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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3 votes
2 answers
76 views

• 3,171
0 votes
1 answer
24 views

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 ...
• 630
-1 votes
0 answers
15 views

• 1,780
0 votes
2 answers
37 views

• 625
0 votes
0 answers
31 views

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 ...
• 3,443
2 votes
1 answer
112 views

• 41
-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 $R>1$, $r := \inf \{|w|: w \notin f(\mathbb{D})\}.$ ...
-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 ...
-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’...
• 63
3 votes
0 answers
65 views

• 13
0 votes
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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: $$\left\{\begin{array}{l} u''(x)=u''(x)\\ u(0)=u(1)=0 \end{array}\right.$$ ...
• 425
0 votes
0 answers
27 views

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 ...
• 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 $x \in \mathbb{R}_{>0}$. ...
0 votes
1 answer
82 views

• 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, ...
• 8,226
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}...
• 57
0 votes
1 answer
65 views

• 692
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 ...
-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'...
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 ...
• 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} \...
• 2,428