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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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Inequalities and averages

Dikshant writes down $2 k+1$ positive integers in a list where $k$ is a positive integer. The integers are not necessarily all distinct, but there are at least three distinct integers in the list. The ...
aiman's user avatar
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-1 votes
2 answers
64 views

Let $a,b\in \mathbb N$. If $a+b=101$ and $\sqrt{a}+\sqrt{b}$ has its maximum value then calculate $a\cdot b$

The problem Let $a,b\in \mathbb N$. If $a+b=101$ and $\sqrt{a}+\sqrt{b}$ has its maximum value then calculate $a\cdot b$ My idea $(\sqrt{a}+\sqrt{b})^2=a+b+ 2\cdot \sqrt{ab}= 101 + 2\cdot \sqrt{ab}$ ...
IONELA BUCIU's user avatar
0 votes
1 answer
77 views

How to prove this simple inequality based on convexity of $e^{x}$?

Suppose $\theta > 0$ and $x>0$. I would like to show that $$ e^{\theta(x+1)} - e^{\theta x} - \frac{ e^{\theta x}-1}{x} \geq \frac{e^{\theta x}-1}{x} - (1-e^{-\theta}) $$ Another way to put it: ...
unknowngoogle's user avatar
0 votes
1 answer
15 views

fractional power function inequality

I am facing an issue to prove if its true the following inequality and any help is greatly appreciate. I used this inequality $x^\alpha \le \alpha x+1-\alpha$ but still I could not get the result. Let ...
Jabar S. Hassan's user avatar
0 votes
1 answer
72 views

Differential inequation $f'(x) > f(x)/x$

I am interested in the following differential inequation, for $f : \mathbb{R} \to \mathbb{R}$: $$f'(x) > \dfrac{f(x)}{x}$$ I only care about this being satisfied at infinity, ie for $x \gg 1$. ...
Azur's user avatar
  • 2,311
0 votes
1 answer
67 views

Show that: $(\frac{MN}{MA})^2+ (\frac{MP}{MB})^2+ (\frac{MQ}{MC})^2+ (\frac{MR}{MD})^2 \geq \frac{4 }{9}$

The problem Let $M$ be a point inside the tetrahedron $ABCD$. We denote by $N,P,Q,R$ the intersections of the lines $AM,BM,MC,DM$ with the planes $(BCD),(ACD),(ABD)$, respectively $(ABC)$. Show that: $...
IONELA BUCIU's user avatar
0 votes
0 answers
38 views

Same number of lists of integers [duplicate]

Have a following problem for which I'll show my reasoning (the problem is $1.6$ from book Problem solving methods in combinatorics by Pablo Soberon): If we want to write all the lists of length $n$ ...
slomil's user avatar
  • 176
0 votes
2 answers
105 views

better method of solving quadratic / cubic

Problem : Let $$\begin{align} f(x) &= x^4 - 8x^3 + 18x^2 \\ g(x) &= 9x^2 - 64x\end{align}$$ . Define $h : \mathbb{R}^+ \to \mathbb{R}$, $h(x)=f(x)-ag(x)$ for some real number $a$. If $h(x)$ ...
bFur4list's user avatar
  • 2,761
3 votes
0 answers
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AM-GM inequality for non necessary positive numbers

For nonnegative real numbers $x_1,\cdots,x_n$ ($n\geqslant2$), it is well known that : $$n\prod_{i=1}^nx_i\leqslant\sum_{i=1}^nx_i^n\tag{$\star$}$$since this is equivalent to the AM-GM inequality. But ...
Adren's user avatar
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0 votes
1 answer
118 views

Show that $\frac{1-xy-x}{x+y+3} + \frac{1-zy-y}{z+y+3}+ \frac{1-xz-z}{x+z+3} \geq \frac{5}{11}$

The problem a) Show that $\frac{ab}{a+b}+ \frac{cd}{c+d} \leq \frac{(a+c)(b+d)}{a+b+c+d}$, with equality for $ad=bc$ (solved already) b) Let the real numbers $x,y,z \in (0, \infty)$ with $x+y+z=1$. ...
IONELA BUCIU's user avatar
1 vote
1 answer
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Finding a value $n$ such that $\sqrt[n]{x^{x^{2}}} \le x^{\sqrt[n]{x^{2}}}$ is true.

For what value of $n \in \mathbb{N}$ such that the following inequality is true. $$\sqrt[n]{x^{x^{2}}} \le x^{\sqrt[n]{x^{2}}}$$ Where $0<x\le \sqrt[5]{216}$ ATTEMPT: This is my first time tackling ...
JAB's user avatar
  • 199
0 votes
1 answer
31 views

Proof of Khintchine’s inequality for sub-gaussians

I am trying to prove the exercise 2.6.6 of HDP book by Roman Vershynin. The exercise is as follows. Actually the right hand side is easy to find, and the left hand side is also easy if given the red-...
dhliu's user avatar
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0 answers
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The diameter of the union of two sets in a metric space cannot exceed the sum of the diameters of the two sets and the distance between them

Let $A$ and $B$ be any two (nonempty) sets in a metric space $(X, d)$. Then how to show that $$ d (A \cup B) \leq d(A) + d(B) + d(A, B)? \tag{0} $$ Here we have the following definitions: For any (...
Saaqib Mahmood's user avatar
0 votes
0 answers
44 views

Estimation of Sum of Binomial Coefficients that doesn't Start from Zero

I am having some trouble showing the following estimation: Let $\mu \in (0, 1)$ be an absolute constant. There exists an absolute constant $\eta \in (0, 1)$ small enough such that the number of ...
Partial T's user avatar
  • 583
3 votes
0 answers
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An integral inequality involving the Bernoulli polynomials

The classical Bernoulli polynomials $B_j(t)$ are generated by \begin{equation*} \frac{z\operatorname{e}^{t z}}{\operatorname{e}^z-1}=\sum_{j=0}^{\infty}B_j(t)\frac{z^j}{j!}, \quad |z|<2\pi. \end{...
qifeng618's user avatar
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0 votes
1 answer
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A problem on dirt displacement

Definition. Given a function $f\in L^1(\mathbb{R})$ such that $xf\in L^1(\mathbb{R})$, the quantity $\int_\mathbb{R}xf(x)\,dx$ is called the unnormalized center of mass of $f$ and is denoted $UCM(f)$. ...
aleph2's user avatar
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3 votes
0 answers
86 views

A trigonometric maximum problem involving trigonometric constraints

Let $a,b,c,\alpha,\beta\in\mathbb{R}^+$ and $\alpha+\beta<2\pi$. Prove that if and only if $$\frac{\sin\alpha}{a\sqrt{b^2+c^2-2bc\cos\alpha}}=\frac {\sin\beta}{b\sqrt{a^2+c^2-2ac\cos\beta}}=-\frac{\...
Mr.He's user avatar
  • 579
4 votes
2 answers
284 views

Show that $\frac{a^2}{b^2-2b+2024}+ \frac{b^2}{c^2-2c+2024}+ \frac{c^2}{a^2-2a+2024} \geq \frac{3}{2023}$ [closed]

Let the real numbers $a,b,c \in \mathbb{R}$ with $a+b+c=3$. Show that: $\frac{a^2}{b^2-2b+2024}+ \frac{b^2}{c^2-2c+2024}+ \frac{c^2}{a^2-2a+2024} \geq \frac{3}{2023}$. My idea: First of all, I thought ...
IONELA BUCIU's user avatar
3 votes
1 answer
54 views

$a+b\sqrt{2}>1, a^2-2b^2=\pm 1 (a,b\in \mathbb{Z}) \implies a+b\sqrt {2}\geq 1+\sqrt {2}$

$\textbf{Example}:$ Let $K=\mathbb{Q}(\sqrt 2)$. We claim that $1+\sqrt 2$ is the fundamental unit of $K$. Easy to show that $N(1+\sqrt 2)=-1$ and thus a unit. Remain to show that if $v>1$ is any ...
Bowei Tang's user avatar
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1 vote
0 answers
55 views

Reference for $\sum_{m=1}^p\sum_{n=1}^q\frac2{\cos(2m\pi/p)+\cos(2n\pi/q)}\le pq(|p-q|+1)$ with coprime odd positive integers $p$ and $q$? [closed]

I am having trouble with the following problem that I found in this Art of Problem Solving post, and I would like some help to find a reference for it. Let $p$ and $q$ be coprime odd positive integers....
Curious's user avatar
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Prove that $\displaystyle \sum\limits_{i=1}^3\sqrt{ \sum\limits_{j=1}^3a_{ji}^2}\leq\sqrt{2}f(a_{11},\cdots,a_{33})$

For a $3\times3$ matrix $A=(a_{ij})$, let \begin{aligned}&f(a_{11},a_{21},a_{31},a_{12},a_{22},a_{32},a_{13},a_{23},a_{33})\\=&\text{max}\{|a_{11}+a_{21}+a_{31}|+|a_{12}+a_{22}+a_{32}|+|a_{13}+...
grj040803's user avatar
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-1 votes
0 answers
19 views

Doubt regarding usage of inequalities in strong induction [closed]

Please click to see the proof In this proof of lucas sequence using strong induction, in the third line why did we try to contend the inequality, specifically using (7/4)^n ? Why not anything else ...
Krishna Rao's user avatar
2 votes
2 answers
75 views

Show that : $\sqrt{[x]\cdot \{x\}} +\sqrt{x \cdot \{x\}} + \sqrt{[x]\cdot x} \leq 2x$

Show that for any positive real number $x$ the inequality holds: $\sqrt{[x]\cdot \{x\}} +\sqrt{x \cdot \{x\}} + \sqrt{[x]\cdot x} \leq 2x$ where by $[a], \{a\}$ we mean the whole par and fractional ...
IONELA BUCIU's user avatar
5 votes
1 answer
186 views

Finding the integer part of a sum

I want to find the integer part of $$\sum_{n=1}^{10^9}\frac{1}{n^{2/3}}=S$$ I know there is a way using integration but I tried using a different approach. I saw this approach with square roots but I ...
Vedant Lohan's user avatar
1 vote
0 answers
21 views

Upper bound for distribution function for variable with zero expectation. [duplicate]

A problem from final Year 1 probability exam. Is it true for any random variable $Y$ s.t. $E[Y]=0$ and $E[Y^2]<\infty$ that: $P(Y>x)\leq\frac{E[Y^2]}{E[Y^2]+x}$ ? I thought we can rewrite it ...
innerproduct's user avatar
1 vote
2 answers
73 views

Find min and max of $P = (|a − b| + 3)(|b − c| + 3)(|c − a| + 3)$

For $a,b,c \in \mathbb{R}$, $a^2 + b^2 + c^2 ⩽ 2$ Find min and max value of $P = (|a − b| + 3)(|b − c| + 3)(|c − a| + 3)$ I don't understand how to find min and max value of an absolute value sign. ...
trum fi fai's user avatar
0 votes
0 answers
17 views

Local property of an integration inequality to global result

Here is the question. Let $f$, $g$ be locally integrable functions on $\mathbb{R}^n$ such that $$\inf_{a \in \mathbb{R}} \int_{B} |f(x) - a|\ dx \leqslant \int_B|g(x)|\ dx$$ for all balls $B$ in $\...
ZYZ's user avatar
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6 votes
0 answers
164 views

Show that $(1-z_1\overline{w_1})(1-z_2\overline {w_2})+(1-z_1\overline {w_2}) (1-z_2\overline {w_1})$ is non-vanishing.

Show that $\left (1-z_1\overline{w_1} \right ) \left (1-z_2\overline {w_2} \right )+ \left (1-z_1\overline {w_2} \right ) \left (1-z_2\overline {w_1}\right )$ is non-vanishing (does not take the value ...
Anacardium's user avatar
  • 2,610
1 vote
1 answer
48 views

solution-verification | Find the minimum of the expression $E(x)= \frac{-2x^4+8x^3-8x^2-9}{x^4-4x^3+4x^2+3}$

the problem a) Show that, for any real number $x$, $x^4-4x^3+4x^2+3>0$ b) Find the minimum of the expression $E(x)= \frac{-2x^4+8x^3-8x^2-9}{x^4-4x^3+4x^2+3}$, where x is a number real my idea So ...
IONELA BUCIU's user avatar
-1 votes
4 answers
88 views

How to find the range of a quadratic function we can't use the quadratic formula?

In the question $f(x) = \frac {x}{(1+x^2)}$ $ yx^2 - x + y = 0 $ $x = \frac {1 ± \sqrt {1-4y^2}}{2y}$ We know that $x$ belongs to $\Bbb R$. So, $\frac {1 ± \sqrt {1-4y^2}}{2y}$ also belong to $\Bbb R$....
The's user avatar
  • 1
0 votes
0 answers
22 views

Why is the von Neumann inequality not always fulfilled for n-tuples of commuting contractions? Why cant we just take the single dilations and get it?

Let $(T_1, \ldots, T_n)$ be an $n$-tuple of contractions in a Hilbert space $H$. We know, that for every single $T_i$, there exists a unitary operator $U_i$ in a Hilbert space $K_i$, such that $$T_i = ...
S-F's user avatar
  • 41
1 vote
1 answer
54 views

Prove/disprove $\sum_{i=1}^n \left\lfloor {x_i}^{y} \right\rfloor \leq \left\lfloor x^y\right\rfloor$ for $x_i,y\geq 1.$

If $x_i,y\in\mathbb{R}_{\geq 1},\ \displaystyle\sum_{i=1}^n x_i = x,$ then it is obviously true that $\displaystyle \sum_{i=1}^n {x_i}^y \leq x^y,$ due to the Binomial theorem. After trying various ...
Adam Rubinson's user avatar
0 votes
0 answers
73 views

Show that $\frac{(a+1)(a-b)}{b+1}+ \frac{(b+1)(b-c)}{c+1} + \frac{(c+1)(c-a)}{a+1} \geq 0$

The problem Let $a,b$ be some real nonzero numbers. Show that: a) $a^2 \geq \frac{b(a+1)^2}{b+1}-b$ b) $\frac{(a+1)(a-b)}{b+1}+ \frac{(b+1)(b-c)}{c+1} + \frac{(c+1)(c-a)}{a+1} \geq 0$ my idea I was ...
IONELA BUCIU's user avatar
-2 votes
0 answers
30 views

System of nonlinear equations with squares and cubes [closed]

I am looking for solutions in $\mathbb{R}^+$ to the following system of equations: $a^2+b^2+c^2=2025$ and $\frac{a^3}{b}+c+\frac{b^3}{a}+c+\frac{c^3}{a}+b \leq\frac{1}{2}(a^2+b^2+c^2)$.
Ana's user avatar
  • 1
3 votes
1 answer
93 views

How to prove Bernstein's inequality?

I am currently learning the basics of machine learning and have come across Bernstein's inequality. This inequality is particularly useful in understanding the behavior of averages of random variables....
Jimmy Zhao's user avatar
2 votes
4 answers
142 views

Prove $\sum_{i=1}^{30}\frac{1}{10+i} >\frac{13}{12}$

Prove $$\sum_{i=1}^{30}\frac{1}{10+i} >\frac{13}{12}$$ My effort: $$\begin{aligned} & \frac{1}{11}>\frac{1}{42} \\ & \frac{1}{12}>\frac{1}{42} \\ & \frac{1}{13}>\frac{1}{42} \\...
LifeIsMath's user avatar
-3 votes
0 answers
32 views

$\sqrt{a^2 - (b-c)^2}+\sqrt{b^2 - (a-c)^2}+\sqrt{c^2 - (a-b)^2} \leq \sqrt{ab} + \sqrt{ac} + \sqrt{bc} $ [closed]

Let $a;b;c$ are the 3 -sided length of a triangle. Prove that: $\sqrt{a^2 - (b-c)^2}+\sqrt{b^2 - (a-c)^2}+\sqrt{c^2 - (a-b)^2} \leq \sqrt{ab} + \sqrt{ac} + \sqrt{bc} $
Lunatic's user avatar
1 vote
2 answers
105 views

Upper bound of central probability of a binomial variable [closed]

Given an integer $k \geq 0$, if random variable satisfies the following condition: $$ \mathbb{P}\left(X = i\right) = \frac{\binom{k}{i}}{2^{k}}, \quad i \in \left\lbrace 0,1,\ldots,k\right\rbrace $$ ...
Jimmy Zhao's user avatar
4 votes
1 answer
92 views

Upper bound for $\sum_{k=1}^n a_{k}b_{n+1-k}$ with $\sum_{k=0}^{j} a_k b_{j-k} =1$

Problem 1: Set $a_0 = b_0 =1$. Given $n \in \mathbb{N} $ and real numbers $a_i, b_i \in [0,1)$ for $i\in \{ 1,2,3~ ...~n\}$ with $$\sum_{k=0}^{j} a_k b_{j-k} =1$$ all $j \le n$. Then I conjecture that ...
M.E.W.'s user avatar
  • 198
1 vote
1 answer
97 views

$\sum_{cyc}\sqrt[3]{\frac a{b(b+2c)}}\ge{\frac3{\sqrt[3]{3}}}$, for positive $a,b,c$ with $ab+bc+ca=3$

Let $a,b,c$ be positive real numbers satisfying $ab+bc+ca=3$. Prove that: $$\sqrt[3]{\frac{a}{b(b+2c)}}+\sqrt[3]{\frac{b}{c(c+2a)}}+\sqrt[3]{\frac{c}{a(a+2b)}}\ge{\frac{3}{\sqrt[3]{3}}}$$ I think the ...
bestty's user avatar
  • 147
8 votes
2 answers
663 views

Positive sum can always be presented as a sum with strictly positive incremental sub-sums

I am trying to prove the following (which to my knowledge is just a "conjecture", I have not seen this proved anywhere or mentioned as a theorem so it very well may be a false statement): ...
giorgio's user avatar
  • 583
-1 votes
0 answers
30 views

PDEs inequality solution [closed]

i have a PDE inequality that i tried a lot to get a general analytical solution or at least a particular family of solutions. The PDE inequality is: $$ \frac{\partial }{\partial r}\left( y\frac{\...
Soufiane Fares's user avatar
2 votes
1 answer
134 views

Order between the solutions of linear recursions [closed]

We consider the following 4 linear recursions: $$ \begin{array}{llll} u_{n+3}& = \frac 12 u_{n+2} & + \frac 14 u_{n+1} & + \frac 18 u_n \\ v_{n+3} & = \frac 12 v_{n+2} &+ \frac ...
Olivier's user avatar
  • 1,323
-1 votes
1 answer
55 views

Problem in proving the equality case in the Hölder's inequality as presented here.

We let $\mathbb{K}$ stand for $\mathbb{R}$ or $\mathbb{C}$. Let $X$ be $\mathbb{K}^n$ and, for $1 \leq p < \infty$, let $\| x \|_p := \left( \sum_{i=1}^n |x_i|^p \right)^{1/p}$ and for $p = \infty$,...
Thomas Finley's user avatar
1 vote
0 answers
112 views

For what $(a, b)$ does $(a + \sqrt{b})^{1000}$ come within $10^{-7}$ of an integer?

'For what $(a, b)$ does $(a + \sqrt{b})^{1000}$ come within $10^{-7}$ of a whole number?'. (a,b) are natural numbers. (may use calculator) I came across the question 'Prove that $(1 + \sqrt{2})^{1000}$...
Ellison's user avatar
  • 11
1 vote
1 answer
107 views

Show that $\frac{1}{a+b+c}+\frac{1}{a+b+d}+ \frac{1}{a+d+c}+\frac{1}{d+b+c}< \frac{6}{a+b+c+d}$

the problem Let $a,b,c,d\in R^*_+$ so every of the 4 number is smaller than the sum of the other 3. Show that $\frac{1}{a+b+c}+\frac{1}{a+b+d}+ \frac{1}{a+d+c}+\frac{1}{d+b+c}< \frac{6}{a+b+c+d}$ ...
IONELA BUCIU's user avatar
0 votes
1 answer
61 views

Find the dominating term

I know that, $$a+b=\Theta(\max\{a, b\}).$$ I need to understand the dominating term in this expression, $$\mathcal{O}(m^{\frac{2}{3}}n^{\frac{2}{3}}+m+n).$$ We know that,$$m^{\frac{2}{3}}n^{\frac{2}{...
D. S.'s user avatar
  • 138
1 vote
1 answer
58 views

solution-verification | Show that $\frac{3x^2+6xy+2y^2}{x+y}+ \frac{3x^2+6xz+2z^2}{x+z}+ \frac{3z^2+6zy+2y^ 2}{z+y} \leq \frac{25}{4}(x+y+z)$.

the problem Let $x,y,z$ be three strictly positive real numbers. Show that $\frac{3x^2+6xy+2y^2}{x+y}+ \frac{3x^2+6xz+2z^2}{x+z}+ \frac{3z^2+6zy+2y^ 2}{z+y} \leq \frac{25}{4}(x+y+z)$. my solution ...
IONELA BUCIU's user avatar
2 votes
0 answers
57 views

Proof of inequality involving matrices: $\operatorname{tr}(I-\Lambda) + \log \det \Lambda \leq 0$

Notation Let $I_n$ be an $n$th order unit matrix. Problem We want to show that $$ \operatorname{tr}(I_n-\Lambda) + \log \det \Lambda \leq 0, $$ where $n\times n$ matrix $\Lambda$ is a positive ...
ytnb's user avatar
  • 588
5 votes
0 answers
169 views

Prove or disprove the limit of a sequence is negative.

I have a sequence of positive numbers $\{f_k\}$ such that all the odd terms sum up to 1 and so do all the even terms, i.e. $\sum_{k=1}^{\infty}f_{2k-1}=\sum_{k=1}^{\infty}f_{2k}=1$, and $1>\sum_{i=...
Jake ZHANG Shiyu's user avatar

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