i\in\{1,2,\cdots,n\}$ be distinct real numbers. Find the minimum of the value of $$\sum_{1\le i<j\le n}\left(\dfrac{1-x_{i}x_{j}}{x_{i}-x_{j}}\right)^2$$ It is clear when $...
16
votes
2answers
189 views

How prove this Lower bound with $\sum_{i=1}^{n}\prod_{j=1}^{i}a_{j}$

Let $a_{i}\in [0,1]$,and $a_{1}\ge a_{2}\ge\cdots \ge a_{n}$,show that $$a_{1}+a_{1}a_{2}+a_{1}a_{2}a_{3}+\cdots+a_{1}a_{2}\cdots a_{n}\ge \sum_{i=1}^{n}\left(\dfrac{a_{1}+a_{2}+\cdots+a_{n}}{n}\...
16
votes
2answers
311 views

This general inequality maybe is true? $\sum_{i=1}^{n}\frac{i}{1+a_{1}+\cdots+a_{i}}<\frac{n}{2}\sqrt{\sum_{i=1}^{n}\frac{1}{a_{i}}}$

Let $a_{1},a_{2},\ldots,a_{n}>0$ and prove or disprove $$\dfrac{1}{1+a_{1}}+\dfrac{2}{1+a_{1}+a_{2}}+\cdots+\dfrac{n}{1+a_{1}+a_{2}+\cdots+a_{n}}\le\dfrac{n}{2}\sqrt{\dfrac{1}{a_{1}}+\dfrac{1}{a_{...
16
votes
1answer
527 views

If $\{a,b,c,d,e\}\subset[0,1]$ so $\sum\limits_{cyc}\frac{1}{1+a+b}\leq\frac{5}{1+2\sqrt[5]{abcde}}$

Let $\{a,b,c,d,e\}\subset[0,1]$. Prove that: $$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+d}+\frac{1}{1+d+e}+\frac{1}{1+e+a}\leq\frac{5}{1+2\sqrt[5]{abcde}}$$ I tried C-S, convexity and more, but ...
16
votes
1answer
440 views

How to prove this inequality in Euclidean space?

Prove that $$\begin{align*}&|a+b||a+c|+|a+b||b+c|+|a+c||b+c|\\ \leq &(|a|+|b|+|c|) \cdot |a+b+c|+|a||b|+|a||c|+|b||c|\end{align*}$$ in Euclidean space $\mathbb{R}^n$. I have been ...
16
votes
5answers
6k views

Prove $\sqrt{3a + b^3} + \sqrt{3b + c^3} + \sqrt{3c + a^3} \ge 6$

If $a,b,c$ are non-negative numbers and $a+b+c=3$, prove that: $$\sqrt{3a + b^3} + \sqrt{3b + c^3} + \sqrt{3c + a^3} \ge 6$$ Here's what I've tried: Using Cauchy-Schawrz I proved that: $$(3a + b^3)...
16
votes
2answers
1k views

Comparing sums of surds without any aids

Without using a calculator, how would you determine if terms of the form $\sum b_i\sqrt{a_i} $ are positive? (You may assume that $a_i, b_i$ are integers, though that need not be the case) When there ...
16
votes
1answer
306 views

An elementary inequality

Assuming $x,y,z\in [-1,1]$, suppose that $$1+2xyz\geqslant x^2 + y^2 + z^2$$ Can we infer from this that $1+2(xyz)^n\geqslant x^{2n} + y^{2n} + z^{2n}$ for any positive integer $n$?
16
votes
1answer
324 views

Prove that $\sum\limits_{cyc}\frac{a}{\sqrt{a+3b}}\geq\sqrt{a+b+c+d}$

Let $a$, $b$, $c$ and $d$ be positive numbers. Prove that: $$\frac{a}{\sqrt{a+3b}}+\frac{b}{\sqrt{b+3c}}+\frac{c}{\sqrt{c+3d}}+\frac{d}{\sqrt{d+3a}}\geq\sqrt{a+b+c+d}$$ I tried Holder, AM-GM and more, ...
16
votes
1answer
654 views

prove $f(x,y) \le 3 $

$x \ge 0, y>0$ $f(x,y)=\sqrt{\dfrac{y}{y+x^2}}+4\sqrt{\dfrac{y}{(y+(x+1)^2)(y+(x+3)^2)}}+4\sqrt{\dfrac{y}{(y+(x-1)^2)(y+(x-3)^2)}}$ prove $f(x,y) \le 3$ I can prove when $x\ge 2, f(x,y) < 3$, ...
15
votes
7answers
4k views

If $a > b$, is $a^2 > b^2$?

Given $a > b$, where $a,b ∈ ℝ$, is it always true that $a^2 > b^2$?
15
votes
7answers
5k views

Proving that the sum of any two sides of the triangle is greater than the third side

I know it's easy to prove with the help of linear inequalities, but this time I want to prove it with the help of trigonometry. Is it possible? If yes, then how?
15
votes
8answers
2k views

Variation on Pythagoras: If $a^2 + b^2 = c^2$, then $a + b \leq c\sqrt{2}$

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with derivative of the Pythagorean Theorem using calculus, trigonometry, ...
15
votes
3answers
893 views

Showing $\sqrt{2}\sqrt{3} $ is greater or less than $ \sqrt{2} + \sqrt{3} $ algebraically

How can we establish algebraically if $\sqrt{2}\sqrt{3}$ is greater than or less than $\sqrt{2} + \sqrt{3}$? I know I can plug the values into any calculator and compare the digits, but that is not ...