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Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

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1answer
26 views

Describe the ‘intervals’ $[a, b]$ and $(a, b)$, in the case where $a = b.$

This is a question from my textbook, however there are no solutions, would $[a,b] = a$ and $(a,b) = \text{undefined}$? I'm not sure if $(a,b)$ is right.
0
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1answer
94 views

Hermite-Hadamard's integral inequality for improper integrals

Let's say that $f(x)$ convex in the interval $[b,\infty)$, write the improper integral $$\int_b^\infty f(x) dx.$$ Do Hermite-Hadamard's integral inequalities holds for this improper integral? Is ...
0
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1answer
38 views

integral inequality: midpoint of interval vs. expectation over interval

I have been thinking about an inequality that should be self-evident, but which I have difficulties proving formally. It looks like the following: Take a function $f(x)$ with $f'(x)>0$ and scalars ...
2
votes
2answers
921 views

Discreet weighted mean inequality

Let ${p_{1}},{p_{2}},\ldots,{p_{n}}$ and ${a_{1}},{a_{2}},\ldots,{a_{n}}$ be positive real numbers and let $r$ be a real number. Then for $r\ne0$ , we define ${M_{r}}(a,p)={\left({\...
29
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5answers
2k views

Computing the best constant in classical Hardy's inequality

Classical Hardy's inequality (cfr. Hardy-Littlewood-Polya Inequalities, Theorem 327) If $p>1$, $f(x) \ge 0$ and $F(x)=\int_0^xf(y)\, dy$ then $$\tag{H} \int_0^\infty \left(\frac{F(x)}{x}\right)^...
6
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3answers
141 views

Find multivariable limit $\frac{x^2y}{x^2+y^3}$

Find multivariable limit of: $$\lim_{ \left( x,y\right) \rightarrow \left(0,0 \right)}\frac{x^2y}{x^2+y^3}$$ How to find that limit? I was trying to do the following, but i am not able to find a ...
1
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0answers
177 views

Prove $\frac{x}{y}+ \frac{y}{z}+ \frac{z}{x}\geqq \frac{2\,x}{y+ z}+ \frac{2\,y}{z+ x}+ \frac{2\,z}{x+ y}$ for $x,\,y,\,z\in [\,j,\,k\,j\,]$

For $x,\,y,\,z\in [\,j,\,k\,j\,]$ $$\frac{x}{y}+ \frac{y}{z}+ \frac{z}{x}\geqq \frac{2\,x}{y+ z}+ \frac{2\,y}{z+ x}+ \frac{2\,z}{x+ y}$$ is true with $j= constant,\,k= constant> 8$ and $k_{\,\max}$...
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2answers
907 views

two variables quadratic inequalities solution

Suppose there are $n$ quadratic inequalities, the form is $A_i x^2 + B_i y^2 + C_i xy + D_i x + E_i y + F_i \leq 0$, $(\forall i \in [1,n])$, where $x,y$ are two variables and $(A_i, B_i, C_i, D_i, ...
2
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1answer
54 views

Proving an inequality given conditions.

Let real numbers $x_1, x_2, x_3, x_4, x_5, x_6$ satisfy $x_1+x_2+x_3+x_4+x_5+x_6=0, $ and $x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=6.$ Prove $x_1x_2x_3x_4x_5x_6\leq\frac{1}{2}.$ I am trying to figure out ...
2
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4answers
182 views

For $x+ y+ z= 3$ prove that $4\geqq x^2y+y^2z+z^2x$

Given $3$ positive real numbers $x,\,y,\,z$ and $x+ y+ z= 3$. Prove that $$4\geqq x^2y+ y^2z+ z^2x$$ This problem was homogenized so I set $x+ y+ z= 3$ to cancel stuff. Now I'm stuck. I have noticed ...
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0answers
68 views

$W= k\,P\left ( a- b \right )\left ( a- c \right )+ Q$ and $W= -\,k\,R\left ( a- b \right )\left ( a- c \right )+ S$

If you're interested in, please help me a little: For $$W= AB+ C \tag{drive s.o.s X}$$ $$W= -\,AD+ E \tag{drive s.o.s Y}$$ I can write $$\therefore\,W= \frac{BE+ CD}{B+ D}\geqq 0 \tag{29}$$ with $B,\,...
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3answers
54 views

proof that if $a,b \ge 2$ then $ab \ge a+b$

how is my proof that if $a,b \ge 2$ then $a+b \le ab$ so if $a \ge 2$ and $b \ge 2$ then $a-1 \ge 1$ and $b-1 \ge 1$ $(a-1)(b-1) \ge b-1$ $(a-1)(b-1) \ge 1$ $(a-1)(b-1) - 1 \ge 0$ $ab -a -b\ge 0$ ...
2
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6answers
94 views

Prove that $5x^2−2xy−8x+ 2y^2−2y+ 5 \ge 0$ for all $x, y\in\mathbb R$. When does equality occur?

I tried grouping the $x$'s and the $y$'s but that didn't get me anywhere. I know that $5x^2, 2y^2$, and $5$ are always positive. I am not sure what to try next.
1
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1answer
20 views

Proving that $\int_0^1 \left(\frac{\partial T}{\partial z}(t,z)\right)^2\mathrm{d}z \geq 2 \int_0^1 T^2(t,z)\mathrm{d}z$

Exercise : Assume that $T$ satisfies the equation $T_t(t,z) = aT_{zz}(t,z)$ for $t>0, z \in (0,1)$ and $a > 0$ a constant. Moreover, suppose that $T(0,z) = T_0(z)$ for $z \in [0,1]$, where $...
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1answer
30 views

Limits and inequality

We want to find the largest $p>0$ such that for every $t>0$. The official solution says, Let $t$ go to infinity. Then $LHS\to \frac{1}{p}$, so $\frac{1}{p}\geq \frac{3}{2p+2}$. Why does this ...
0
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1answer
57 views

Help with inequality with one unknown

Please could you help how to solve the inequality $(\sqrt{x-9})(2^{x-8}+3^{x-9}-9)\geq 0$
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0answers
42 views

Proving these inequalities for a symmetric random walk

I would like to prove the following inequalities. Here $S_n = \sum_{i=1}^n X_i$ where $x_i$ are i.i.d and symmetric. $$P(|S_n|>x) \geq \frac{1}{2}P(\max_{k\leq n}|X_k|>x)\geq \frac{1}{2}(1-e^{...
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6answers
270 views

Inequality question­

$$a,b,c,d\ge 0$$ $$a\le 1$$ $$a+b\le 5$$ $$a+b+c\le 14$$ $$a+b+c+d\le 30$$ Prove that $\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}\le 10$. We can subtract inequalities to get the answer, but that is wrong......
5
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3answers
240 views

Maximize $x_1^3+x_2^3+\cdots + x_n^3$

This is from a Brazilian math contest for college students (OBMU): Given a positive integer $n$, find the maximum value of $$x_1^3+x_2^3+ \cdots + x_n^3$$ where $x_j$ is a real number for all $j \...
1
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0answers
27 views

Matrix inequality related to Minkovski space.

$M=\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-1\end{pmatrix}$ $L\in M_4(\mathbb{R})$ s.t. $L^tML=M$ ($L^t$ is the transposition of $L$) Please ...
2
votes
3answers
29 views

logarithmical inequality with equality case proof

If 10$\leq$a$_1$$\leq$a$_2$$\leq$...$\leq$a$_{2020}$$\leq$100 with a$_i$ are real numbers, i=$\overline{1,2020}$ then prove that $$\sum_{i=1}^{2019} log_{a_i}a_{i+1}\leq2020 $$ and the equality case ...
1
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1answer
42 views

lower bounding logarithm of sums

Is it true that $$ \log\left(\sum_{i=1}^{n} \alpha_i\right) = \log\left(n \frac{1}{n}\sum_{i=1}^{n}\alpha_i\right) = \log(n) + \log\left(\frac{1}{n}\sum_{i=1}^{n} \alpha_i\right) \\ \geq \log(n) + \...
3
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3answers
73 views

Help understanding Polya's proof of the Arithmetic-Geometric Mean Inequality

Background Steele, in The Cauchy-Schwarz Master Class, states the general Arithmetic-Geometric Mean Inequality as follows (pg 23): $a_{1}^{p_{1}}a_{2}^{p_{2}} \ldots a_{n}^{p_{n}} \leq p_{1}a_{1} ...
1
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1answer
35 views

Inequality of double sum with max operator and squared

I am trying to understand a proof, however one inequality remains unclear: $\mathbb{E} \big( \frac{1}{k_1 k_0} \sum_{i=1}^{k_1} \sum_{j=1}^{k_0} a_{j,i} -b_{j,i} \big)^2 \leq \max_i \frac{1}{k_m^2}\...
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0answers
24 views

Proving equivalence with Farkas lemma

The problem I am trying to solve: trying to show that the two systems are equivalent. $$\exists x : Ax=a$$ $$Bx \leq b $$ second system: $$\nexists y,z: y^TA+z^TB=0$$ $$y^Ta+z^Tb<0$$ $$z\geq 0$$...
9
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1answer
333 views

Prove $3\sum\limits_{cyc}\,a^{\,2}b(\,a- b\,)\geqq b(\,a+ b- c\,)(\,a- c\,)(\,c- b\,)$

If you are interested in IMO 1983 We have $$\sum\limits_{cyc}\,a^{\,2}b(\,a- b\,)= \frac{bc(\,a+ b- c\,)^{\,2}(\,a- b\,)^{\,2}}{b(\,a+ b- c\,)+ 3\,a(\,c+ a- b\,)}+ $...
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2answers
78 views

How many tuples of {$a, b, c, …$} satisfy $abc… \leq n$?

Let $n$ and $k$ be positive integers. Let $a, b, c, ...$ be $k$ positive integers such that $abc... \leq n$. How many tuples of {$a, b, c, ...$} satisfy the inequality? Note that the tuples {$a=1, ...
1
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2answers
47 views

Proving $\sqrt{a^2+c^2}+\sqrt{b^2+d^2}\ge \:\sqrt{\left(a+b\right)^2+\left(c+d\right)^2}$ [duplicate]

The inequality: $$\sqrt{a^2+c^2}+\sqrt{b^2+d^2}\ge \:\sqrt{\left(a+b\right)^2+\left(c+d\right)^2}$$ But can someone help me with a nice elegant solution. This is an olympiad question I was trying to ...
0
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1answer
61 views

How can I prove this general case of the inequality?

I think I need some help with this problem. According to p.544 of Feller's An Introduction To Probability Theory. Vol II if $|a_k|\leq c_k$, $|b_k|\leq c_k$ then we have $|\prod_{i=1}^na_i -\prod_{i=...
1
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1answer
74 views

Proof idea for sum of square / square of sum inequality

This problem came up in a long proof and it is not clear to me how to show this. I tried to apply Cauchy-Schwarz but it is not tight enough. Any idea? Let $\Delta$ be a matrix of real values with $B$ ...
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1answer
59 views

Prove $\sum\limits_{cyc}\,\frac{a^{\,2}+ k\,b^{\,2}}{k\,a^{\,2}+ b^{\,2}}\geqq 3$

For $a\geqq b\geqq c> 0$. Prove $$\sum\limits_{cyc}\,\frac{a^{\,2}+ k\,b^{\,2}}{k\,a^{\,2}+ b^{\,2}}\geqq 3 \tag{SHED}$$ with $k= \frac{b}{c}\geqq 1$. I used SHEDtechniQ to find and I want to ...
0
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3answers
124 views

Prove $1+ a^{\,2}+ b^{\,2}+ c^{\,2}+ 4\,abc\geqq a+ b+ c+ ab+ bc+ ca$ for $a,\,b,\,c\geqq 0$

Given that $a,\,b,\,c$ are $3$ non-negatve numbers, prove $$1+ a^{\,2}+ b^{\,2}+ c^{\,2}+ 4\,abc\geqq a+ b+ c+ ab+ bc+ ca$$ Let $X= a+ b+ c$$,$ we have to prove $$\left ( \frac{1}{X^{\,3}}- \frac{1}{X^...
68
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3answers
6k views

Fastest way to check if $x^y > y^x$?

What is the fastest way to check if $x^y > y^x$ if I were writing a computer program to do that? The issue is that $x$ and $y$ can be very large.
1
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2answers
76 views

How do I prove the inequality $(\sum a^3)^2 \leq (\sum a^2)^3$?

Let $a_1, \dots, a_n \in \mathbb{R}.$ I wish to show that $(\sum_{i=1}^n a_i^3)^2 \leq (\sum_{i=1}^n a_i^2)^3$ in order to prove another statement. But I cannot see how to prove this, if at all the ...
1
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6answers
56 views

Prove $5^n + 3^n - 2^{2n+1} > 0$ by induction

I am not sure how to deal with the $-2^{2n+1}$ term. I did the basis proof for n=1 I am stuck at this step: $$ 5^{k+1}+3^{k+1}-2^{2(k+1)+1} = 5\cdot 5^k + 3 \cdot 3^k -2^3 \cdot 2^{2k} $$ Any ...
0
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2answers
51 views

Inequality in double integral [on hold]

Given The square $D=\{(x,y)\mid0\le x\le1 ; 0\le y\le 1\}$, prove that $$\iint_D \left(x^3+y^3\right)^{1/3} \ dA \le \iint_D \left(x^2+y^2\right)^{1/2} \ dA.$$
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0answers
31 views

$A$ is continuous in $0_X$ => $\exists c \in \mathbb{R}$, $ c \geq 0$ : $||Ax||_Y \leq c ||x||_X $

the map $A : X \rightarrow Y $ is linear, where $(X,||.||)$ and $(Y,||.||)$ are normalized vector spaces. I already have a solution, which is correct but a friend of mine showed me his solution and ...
2
votes
3answers
75 views

Find minimum and maximum of $P=a+b+c$

Let $a,b,c\ge 0$ such that $a^2+b^2+c^2+abc=4$. Find minimum and maximum of $$P=a+b+c$$ +)Maximum: Let $x=\frac{2\sqrt{ab}}{\sqrt{\left(c+a\right)\left(c+b\right)}};y=\frac{2\sqrt{bc}}{\sqrt{\left(a+...
-1
votes
3answers
43 views

Prove that $n(n+2)$ lies between $n^2$ and $(n+1)^2$, given n is a positive integer

I'm currently trying to prove the inequality $$n^2<n(n+2)<(n+1)^2$$ Is it possible to solve this without induction?
0
votes
1answer
76 views

Prove $\sum\limits_{cyc}\frac{1}{x^{2017}+ x^{2015}+ 1} \geq 1$ with $x,\,y,\,z>0,\,xyz= 1$ [closed]

Prove $$\sum\limits_{cyc}\frac{1}{x^{2017}+ x^{2015}+ 1} \geq 1$$ with $x,\,y,\,z>0,\,xyz= 1$ I try to use Jensen inequality, then: $$f\left ( x \right )+ f\left ( y \right )+ f\left ( z \right )\...
-2
votes
1answer
46 views

Prove $\sum\limits_{cyc}\frac{ab}{b^{\,2}+ c^{\,2}}\geqq \frac{3}{2}$

For $a\geqq b\geqq c> 0$. Prove $$\frac{ab}{b^{\,2}+ c^{\,2}}+ \frac{bc}{c^{\,2}+ a^{\,2}}+ \frac{ca}{a^{\,2}+ b^{\,2}}\geqq \frac{3}{2}$$ I used discrim to find and I want to see a solution ...
1
vote
2answers
49 views

Prove $2\sum\limits_{cyc}\,a^{\,3}+ 3\,abc\geqq 3\sum\limits_{cyc}\,a^{\,2}b$ [on hold]

For $a,\,b,\,c\geqq 0$ and $b\equiv {\rm mid}\,\{\,a,\,b,\,c\,\}$. Prove $$2\sum\limits_{cyc}\,a^{\,3}+ 3\,abc\geqq 3\sum\limits_{cyc}\,a^{\,2}b$$ Inspried from $\lceil$ Prove $k=0$ is the only non-...
0
votes
1answer
57 views

Prove $k=0$ is the only non-negative $k$ such that $\sum\limits_{cyc}\,a^{\,3}- \sum\limits_{cyc}\,a^{\,2}b\geqq k(\,a- b\,)(\,a- c\,)(\,b+ c\,)$

Prove with $a+ b,\,b+ c,\,c+ a\geqq 0$ $$\begin{equation}\begin{split} k= constant= 0 \end{split}\end{equation}$$ is the only non-negative $k$ such that $$\begin{equation}\begin{split} \sum\limits_{...
1
vote
2answers
90 views

Prove $\frac{a}{b+ c}+ \frac{b}{c+ a}+ \frac{c}{a+ b}+ \frac{63}{5}\left [ \frac{2\,c^{\,2}}{(\,a+ b\,)^{\,2}}- \frac{c}{a+ b} \right ]\geqq 0$

Given that $a,\,b,\,c> 0$, prove: $$\frac{a}{b+ c}+ \frac{b}{c+ a}+ \frac{c}{a+ b}+ \frac{63}{5}\left [ \frac{2\,c^{\,2}}{(\,a+ b\,)^{\,2}}- \frac{c}{a+ b} \right ]\geqq 0$$ I only tried Buffalo ...
0
votes
1answer
45 views

On inequality of exponentially activated, unit transform of vector

Let $x$ be a vector and $A$ a matrix. Let $$ y := \dfrac{Ax}{\lVert A \rVert} $$ bet a unit transform of vector $x$, and $$ \widehat{y}:= y/\lVert x\rVert $$ the same transformation with ...
1
vote
1answer
76 views

Prove $F(\,k\,)=(\,16\,X^{\,}- 24\,X+ 18\,)k^{\,2}- 11\,X+ 1\geqq 0$

Given that $1\leqq X\leqq k,$ prove that $$F(\,k\,)=(\,16\,X^{\,}- 24\,X+ 18\,)k^{\,2}- 11\,X+ 1\geqq 0 \tag{29}$$ Origin For $a,\,b,\,c\geqq 0$ and $a+ b+ c= 3$, prove that $(2+a^2)(2+b^2)(2+c^2)+...
2
votes
1answer
32 views

Gaussians and Young's inequality for convolutions

Consider a simplified version of Young's inequality: $$ ||f\ast g||_p\leq ||f||_1||g||_p, \quad 1< p\leq\infty $$ $$ f\ast g\equiv \int_{\mathbb R}dy f(y)g(x-y). $$ What strategy one should follow ...
0
votes
1answer
24 views

Norm 2 against norm inf

We know from basic linear algebra that $\forall x \neq 0, \frac{||x||_2}{||x||_{\infty}} \leq \sqrt{n}$ (where $n$ is the dimension).We also know that the equality occurs if and only if all ...
3
votes
2answers
57 views

Given $3$ positive reals $a$, $b$ and $c$ such that $a+b+c = 1$, show that $a^a b^b c^c + a^b b^c c^a + a^c b^a c^b \le1$.

Good Day! How are you doing? I was learning about the awesome A.M - G.M. inequality from the Brilliant Wiki. There was a question in the exercises: Given $3$ positive reals $a$, $b$ and $c$ such ...
0
votes
1answer
91 views

Is $\|\mathbf{u} -\mathbf{v}\|\leq \|\mathbf{u}\| + \|\mathbf{v}\| $?

Here's my working: $\|\mathbf{u} -\mathbf{v}\|^2 = \|\mathbf{u}\|^2 + \|\mathbf{v}\|^2 + (- 2\, \mathbf{u}\,\bullet\mathbf{v})$ Since, by the Cauchy-Schwarz theorem, $|\mathbf{u}\,\bullet\mathbf{v}| ...