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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19 views

Logarithm equation, maybe inequality related? [closed]

Solve on $\mathbb{R}$ the equation $\log_3(4 + 3x^4) + \log_5(1 + \sqrt[3]{x^2}) + \log_2(1+x^2) = \log_3 4$.
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16 views

Usefull inequality for p-laplacian study of form $|x-y|^{p-1}y + |x-y|^p \leq C |x|^{p-2}x(x-y) $

I've been looking for an inequality of the kind $|x-y|^{p-1}y + |x-y|^p \leq C |x|^{p-2}x(x-y) \ $, for all $x, y\in \mathbb{R}$, $p>2$ and $C>0$ a constant. or results to help me demonstrate ...
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2answers
55 views

Is it true that for $x>0$, one has $x\le\frac{1}{\log x}$?

I know that for $x>0$, one can quite easily prove that $$\frac{1}{x}\le\frac{1}{\log{x}},$$ which follows from the trivial identity that $x\le e^x\iff \log x\le x\iff\frac{1}{\log x}\ge\frac{1}{x}$,...
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0answers
46 views

Solving simultaneous linear inequalities over the integers

Find the number of integral points $(x,y)$ such that $x_1<x<x_2$ and $y_1<ay+bx<y_2$. I encountered this pattern while studying linear congruence of type $a x + b y \equiv c \pmod m$.
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1answer
149 views

Prove that $\frac{(3 a+3 b) !(2 a) !(3 b) !(2 b) !}{(2 a+3 b) !(a+2 b) !(a+b) ! a !(b !)^{2}}$ is an integer.

Prove that $$\frac{(3 a+3 b) !(2 a) !(3 b) !(2 b) !}{(2 a+3 b) !(a+2 b) !(a+b) ! a !(b !)^{2}}$$ is an integer for all pairs of positive integers $a, b$ (American Mathematical Monthly) My work - $ v_{...
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25 views

Poincaré inequality in the space $H_{r}^{2}$

I know that the below Poincare's inequality is valid in the space $$H_{E}^{2}=\left \{ y\in H^{2}(0,1)\mid y(0)=y_{x}(0)=0 \right \}$$ Let $w(x, t)$, $x ∈ (0, 1)$, $t ∈ (0, ∞)$ satisfies the boundary ...
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2answers
40 views

About the smallest value of B

I am trying to solve this problem: We know that there's a inequality: $$(3n-1)(n+B)\geq A(4n-1)n$$ When $A=\frac{3}{4}$, what is the smallest possible value of B. So, what I did is that: $$B\geq \frac{...
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0answers
39 views

From one to double integral inequality

Let $f$ be a continuous function on $[0,1]$ and let $a\in (0,1)$. Prove that there exists a positive constant $c$ such that $$\int_a^1\int_{x-a}^xf^2(s)dsdx\geq c\int_0^1f^2(s)ds.$$ I cannot see how I ...
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36 views

Smallest $c$ such that $f'<cf$ holds for all $f$ such that $f,f',f'',f'''>0$ and $f''' \le f.$

Let $f: \mathbb{R} \to \mathbb{R}$ be a $C^3$ function such that $f,f',f'',f'''>0$ and $f''' \le f.$ What is the smallest $c$ such that we can guarantee $f'<cf$? Since $f(x)=e^x$ works, we must ...
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28 views

Verification of an Inequality

Reading the paper: "A Continuous Feedback Approach to Global Strong Stabilization of Nonlinear Systems", I came up to the following inequality: If $ p\geq 1 $ is an odd integer, then $$ |x-y|...
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2answers
56 views

Prove $a^4+b^4 \geq a^3b$ [closed]

How would you prove that $a^4+b^4 \geq a^3b$ for every real $a$ and $b$? I solved it using the convexity of the $x^4$ function, but I am wandering for a more direct solution. Thanks
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2answers
69 views

If$0 < q < p < 1$and $\lambda > 0$, there is $\delta(\lambda)$ s. t. $\frac{t^q}{(t + \varepsilon)^{q + \beta}} \geq \lambda t^p$ for $0 < t < \delta$

The following is a claim in a paper I am reading (Montenegro, M. - Existence of Solutions to a Singular Elliptic Equation - Milan Journal of Math. 2011): If $0 < q < p < 1$ and $\lambda > ...
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1answer
29 views

Weak Bernstein inequality for trigonometric polynomials

Could somebody help me to complete or fix my solution to the problem 6.7 from the book Number Theory, Fourier Analysis and Geometric Discrepancy by Giancarlo Travglini. Problem: Let $P_{N}=\sum_{k=0}^{...
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1answer
26 views

Let $X$ and $Y$ be points contained in the disk of radius $r$ around the point P. Explain why $d(X, Y) \leq 2r$.

Working on the book: Lang, Serge & Murrow, Gene. "Geometry - Second Edition" (p. 23) Let $X$ and $Y$ be points contained in the disk of radius $r$ around the point P. Explain why $d(X, ...
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1answer
18 views

Higher order Jensen-like expansion upper bound

If $Z$ is a random variable with fine moment generating function, what is a good way to upper bound $$|\log \mathbb{E}e^Z- \mathbb{E}Z- \frac{1}{2}\mathbb{E}Z^2|$$ This looks like a third offer Taylor ...
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1answer
116 views

Inequality I'm sure to be true

While solving a bigger problem, I've reduced it to an inequality $$\left(1+2^{b^\frac{1}{b-1}-1}\right)^b < 1+2^{b^\frac{b}{b-1}-1}$$for $b>2$, which looks plausible when looking at the plots. I'...
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3answers
111 views

How prove this $\sum_{i=n+2}^{+\infty}\frac{1}{i^2}>\frac{2n+5}{2(n+2)^2}$

let $n$ be postive integer,show that $$\sum_{i=n+2}^{+\infty}\dfrac{1}{i^2}>\dfrac{2n+5}{2(n+2)^2}\tag{1}$$ I know $$\sum_{i=n+2}^{+\infty}\dfrac{1}{i^2}>\int_{n+2}^{+\infty}\dfrac{1}{x^2}dx=\...
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Integral inequality in space-time domain

Let $f$ be a continuous function on $(0,1)$ and let $T\geq a$ be any positive number with $0<a<1$ be a real number. I want to prove the following inequality for some positive constant $c$: $$\...
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1answer
48 views

Prove that $\liminf_{n\to\infty}a_{n}\leq\liminf_{n\to\infty}\frac{s_{n}}{n+1}\leq\limsup_{n\to\infty}\frac{s_{n}}{n+1}\leq\limsup_{n\to\infty}a_{n}$ [duplicate]

Suppose that $(a_{n})_{n=0}^{\infty}$ is a bounded sequence. Let $s_{n} = a_{0} + a_{1} + \ldots + a_{n}$. Show that \begin{align*} \liminf_{n\to\infty}a_{n}\leq\liminf_{n\to\infty}\frac{s_{n}}{n+1}\...
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0answers
30 views

Cant obtain the following inequality

Given the following two inequalities, how do I obtain the third inequality? Sorry I could not get the images being displayed. Also $$f(O)\geq d$$ I have been sitting the whole day trying to solve it ...
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2answers
37 views

solve an exponential inequality $2a^x-a^{x+1} \le b$

Given two scalars $0<a<1$ and $0<b<1$, how to solve the unknown $x$ in the following equation: $2a^x-a^{x+1} \le b$
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0answers
26 views

Proof help: Log Inequalities

I have been working on a lemma (page 419 Understanding Machine Learning) and I understand every component of the proof but I do not understand how "the proof follows". Essentially I need ...
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5answers
59 views

For $a,b,c>0$ prove that $abc(a+b+c) \le a^3 b + b^3 c + c^3 a$

if $a,b,c > 0$ then prove that $abc(a+b+c) \le a^3 b + b^3 c + c^3 a$ My attempt The hint given for this question was to use the Cauchy-schwarz inequality. But if you look at the expression given, ...
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2answers
49 views

Why solving the inequality $\text{arcsec}(x)>\frac{\pi}{4}$ is giving only half of the answer

$$\text{arcsec}(x)>\frac{\pi}{4}$$ Taking $\sec$ on both sides yields $$\sec(\text{arcsec}(x))>\sec(\frac{\pi}{4})$$ $$x>2^{\frac{1}{2}}$$ But from this desmos graph one can easily see that ...
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2answers
36 views

With $\vec{x}=(x_1,\ldots,x_n)$, find all the min and max of $\prod_{i=1}^{n} x_{i}^i$ given that $||\vec{x}||=1$

With $\vec{x}=(x_1,\ldots,x_n)$, find all the min and max of $\prod_{i=1}^{n} x_{i}^i$ given that $||\vec{x}||=1$ Now clearly this is Lagrange multiplier. So one might take $\prod_{i=1}^{n} x_{i}^i-\...
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2answers
30 views

If $ABC$ is a right angled triangle at $C$ then prove that $a^n+b^n<c^n$ for all $n>2$

If $ABC$ is a right angled triangle at $C$ then prove that $a^n+b^n<c^n$ for all $n>2$ This is an olympiad book problem. I know that $a+b>c$ and $a^2+b^2=c^2$, $a^n+b^n≠c^n$ for all $n>2$,...
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3answers
69 views

Olympiad inequality proof issue

Prove that $(a^2+b^2)^2\geq(a+b+c)(a+b-c)(b+c-a)(c+a-b)\ \forall \ a,b,c\in\mathbb{R^+} $. I, forgetting to consider whether $a_1$ and $a_2$ are strictly non-negative (don't think they are), found a ...
2
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1answer
11 views

Proving two different expressions of non-centrality parameters are equivalent

I am stuck in proving $$\sum_{i=1}^{K}\xi_i(\mu_i - \bar{\mu})^2 = \sum_{i,j}\xi_i\xi_j(\mu_i - \mu_j)^2,$$ where $\bar{\mu} = \sum_{i=1}^{K}\xi_i\mu_i$ and $\sum_{i=1}^{K}\xi_i = 1$. I am not sure ...
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2answers
38 views

Addition of Inequalities

I've got an inequality addition question: If $5 + 3x \lt 14$ and $-x \lt 1$, one can find a range for $x$ by solving each equation independently. However, why can I not add the equations? For example: ...
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0answers
22 views

On bounds for the deficiency of $m^2$, where $p^k m^2$ is an odd perfect number with special prime $p$ - Part II

(Note: This question is a sequel to this earlier post.) Hereinafter, call a number $N$ perfect if $N$ satisfies $\sigma(N)=2N$, where $$\sigma(x)=\sum_{d \mid x}{d}$$ is the sum of divisors of the ...
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1answer
14 views

Does the mathematical expectation inequality $E\{g(V)\}<c$ hold?

Assume that $V$ is a stochastic variable, $g(V)\geq 0$ is a function related to $V$. If the upper bound of $g(V)$ can be determined, i.e. $g(V)<c$, where $c$ is a constant, does the mathematical ...
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0answers
34 views

Hoeffding's Inequality for sum of Bernoulli random variables

In the book High-Dimensional Probability, by Roman Vershynin, the Hoeffding's Inequality is stated as the following: Let $X_1,...,X_N$ be independent symmetric Bernoulli random variables (e.i $P(X=-1)=...
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2answers
63 views

How to prove that for every integer $k \geq 2$ we have $k^{1/k} \leq e^{1/e}$ without using the first derivative test?

I just stumbled across this cool property, by doing some calculus I could prove it, the function $f(x) = x^{1/x}$ has a local maximum at $x = e$ and the derivative changes sign at that point, but I ...
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0answers
32 views

Nice inequality with exponents $a^{2b}+b^{2a}\leq a^{\Big(\frac{a(1-a)(\frac{1}{2}-a)}{4}\Big)^2}$

Hi it's a little refinement to play with a hard inequality of Vasile Cirtoaje : Let $a\geq b>0$ such that $a+b=1$ then we have : $$a^{2b}+b^{2a}\leq a^{\Big(\frac{a(1-a)(\frac{1}{2}-a)}{4}\Big)^2}...
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0answers
28 views

Upper bound of sequence

I have the sequence a non-negative sequence $u_k$ that satisfy the following relation $$ u_{k+1} \leq u_k + \alpha \sum_{i=1}^d (\Delta_{i,k+1})^2$$ where $\alpha > 0$, $d \in \mathbb{N}$ (fixed) ...
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1answer
23 views

What is meant by ordered pair in trigonometric inequalities with two variables???? [closed]

Example $2^{\csc^2(x)}\sqrt {(y²-1)+1} \le 2$ Find the number of ordered pairs for x and y Is $2^{\csc^2(x)}$ any particular function of x And $\sqrt {(y²-1)+1}$ any particular function for y??? How ...
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2answers
86 views

Number of Real Solutions of $\frac{7^{1+\cos(\pi x)}}{3}+3^{x^2-2}+9^{\frac{1}{2}-|x|}=1$

Find the number of real solutions of the equation $$\frac{7^{1+\cos(\pi x)}}{3}+3^{x^2-2}+9^{\frac{1}{2}-|x|}=1\,.$$ By hit and trial i got the solution at $x=\pm 1$ but i am not able to solve it as ...
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1answer
26 views

For lambda eigenvalue of G , how to prove this statement

This is an exercise from a textbook. I have no clue how to start this . If G is a graph and $\lambda$ is an eigenvalues of G then prove : $$|\lambda|\le\Delta(G)$$
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1answer
21 views

Inequality on ratio of norms

I would have a question that is a specific case of this post - Bounding the ratio of the $l^1$ norms of two vectors to the ratio of their $l^2$ norms - with $x_j=y_j^2$ and $c_1=1$, $c_2=1/2$. ...
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0answers
67 views

How to prove the inequality $x^x+y^y\geqslant x^y+y^x$ for $x>0,y>0$? [duplicate]

$x>0,y>0$ How to prove the inequality: $x^x+y^y\geqslant x^y+y^x$. when $x\geqslant 1$ and $ y\geqslant 1$,or $y\geqslant 1\geqslant x $ or $x\geqslant 1\geqslant y$,it's easy to proof. I can't ...
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2answers
25 views

Proving an inequality of real numbers [closed]

For all real numbers $x, y > 0,$ prove that $(x + y)(x^{-1} + y^{-1}) \geq 4.$ I know this is true, but I don't know how to prove it.
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0answers
47 views

Prove the trigonometric inequality involving a finite sum. [closed]

Prove that for all real numbers $x$ and any non-negative integer $n,$ we have that $$(1-\cos x) \, |\sin(x)+\sin(2x) +\dotsc +\sin(nx)| \cdot | \cos(x)+\cos(2x)+\dotsc+\cos(nx)| \le 2.$$
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0answers
17 views

Proving an energy estimate of $2 \delta \|u\|_{L^2(\Omega)}\|u_t\|_{L^2(\Omega)}+\|b\|_{L^2(\Omega)} \|u_t\|_{L^2(\Omega)}$

I want to prove that $$2 \delta \|u\|_{L^2(\Omega)}\|u_t\|_{L^2(\Omega)}+\|b\|_{L^2(\Omega)}\|u_t\|_{L^2(\Omega)}\leq \frac{1+2\sqrt{\delta}}{2}(\|u_t\|^2_{L^2(\Omega)}+\|u_x\|^2_{L^2(\Omega)}+\delta\|...
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2answers
60 views

Prove that ,$\left(a+\frac{1}{a}\right)^{2}+\left(b+\frac{1}{b}\right)^{2} \geq 8$

For any positive a, b prove that $$\left(a+\frac{1}{a}\right)^{2}+\left(b+\frac{1}{b}\right)^{2} \geq 8$$ My approach: Using the well known inequality, $ \boxed{\mathrm{AM} \geq \mathrm{GM}}$ $\left(...
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0answers
34 views

How to prove the amount of the objects that this set has is larger than $p$? [closed]

The set is $\{(a,b)|0\le a,b<\sqrt p\}$ while $p$ is a prime number.
1
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1answer
21 views

Show $(Y_{n}-a)_{+}\leq (Y_{n})_{+}+\lvert a\rvert$

In a proof, I saw the use of the following inequality $(Y_{n}-a)_{+}\leq (Y_{n})_{+}+\lvert a\rvert(*)$ without any explanation, where $Y_{n}$ is some random variable and $a$ a constant. Note the ...
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1answer
26 views

Help verifying a numerical inequality

Suppose that $\gamma_1, \dots, \gamma_n$ are positive, increasing, real numbers. Is it true that if $\alpha_i$ are nonnegative and sum to $A$, then $$ \frac{\sum_{i=1}^n \gamma_i^2 \alpha_i}{\sum_{i=1}...
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2answers
47 views

Inequality by MVT [closed]

Prove using MVT (Mean Value Theorem): $\frac{1}{n+1} < \ln(n+1)-\ln(n)<\frac{1}{n}$ I managed to prove that right inequality, how could i prove the left one? Thanks in advance!
2
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1answer
40 views

Showing $−1 − r^2 \leq \cos^3\theta + r^2 \cos^5\theta \leq 1 + r^2$

I was studying my textbook in advanced calculus when I encountered an inequality I can't seem to justify: Now $−1 \leq \cos\theta \leq 1$ , which implies that $$−1 − r^2 \leq \cos^3\theta + r^2 \cos^...
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1answer
29 views

Proving an inequality in non-linear programming

I'm solving a series of exercises from non-linear programming problems. In my convexity section of study, I've found this problem that I have no idea on how to solve it, can you please help me giving ...

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