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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Prove $(a+b+c-3)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-3\right)+abc+\frac{1}{abc}\ge 2.$

For any positive real numbers $a,b,c$ then prove $$(a+b+c-3)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-3\right)+abc+\frac{1}{abc}\ge 2.$$ I've try to assume $a+b+c\ge 3.\quad(1)$ By AM-GM $$abc+\frac{1}...
Hello world's user avatar
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Solving a system of modular inequalities

Let $N$ be an even positive integer. Let $f$ be a function on integers. I am attempting to solve integer inequalities of the form: $$ n_1,n_2,n_3,\ldots \in \{0,1\} $$ $$ 0\le f_j(n_1,n_2,\ldots)\mod{...
Bobby Ocean's user avatar
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How to show the following inequality: $|x^p - y^p| \leq |x - y| \cdot p \cdot (|x| + |x - y|)^{p - 1}$ ($1 \leq p < \infty$)

I would like to show the following inequality for all $1 \leq p < \infty$ and $x, y \geq 0$: $|x^p - y^p| \leq |x - y| \cdot p \cdot (|x| + |x - y|)^{p - 1}$ For $p = n \in \mathbb{N}$, this is ...
Smiley1000's user avatar
-1 votes
0 answers
15 views

Upper bound on the argmin of an information theoretic function

Given non-random constants $b$, $\delta$, $\sigma$, and random variables $\beta\sim\text{Bernoulli}(\pi)$ and $G\sim\mathcal{N}(0,1)$, I have the following information-theoretic function $$ U(b;\delta)...
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Prove: $\frac{1}{\sqrt{a+b^2}}+\frac{1}{\sqrt{b+c^2}}+\frac{1}{\sqrt{c+a^2}} \geq \frac{2\sqrt{3}+9}{6}$ for $ab+bc+ca=2$

Problem: Let $a,b,c$ are non-negatives such that $ab+bc+ca=2$. Prove: $$\frac{1}{\sqrt{a+b^2}}+\frac{1}{\sqrt{b+c^2}}+\frac{1}{\sqrt{c+a^2}} \geq \frac{2\sqrt{3}+9}{6}$$ Equality holds iff $(a,b,c) =(...
Danh Trung's user avatar
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0 answers
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A lemma involving Chebyshev's inequality to prove the weak law of large numbers

Reading Roch's Modern Discrete Probability one finds the following exercise (2.6): (Sums of uncorrelated variables). Centered random variables $X_1,\dots X_n$ are uncorrelated if forr all $r$ and for ...
René Quijada's user avatar
1 vote
1 answer
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Elegant Proof of $\frac{2}{5}\left(\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\right)+\frac{14}{5}\frac{3\sqrt[3]{xyz}}{x+y+z}\geq4$

I've been looking at the inequality: for $x, y, z > 0$, $$\left(1-\frac{3}{5}\right)\left(\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\right)+\left(1+3\cdot\frac{3}{5}\right)\frac{3\sqrt[3]{xyz}}{x+y+z}\...
asomog's user avatar
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2 answers
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Inequality for integers with floor function

I want to show that for any nonnegative integers $l$ and $b$ we have $$ \frac{l}{2^{b+1}} - 1 \leq \left\lfloor \frac{l-1}{2^{b+1}} \right\rfloor. $$ I have a proof where I wrote $l = \alpha\cdot 2^{...
Lereu's user avatar
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0 answers
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How to prove an inequality by induction [closed]

I am having difficulty proving this by induction: $$\sum_{k=1}^n \frac{k}{k^2 +1} \le \frac{n}{2}$$ Any tips on how to approach the problem? Thank you!
elguero's user avatar
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1 answer
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Does a constant $C>0$ exist such that for $\forall\ p\in\mathbb{C}[z_1,z_2]$ we have: $\sup_{z\in rD^2}|p(z_1,z_2)|\le C\sup_{z\in D^2}|p(z_1,z_2)|$?

Question: Does a finite constant $C>0$ exist such that for $\forall\ p\in\mathbb{C}[z_1,z_2]$ we have: $$\sup_{z\in r \mathbb D^2}|p(z_1,z_2)|\le C\sup_{z\in \mathbb D^2}|p(z_1,z_2)|$$ where $r>...
anon's user avatar
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A tail bound similar to Markov inequality [duplicate]

I am trying to prove the following: If $Z$ is s.t. $\mathbb{E}(Z)=0$ and $\mathbb{E}(Z^2)=1$, for any $r >0$, $$\mathbb{P}(Z\geq r)\leq (1 + r^2)^{-1}.$$ I tried a lot using Markov and Chebyshev ...
user115608's user avatar
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2 votes
1 answer
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An inequality with a "Cauchy-Schwarz" flavour

Let $x_1\leq x_2\leq ... \leq x_n$ and $y_1<y_2< ... <y_n$ be positive integers such that $x_1\geq 2$, $x_i < y_i$ and $y_i + 1<y_{i+1}$. Do we have that $$x_ny_n (\sum_{i=1}^n x_i)^2 \...
Juan Moreno's user avatar
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1 vote
1 answer
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How to prove $\lim_{n\to \infty} (1+\frac{1}{1!}+\frac{1}{2!}+...\frac{1}{n!}) = e$ with $x_n = (1+\frac{1}{n})^n$?

I'm very confused about the question below, which I couldn't figure out for days. In Example 5 the author is teaching us proving $$\lim_{n\to \infty} (1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+...\frac{...
John HHU's user avatar
-1 votes
0 answers
38 views

Prove that $\frac{R^2 r}{(R+r)^2} \geqslant \frac{1}{4}.$ [closed]

$f: \mathbb{D} \rightarrow \mathbb{C}$ is analytic with $f(0)=0, f^{\prime}(0)=1$, and satisfies that $|f(z)|<R$ on $\mathbb{D}$ for some $R>1$, $r := \inf \{|w|: w \notin f(\mathbb{D})\}.$ ...
KY LIAO's user avatar
-4 votes
0 answers
22 views

Given a system of n inequalities, prove the following [closed]

x ^ 4 - 2alpha_{l} * x ^ 3 + (alpha_{l} ^ 2 - 2alpha_{l} + 2) * x ^ 2 - 2alpha_{l}*x + (alpha_{l} - 1) ^ 2 <= 0 where each alpha t in [1/2, 5] (i=1,2,3...n). Let x_{l} be an arbitrary solution ...
Jay Khandelwal's user avatar
-1 votes
0 answers
34 views

Prove that $(a + b + c)(ab + bc + ca)(a^3 + b^3 + c^3) \le (a^2 + b^2 + c^2)^3$. [duplicate]

Problem: Let $a$, $b$, and $c$ be positive real numbers. Prove that $$(a + b + c)(ab + bc + ca)(a^3 + b^3 + c^3) \le (a^2 + b^2 + c^2)^3.$$ I have tried expanding everything out, and applying Muirhead’...
solasky's user avatar
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Inequality question(wish to confirm working)

Let $a_{i}$ be a set of positive real numbers such that $\sum_{i=1}^n a_{i}^3=3$ and $\sum_{i=1}^n a_{i}^5=5$. Prove that $\sum_{i=1}^n a_{i} > \frac{3}{2}$. My attempt: Using Titu's lemma, $$\frac{...
A shubh's user avatar
  • 121
3 votes
2 answers
69 views

If $x>0,y>0$ and $4xy=2^{x+y}$ then find the minimum and maximum values of $x+y$.

If $x>0,y>0$ and $4xy=2^{x+y}$ then find the minimum and maximum values of $x+y$. My Attempt I tried by putting $t=x+y$ $\Rightarrow 4x(t-x)=2^t$. On differentiation we have $4t-8x=\frac{dt}{dx}(...
Maverick's user avatar
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1 vote
1 answer
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Relationship between variance and covariance

I know that $$\text{var}(x-y) = \text{var}(x) + \text{var}(y) - 2\text{cov}(x,y)$$ and $$\text{cov}(x,y) = \frac{1}{2}(\text{var}(x) + \text{var}(y) - \text{var}(x-y)).$$ Is it possible to say that $$\...
lela's user avatar
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0 votes
0 answers
33 views

Prove that $\mathop{max}_{x\in [0,1]}|u(x)|\le \frac{1}{8}\mathop{max}_{x\in [0,1]}|{u''(x)}|$ when $u(x)\in C^2[0,1],u(0)=u(1)=0$ [duplicate]

I want to prove the inequality with analysis methods instead of method of PDE Actually we can consider the following PDE: $$\left\{\begin{array}{l} u''(x)=u''(x)\\ u(0)=u(1)=0 \end{array}\right.$$ ...
Gang men's user avatar
  • 425
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0 answers
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Question about proof on inequalities

I have four equations: $n\geq 5k+2$ $n_1\geq 5k_1+1$ $n_2\geq 5k_2 +1$ $n=n_1+n_2$ What I want to get eventually is $k_1+k_2=k$, but it seems like I cannot say that as I get $n\geq 5(k_1+k_2)+2$ and ...
Johny's user avatar
  • 21
3 votes
3 answers
136 views

Elementary proof that $(1+1/x)^x$ is increasing for $x \in \mathbb{R}_{>0}$

All similar proofs I could find show that $(1+1/n)^n$ is increasing for positive integers values of $n$ only, or show that the derivative of $(1+1/x)^x$ is positive for all $x \in \mathbb{R}_{>0}$. ...
Hussein Aiman's user avatar
0 votes
1 answer
82 views

Convex function derivatives inequality

Decide whether there exists function $f: \mathbb{R} \to (0,+\infty)$ such that for all $x \in \mathbb{R}$ we have $f''(x)f(x)>(f'(x))^{2}$. I know that if such function would exist then $f''(x) &...
Math_man's user avatar
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2 votes
7 answers
86 views

Showing that for every acute angle $x$ in a right triangle $\frac{1}{\sin x}+\frac{1}{\cos x}\ge 2\sqrt 2$ is always true

The problem is to show that in a right-angle triangle with hypothenuse K and sides M and N, the inequality $\frac{K}{M}+\frac{K}{N}\ge 2\sqrt 2$ is always true. My approach: I tried to simplify the ...
Billy's user avatar
  • 165
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0 answers
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Exercise 8.15 Brezis - Interpolation inequality

I have a problem with this exercise (see the text in the following link). Interpolation like inequality ,Question from Brezis' book exercise 8.15 The link practically solves it. Only one last step ...
Seurat's user avatar
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0 votes
1 answer
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why this equality involving matrix holds true?

I am studying Lemma 11 of the paper I am having difficulty understanding on the last step, in particular, I have two questions: The first question (1) On the first equality of the last step on page 15,...
chloe's user avatar
  • 314
-2 votes
2 answers
44 views

Log summation with geometric progression

Problem statement: Show that $\ln\left(x_{n}\right)<\sum_{k=1}^{n}\frac{1}{3^{k}}$ where $x_{n}$ = $\left(1+\frac{1}{3}\right)\left(1+\frac{1}{3^{2}}\right)...\left(1+\frac{1}{3^{n}}\right)$ I ...
LÜHECCHEgon's user avatar
2 votes
1 answer
31 views

Inequality of entropies for Bernoulli plus Gaussian

Question: Given $X\sim\text{Bernoulli}(\alpha)$, $Y\sim\mathcal{N}(0,1)$, and two non-random constants $C_1$ and $C_2$ such that $C_1>C_2$. What can we say about the inequality between the two ...
Resu's user avatar
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0 votes
0 answers
28 views

Subgaussian concentration implies bound on probability density

Very simple question: Suppose $X$ is a subgaussian RV with a density $p_X(x)$. Does this imply any nontrivial upper bounds on $p_X(x)$ for large $x$? Some context: It is not hard to show that bounds ...
student566's user avatar
0 votes
1 answer
28 views

Inequality regarding inner product and functions with zero integral

Suppose $X$ is a finite set and $f:X\rightarrow \mathbb{R}$ satisfies $\sum_{x\in X}f(x)=0$. Let $p\in\Delta(X)$ be a probability measure on $X$. Does the following statement hold? $$ \sum_{x\in X} f(...
Lemma1's user avatar
  • 3
1 vote
2 answers
139 views

Prove $(\Re{z_1})^2+(\Re{z_2})^2+(\Re{z_3})^2 < \frac{3}{2}$ for solutions of $4z^3-4z^2+12z-1=0$

The statement of the problem : Let $z_1, z_2, z_3$ be the solutions of the equation $$4z^3-4z^2+12z-1=0$$ Prove that : $(\Re{z_1})^2+(\Re{z_2})^2+(\Re{z_3})^2 < \frac{3}{2}$ ($\Re x$ is the real ...
Last X's user avatar
  • 179
1 vote
0 answers
44 views

Suppose functions $y$ and $f$ are non-negative and continuous on the interval, prove this inequality

Suppose functions $y$ and $f$ are non-negative and continuous on the interval $<a,b>$ (where $<$ can be $( $ or $[ $ and $>$ can be $] $ or $) $), and let $\lambda > 0$. If the ...
Arbatus's user avatar
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0 votes
1 answer
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How to prove $a^{b}+b^{a} \geq 1$ for all $a, b \geq 0$? [duplicate]

It is clear if $a \geq 1$ or $b \geq 1$, but how can I show it when $0< a,b <1$? I want hints.
Mahmoud albahar's user avatar
0 votes
0 answers
32 views

Upper bound on matrix vector multiplication

For a matrix $A$, let $\|A\|_2$ denote the spectral norm (operator norm with euclidean norm used in both spaces). And for a vector $x$, let $\|x\|_2$ denote the euclidean norm. Is the following ...
Dylan Dijk's user avatar
-2 votes
0 answers
38 views

Upper Bound of a product of sines

Consider the function $$ f_n(t)= \prod_{1 \leq k \leq n-1,\\gcd(k,n)=1} \sin(t-\frac{k \pi}{n}),t \in [0,\pi].$$ I wonder whether it is possible to compute some nontrivial uppper bounds of the ...
AgnostMystic's user avatar
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0 votes
0 answers
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Estimating Difference of Two Averages of Function by Its BMO Norm

Assume $f\in \text{BMO}(\mathbb{R}^d)$. Denote $f_Q=\displaystyle\dfrac{1}{|Q|}\int_Q f(x)\,dx$. I want to show for $\alpha>2$ and any cube $Q$ with positive volume, we have $|f_{\alpha Q}-f_Q|\leq ...
Laurence PW's user avatar
0 votes
0 answers
29 views

Bounds for Gamma function expression

Let $a,b,c,d \in \mathbb R$ such that $c\ge 1$ and $0 < d < 2$. Let $i$ be the imaginary unit and $\Gamma$ the Euler Gamma. Numerically it looks that $$2^{(a+bi)}\Gamma\left(\frac{c-(a+bi)}{d} \...
zelda's user avatar
  • 1
1 vote
0 answers
41 views

Prove the following inequality for $a,b,c,d>0$.

Let $a,b,c,d>0$. Is there an easy way to see that the following inequality is true: $$4(a+b+c+d)+2(a^2+b^2+c^2+d^2)-4(ab+ac+ad+bc+bd+cd)-(a^2b+ab^2+a^2c+ac^2+a^2d+ad^2+b^2c+bc^2+b^2d+bd^2+c^2d+cd^2)...
Ryan Hendricks's user avatar
4 votes
3 answers
162 views

If $x,y,z>0$ and $x+y+z=1$,then find the maximum value of $(1-x)(2-y)(3-z)$.

If $x,y,z>0$ and $x+y+z=1$,then find the maximum value of $(1-x)(2-y)(3-z)$. My Attempt: We have $0<1-x<1,1<2-y<2,2<3-z<3$ so A.M-G.M inequality cannot be used since we can never ...
Maverick's user avatar
  • 9,046
0 votes
0 answers
138 views

Minimize $P=\sqrt{a^2b+b^2c+c^2a+abc}+\sqrt{a+b+c-1}$ when $ab+bc+ca=3.$

Let $a,b,c\ge 0:ab+bc+ca=3.$ Find the minimal value $$P=\sqrt{a^2b+b^2c+c^2a+abc}+\sqrt{a+b+c-1}.$$ I think $2+\sqrt{2}$ is desired result. $P$ attain this value when $a=b=c=1.$ I use $a+b+c\ge \sqrt{...
Hello world's user avatar
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0 answers
40 views

Inequality with Products and Sums

I need help to find a proof for the following inquality. Assuming that $ 0 \leq c_i \leq 1 $ and $ 0 \leq d_i \leq 1 $, show that $$ \prod_{i=1}^N (c_i + d_i - c_i d_i) \geq \prod_{i=1}^N c_i + \prod_{...
Duns's user avatar
  • 758
0 votes
1 answer
26 views

p-variation of x

Let us consider data $(k \in \mathbb{Z}_{\ge 1}, (s_1, \ldots, s_k) \in \mathbb{R}^k_{\ge 0})$ such that $s_1 + \ldots + s_k = 1$. The mesh of a datum like that is $\max (s_1, \ldots, s_k)$. Given $p &...
Sasha's user avatar
  • 1,113
2 votes
2 answers
77 views

The total number of non-negative integers $n$ satisfying the equations $n^2=p+q$ and $n^3=p^2+q^2$, where $p$ and $q$ are integers, is...

The total number of non-negative integers $n$ satisfying the equations $n^2=p+q$ and $n^3=p^2+q^2$, where $p$ and $q$ are integers, is... Case$1$: If both $p$ and $q$ are non-negative integers then, ...
aarbee's user avatar
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3 votes
0 answers
62 views

A self-proof of Vapnik - Chervonenkis theorem

Theorem: For every $\varepsilon >0$, with the probability greater than $1-\varepsilon$ \begin{align*} R_p(\hat{g}_{n,\mathcal{G}}) - R_{p}(g^*_{p,\mathcal{G}}) \le 2 \sqrt{\dfrac{2V_{\mathcal{G}...
Eto's user avatar
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0 votes
1 answer
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I'm wondering if we can prove the following inequality with those ideas.

We aim to prove that $\tan\left(\frac{|x+y|}{1+|x+y|}\right)$ $\leq \tan\left(\frac{|x|}{1+|x|}\right) + \tan\left(\frac{|y|}{1+|y|}\right)$ for all $x, y \in \mathbb{R}$. Let's assume that $\tan\left(...
impact21's user avatar
2 votes
1 answer
74 views

Prove that $\int f \ln(f) d \mu =\sup \left \{ \int f \phi d \mu : \int e^{\phi} d\mu \leq 1 \right \}$

Question Prove that $\int f \ln(f) d \mu = \sup \left \{ \int f \phi d \mu : \int e^{\phi} d\mu \leq 1 \right \}$ With $f$ verifying $ \int f d \mu = 1 $ and $ f \cdot \ln(f) $ is integrable, with $ \...
OffHakhol's user avatar
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3 votes
1 answer
270 views

Proof of Triangle Inequality for $d(g; x, y) = \left(|x-y|^4 + g\,| x \times y |^2\right)^{\frac{1}{4}}$

I am seeking assistance in proving that a function, denoted as $d(g; x, y)$, defined on $\mathbb{R}^2 \times \mathbb{R}^2$ and parameterized by the non-negative real number $g$, may satisfy the ...
roiban12096's user avatar
-4 votes
0 answers
56 views

A very old problem on Maxima and minima [closed]

Find the maximum value of the expression $$ |\left|x_1-x_2\right|-x_3\left|-\ldots-x_{1990}\right|, $$ where $x_1, x_2, \ldots, x_{1990}$ are distinct natural numbers between 1 and 1990. (O Bogopol'...
SquïdÆir's user avatar
1 vote
1 answer
74 views

Prove that $a+b+c+d \geq \frac{2}{a+1}+\frac{2}{b+1}+\frac{2}{c+1}+\frac{2} {d+1}$

the question Let $a,b,c,d>0$ with $a+b+c+d \geq \frac{1}{a}+ \frac{1}{b}+\frac{1}{c}+ \frac{1}{d}$. Prove that $a+b+c+d \geq \frac{2}{a+1}+\frac{2}{b+1}+\frac{2}{c+1}+\frac{2} {d+1}$ the idea Maybe ...
IONELA BUCIU's user avatar
  • 1,029
7 votes
2 answers
233 views

Prove ${2n \choose n+i} \geq e^{-8 i^2/n} {2n \choose n}$

I am trying to prove ${2n \choose n+i} \geq e^{-8 i^2/n} {2n \choose n}$ for $0\leq i \leq n$. My attempt: I rewrote ${2n \choose n+i}$ to $${2n \choose n+i} = {2n \choose n} \prod_{1\leq j \leq i} \...
AspiringMat's user avatar
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