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Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

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2answers
29 views

How to prove that $a(1-a^N)/(1-a)<N$?

I derive the expectation for a problem and get $$N-\frac{a(1-a^N)}{1-a}$$ where $0<a<1$, and $N>0$. The physical meaning of this value determines it should be positive. But I intuitively ...
-1
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1answer
19 views

Prove 0<=entropy<=1

The entropy of a Bernoulli random variable X with $P(X=1)=q$ is given by B(q)=-qlog(q)-(1-q)log(1-q) How do we prove 0<=B(q)<=1?
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0answers
24 views

Showing monotonicity for ratio of binomial pmf and tail cdf

I'm interested in showing for $X\sim\text{Bin}(n,p)$ that when $x\geq np$, $$ \frac{P(X=x)}{P(X\geq x)}\leq \frac{P(X=x+1)}{P(X\geq x+1)} $$ I've verified using numerical simulations, but can't seem ...
-1
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2answers
26 views

For which values of $m$ I get this for any $x$?

For which values of $m$ I get this for any $x$? $$ (2m-4)x^2 + (m+1)x -1 > 3x - 2 $$
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2answers
39 views

Inequality with $ax^2+bx+c$

Let $a,b,c,x,y > 0$ reals prove that: $$(ax^2+bx+c)(ay^2-by+c) \geq (4ac-b^2)xy$$ What I have done is this: $$ax^2+bx+c=a \left (x+\frac{b}{2a} \right)^2+\frac{4ac-b^2}{4a}$$ $$ay^2-by+c=a \left (...
1
vote
3answers
33 views

Find range of $x$ if $\log_5\bigg(6+\dfrac{2}{x}\bigg)+\log_{1/5}\bigg(1+\dfrac{x}{10}\bigg)\leq1$

If $\log_5\bigg(6+\dfrac{2}{x}\bigg)+\log_{1/5}\bigg(1+\dfrac{x}{10}\bigg)\leq1$, then $x$ lies in _______ My Attempt $$ \log_5\bigg(6+\dfrac{2}{x}\bigg)+\log_{1/5}\bigg(1+\dfrac{x}{10}\bigg)=\log_5\...
5
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1answer
75 views

to prove this inequality $\sum a^3b+3\ge 2(ab+bc+ca)$

let $a,b,c>0$ and such $a+b+c=3$,show that $$a^3b+b^3c+c^3a+3\ge 2(ab+bc+ac)$$ This problem is from my question when $n=3$ case,I found not to prove it. show this inequality with $\sum_{i=1}^{n}...
-1
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2answers
28 views

Absolute value with factors in it [on hold]

$$ \lvert1+x(1-x)\rvert< \lvert1-x\rvert$$ How do I solve for $x$? I'm having a hard time finding the intervals.
1
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1answer
25 views

How to solve $(…((a - b) \cdot y - b) \cdot y \ldots) \cdot y \geq 0$ if there are $n$ subtractions and multiplications?

It's a programming task, but I'd like to know if there is a way to solve it mathematically. I need to check $(...((a - b) \cdot y - b) \cdot y \ldots) \cdot y \geq 0$ if we subtract $b$ and multiply ...
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0answers
19 views

Is $|x^p-y^p| \leq |x-y|^p$ for $|x-y|<1, x,y \gt 0$ and $p \in (0,1)$? [duplicate]

Does this inequality hold $|x^p-y^p| \leq |x-y|^p$ for $|x-y|<1, x,y \gt 0$ and $p \in (0,1)$? This seems to be true, but I don't know how to prove it. Can someone please validate this, possibly ...
-4
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0answers
55 views

Inequality relating 3 numbers [on hold]

Help me sir $x,y,z$ be real numbers $>0$ and $x(x-1)+y(y-1)+z(z-1)=0$ a) Prove that $\frac{1}{x+2} + \frac{1}{y+2} + \frac{1}{z+2} \geq 1$ b) Find $\max (P) = x^2+y^2+z^2 - (\frac{xy}{x+y} + \...
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2answers
69 views

inequality math 10 [on hold]

Sorry sir, I have a problem and I don't know how to solve this. Please help and solve this with me Let $a,b$ be a real numbers, $$ a,b >0 \qquad  \text{ and } \qquad \frac{a}{a+1} + \frac{3b}{...
1
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0answers
17 views

Help to prove that if C is a copula such that $C(t,1-t) = 0$ for all $t$ in $[0,1]$ then $C = max(u+v-1,0)$

I'm asked to prove that if C is a copula such that $C(t,1-t) = 0$ for all $t$ in $[0,1]$ then $C = max(u+v-1,0)$. So far I have this: Let $u,v$ be in $[0,1]$. If $u+v-1 < 0$, i.e. $u < 1-v$. ...
1
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1answer
58 views

Prove or find a counterexample $\forall x>0: f(2x)-f(x)<g(3x)-g(2x)$, given information about $f, g$.

Let $f,g:(0, \infty) \rightarrow \mathbb{R}$ be two functions that satisfy the following for all $x>0$: $g'(x)>f'(x)$ and $f''(x)>0$. Prove or find a counterexample: $$ \forall x>...
4
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5answers
89 views

Prove the inequality $\frac{e^x+e^{-x}}{2} \leq e^{x^2/2}$ for all real numbers $x$. [duplicate]

How do I prove what's written in the title? I was able to get an incomplete proof for the case $x>2$. Here's my try: Use $e^x = \sum_{j=1}^{\infty} \frac{x^j}{j!}$. Now we can see that if $x$ is a ...
1
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2answers
60 views

Trig Integral Inequality

Show that if $f$ is Riemann integrable on $[a,b]$, then $$\left(\int_{a}^{b}f(x)\sin x\ dx\right)^2+\left(\int_{a}^{b}f(x)\cos x\ dx\right)^2\le(b-a)\int_{a}^{b}f^2(x)\ dx.$$ I know I need to use ...
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0answers
45 views

A refinement of $a^{ab}+b^{bc}+c^{cd}+d^{da} >\pi$

Hello I have this to propose today : Let $a,b,c,d>0$ be real positive numbers such that $a+b+c+d=4$ then we have : $$a^{ab}+b^{bc}+c^{cd}+d^{da} \geq 3+\left(\frac{ab+bc+cd+da}{4}\right)^{(ab+...
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1answer
34 views

Inequalities about area and perimeter

"A gardener is laying out a rectangular lawn. His specifications are that the area $(A)$ must be greater than $40$cm but the perimeter $(P)$ must be less than $40$cm. if the width of the lawn $(w)$ ...
0
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0answers
24 views

can knowing an improper integral dependence on first parameter help in studying dependence from second parameter?

I am interested in studying the dependence on parameter $a$ of integrals of this type $$ \int_{-\infty} ^{\infty} \frac{f(x,k)}{a^2+x^2}dx $$ whereby real $k> 0$ and $a>0$ , while about real $f(...
1
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1answer
34 views

Inequality with square roots where solution found with discriminant is not valid

I have : $3 + \sqrt{x-1} > \sqrt{2x}$ when doing basic algebraic operations I will find that : $ 0 > x^{2} -52x + 100 $ I will use $b^2-4ac$ formula to eventually find out that there are ...
0
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1answer
51 views

Inequality similar to Minkowski

Prove that $(|x_1-z_1|^p+|x_2-z_2|^p)^{\frac{1}{p}} \le (|x_1-y_1|^p + |x_2-y_2|^p)^{\frac{1}{p}}+(|y_1-z_1|^p + |y_2-z_2|^p)^{\frac{1}{p}}$ The form is very close to Minkowski's inequality,but I can'...
0
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3answers
62 views

Prove that for all positive real numbers $x$ and $y$, $(x+y)(\frac1x+\frac1y)\ge4$.

Prove that for all positive real numbers $x$ and $y$, $(x+y)(\frac1x+\frac1y)\ge4$. Any help would be appreciated thanks.
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2answers
50 views

If $x_1, x_2, \ldots , x_n$ and $y_1, y_2, \ldots , y_n$ are nonnegative real numbers…

If $x_1, x_2, \ldots , x_n$ and $y_1, y_2, \ldots , y_n$ are nonnegative real numbers such that $x_i + y_i = 1$ for $i = 1, \ldots, n$, prove $$(1-x_1x_2\ldots x_n)^m+(1-y_1)^m(1-y_2)^m\ldots(1-y_m)^...
3
votes
1answer
39 views

Confused about a mathematical induction detail

Suppose I wish to prove that $\forall n\in\mathbb{N}\ge 5,$ the statement $\mathscr{P}(n): n^2<2^n$ is true. Proof: Proceeding by induction on $n$, we note that $\mathscr{P}(5):(5)^2=25<2^5=...
1
vote
5answers
75 views

Is it true that $\forall n \in \Bbb{N} : (\sum_{i=1}^{n} a_{i} ) (\sum_{i=1}^{n} \frac{1}{a_{i}} ) \ge n^2$ , if all $a_{i}$ are positive? [duplicate]

If $\forall i \in \Bbb{N}: a_{i} \in \Bbb{R}^+$ , is it true that $\forall n \in \Bbb{N} : \big(\sum_{i=1}^{n}a_{i}\big) \big(\sum_{i=1}^{n} \frac{1}{a_{i}}\big) \ge n^2$ ? I have been able to ...
0
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0answers
45 views

$ t(n) = t( x_1 x_2 x_3 …) = t(x_1) + t(x_2) + t(x_3) + … + t( x_1 + x_2 + x_3 + … ) $

Let $ n > 1 $ be an integer. Consider The prime factorization $$ n = x_1 x_2 x_3 ... $$ Now define $$ t(n) = t( x_1 x_2 x_3 ...) = t(x_1) + t(x_2) + t(x_3) + ... + t( x_1 + x_2 + x_3 + ... ) $$...
0
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1answer
10 views

Proving subadditivity of max norm for matrices

I'm having struggle with proving that function given by $max(|a_{ij}|)$ for a 3-by-3 matrix A is norm. Showing positivity and homogeneity is trivial, but I'm struggling with triangle inequality ...
0
votes
3answers
47 views

Is this inequality true? $u^2+v^2+s^2+t^2\geq (u+v)(s+t)$

Is this inequality true in $\mathbb{R}$? $$u^2+v^2+s^2+t^2\geq (u+v)(s+t)$$ I don't know if this is a well-known result. If you have a counterexample or a relevant reference I would appreciate it.
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0answers
10 views

Show that for every $x$ there exists a $n \in \mathbb{N}$ so that $f(x-(\frac{1}{2})^{n} \nabla f(x)) \leq f(x)-(\frac{1}{2})^{n+1} ||\nabla f(x)||^2$

Let $f: \mathbb{R}^{n}\rightarrow\mathbb{R}$ be continuous and differentiable and positive. As stated in the title show that for every $x$ there exists a $n \in \mathbb{N}$ so that $f(x-(\frac{1}{...
0
votes
1answer
23 views

How to prove that$D<0.5 T\;$?

We have the formula of obligation duration: $$D = \frac{C}{P} (1- \frac{1+Tr}{(1+r)^{T}}) \frac {1+r}{r^2} + \frac {C+N}{P} \frac{T}{(1+r)^2},$$ Here: $r\in(0,1)$, $T\in \mathbb N$ (number of ...
1
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1answer
27 views

Let $a;b;c\in R+$ such that $a+\frac{b}{16}+\frac{c}{81}\le \:3;\frac{b}{16}+\frac{c}{81}\le 2;c\le 81$. Find maxima of $A$

Let $a;b;c\in R+$ such that $a+\frac{b}{16}+\frac{c}{81}\le \:3;\frac{b}{16}+\frac{c}{81}\le 2;c\le 81$. Find the maxima of function $$A=\sqrt[4]{a}+\sqrt[4]{b}+\sqrt[4]{c}$$ Wlog $a\le b\le c$ $f''(...
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2answers
54 views

Prove the following logarithm inequality.

If $x, y \in (0, 1)$ and $x+y=1$, prove that $x\log(x)+y\log(y) \geq \frac {\log(x)+\log(y)} {2}$. I transformed the LHS to $\log(x^xy^y)$ and the RHS to $\log(\sqrt{xy})$, from where we get that $...
3
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1answer
22 views

Help bounding a “norm”

In Weak Convergence and Stochastic Processes, the authors introduce the following notation: $$\|\xi\|_{2,1} = \int_0^\infty \sqrt{P(\xi > x)}\,\mathrm dx$$ They then admit that this is technically ...
3
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2answers
54 views

How to prove the following inequality? (related to no-arbitrage conditions)

I'm working through a practice book on mathematical finance, but struggling to prove part of a question on no-arbitrage conditions. In the problem, I'm first given $K_1 < K_2 < K_3 $. Then, the ...
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0answers
37 views

How to show that this inequality is correct? $\sup_{|z|=1} |e^{4z^3}\sin z−2|> 1$

How to show that this inequality is correct? $\sup_{|z|=1} |e^{4z^3}\sin z−2|> 1$ We have formula $\sin z= \frac{e^{iz}-e^{-iz}}{2i}$ and $\sin z<1$
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1answer
26 views

how to use Inequality of arithmetic and geometric means to prove the following inequality [on hold]

Let A, B, C > 0 How can I use inequality of arithmetic and geometric means to prove the following inequality
0
votes
1answer
54 views

Does Cauchy-Schwarz imply $|x^Ty| \leq \|x\|_p\|y\|_p$ for any $p \geq 1$?

Given $x,y \in \mathbb{R}^n$, the Cauchy Schwarz inequality states, $|x^Ty| \leq \|x\|_2\|y\|_2$ And for non-Euclidean (norms other than $l_2$), we have, $|x^Ty| \leq \|x\|_p\|y\|_q$ where $\|\cdot\...
0
votes
2answers
51 views

Inequality for sum of squares

let $a_t$ and $b_t$ be any real number. Is the following inequality true? \begin{equation} \frac{1}{T} \sum_{t=1}^{T}2a_t^2 + \frac{1}{T} \sum_{t=1}^{T}2b_t^2 > \frac{1}{T} \sum_{t=1}^{T}(a_t + b_t)...
0
votes
3answers
32 views

Solutions of $\log_{\frac{1}{4}}(x^2 - 1) < 2 \log_{\frac{1}{4}}(x-2)$?

As in the title, what are the solutions to $$\log_{\frac{1}{4}}(x^2 - 1) < 2 \log_{\frac{1}{4}}(x-2)$$ ? I have tried to use the logarithm's power and subtraction rules, but it seems like the ...
0
votes
0answers
13 views

Inequality of fourier series

Let $a_n$ and $b_n$ the fourier coefficient of a $2\pi-$ periodic functions $f$ we assume the regularity as we want to obtain the convergence of the series ( for example $f$ is $C^2$) How can i ...
1
vote
1answer
36 views

Proving an inequality between $(1, \frac{\alpha}{2})$-Hölder norms of two functions

I have to prove that, given $f\in C^{1, \frac{\alpha}{2}}([a, b])$, such that $\|f\|_{\infty}<L$ for some $L$, $f\geq0$ and $\alpha\in(0, 1)$, there exists a constant $K$ such that $$ \|e^f-1\|_{1,...
0
votes
1answer
49 views

Proof of inequality using Cauchy–Schwarz inequality

How I can prove this inequality? How can I prove it using Cauchy–Schwarz inequality? Let $a_1,...,a_n,b_1,....,b_n$ are any real numbers. $\sqrt {\sum_{k=1}^n (a_k+b_k)^2)} \leq \sqrt {\sum_{k=1}^n ...
2
votes
1answer
47 views

Antiderivative Supremum inequality $F(b)-F(a) \le (b-a)\sup\{f(x):x \in [a,b]\}$

Hello everyone I am suppose to show the following: Let F be the antiderivative of f, show that $$ F(b)-F(a) \le \sup\{f(x):x \in [a,b]\} $$ I figured I could use the Mean-Value-Theorem since $F'(...
0
votes
0answers
10 views

tighter bound for the expectation of max of sets of functions

Let $X\in\mathbb{R}^{n\times d}$ be a random matrix, and $\{f_k\}_{k=1}^K, m_k \in \mathbb{R}^{d}$ be a fixed set of matrices. I was wondering, what is the expectation (or upper bound of the ...
0
votes
1answer
38 views

is the weighted arithmetic mean always larger than the weighted arithmetic mean with squared weights? [closed]

given $W=\{w_1,...,w_n\} $ and $V=\{v_1 ... v_n\}$ with $\forall i: 0\leq v_i \leq 1 $ $\forall i: 0 < w_i \leq 1 $ is this statement valid ? $$\frac{\sum_1^n w^2_i \cdot{} v_i}{\sum_1^n w^2_i} ...
0
votes
1answer
29 views

Is the sum of squares $\sum_1^n w_i^2$ always larger than $w_k$ for any k with $\sum_1^n w_i = 1$ and $\forall i : 0 \leq w_i \leq 1$ [closed]

Given this serie of values: $\{w_i : 0 \leq w_i \leq 1\}$ of size n, with $\sum_1^n w_i = 1$ is the following statement valid ? $\forall k \in [1,n]: w_k \leq \sum_1^n w_i^2$
2
votes
0answers
62 views

A nice power sum inequality

Hello I have this theorem to propose : Let $a_i$ be $n$ real positive numbers such that $\sum_{i=1}^{n}a_i=n$ and $a_{n+1}=a_1$ then we have : $$\sum_{i=1}^{n}a_i^{a_i a_{i+1}+n}\geq n$$ For $n=...
-1
votes
1answer
41 views

Inequality In Real Number [on hold]

Can we find a constant $C$ independent on $a$ such that $$(a+\tau)^{\mu}-a^{\mu}\leq C \tau^{\mu},$$ where $a\in\mathbb{R}^+,\tau\in(0,1)$ and $\mu\in(1,2)$? I have found $C=1$ for $\mu\in(0,1]$, ...
0
votes
3answers
75 views

Cauchy Schwartz Inequality Question: $(a^2+b^2)^3=c^2+d^2 \implies \frac{a^3}{c}+\frac{b^3}{d}\geq 1$

If $(a^2+b^2)^3=c^2+d^2$, prove that $\frac{a^3}{c}+\frac{b^3}{d}\geq 1$. Please help.
1
vote
1answer
34 views

How to show negative entropy function $f(x)=x\log(x)$ is strongly convex?

Let $f: \mathbb{R}_+ \rightarrow \mathbb{R}$ where $f(x)=x\log(x)$. How to show it is strongly convex, i.e., Definition: Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be differentiable. Then $f$ is ...