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Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

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41 views

Inequality sum of exponentiation

I was trying to prove for two positive integers $i$ and $j$ and a natural number $n > 1$ whether the following statement holds: If $a_{0}^{n} + a_{1}^{n} + \ldots + a_{i}^{n} > b_{0}^{n} + b_{1}...
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2answers
37 views

Prove and interpret $|f(x)+g(x)|=|f(x)|+|g(x)| \implies f(x)g(x)\geq0$

$$ |f(x)+g(x)|=|f(x)|+|g(x)| \implies f(x)g(x)\geq0 $$ I don't have any clue of how to prove this ? Can someone give any geometrical interpretation to it, as I really don't want to just bihart it ?
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0answers
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How to show the following function involving the digamma function is increasing [on hold]

Let $f(x)=x \psi(x+1)$, where $\psi$ is the digamma function. Define $$g(x)=(f(ax)+f((1-a)x)-f(x))-(f(ax+by)+f((1-a)x+(1-b)y)-f(x+y)),$$ where $0\le a,b\le 1$ and $x,y\ge 0$. How to show that $g(x)$ ...
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2answers
53 views

Interesting Inequality With Exponents And Base > 1

I had trouble proving the following inequality: $\beta > 1$ $(\alpha_{1}\beta^{2\alpha_{1}} + \ldots + \alpha_{n}\beta^{2\alpha_{n}})(\beta^{\alpha_{1}} + \ldots + \beta^{\alpha_{n}}) \geq (\...
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1answer
53 views

Prove that $\sum_{cyc}\frac{a^2}{ca^2 + 2c^2} \ge 1$ [duplicate]

$a$, $b$ and $c$ are positives such that $ab + bc + ca = 3abc$. Prove that $$ \sum_{cyc}\frac{a^2}{ca^2 + 2c^2} \ge 1$$ Here's what I did. My stupidity has reached a spiritual level. We have that $...
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0answers
22 views

Recurrence with non-fixed number of variables

Given the recurrence $\forall \{a_i\}\subseteq \mathbb N.\ \ T(\sum_i{a_i},m)\geq \sum_i {T(a_i, m-\prod_i (a_i!))}$ with $\forall n,m. T(n,\Theta(1))=\Theta(1),T(\Theta(1),m)=\Theta(1) $ How ...
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3answers
49 views

Proof $|a-b| \lt \frac{|a|}{2} \rightarrow |b| \gt \frac{|a|}{2}$

In this answer to a question of mine, in the second paragraph, the author uses an expression analogous to $|a-b| \lt \frac{|a|}{2}$ , and then in the parenteses says that it implies that $|b| \gt \...
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20 views

Reasoning about inequalities involving floor functions

I am working on the beginning of an inductive argument and I wanted to confirm that my base case is sound. Let $f(x) = \lfloor x\rfloor - \left\lfloor\dfrac{x}{2}\right\rfloor$ where is $x$ is a ...
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28 views

Find this inequality by finding the generalizating one too

Given $x,\,y,\,z$ such that $x,\,y,\,z\in [\,1,\,8\,]$$.$ Prove$:$ $$\frac{x}{y}+ \frac{y}{z}+ \frac{z}{x}\geqq \frac{2\,x}{y+ z}+ \frac{2\,y}{z+ x}+ \frac{2\,z}{x+ y}$$ By computer$,$ I have found ...
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1answer
96 views

Find the minimum value of $\left(\frac{a}{x} + \frac{b}{y} + \frac{c}{z}\right)\sqrt{yz + zx + xy}$

Going back a few more years and you can find more and more interesting problems over the years as time turns back. I am still surprised at how easy this competition has become. Then I come across this ...
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1answer
38 views

$\forall\varepsilon > 0,\exists\ a >0 : |f(x)|\,\le\, a\|f\|_2 + \varepsilon\|f'\|_2$ for $f\in H^1(0,1)$

I know that function evaluation in $H^1(0,1)$ is continuous (see, e.g., Is the Delta distribution a continuous functional on $H^1(\mathbb R)$). So, $\delta_x : H^1(0,1)\to\mathbb C$ is a continuous ...
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21 views

Approximating $\max\frac{x_\tau}{x}$

I have the following delay system: $$x'(t) = g(t,\tau,x)$$ Given that $g(\cdot)$ is smooth and bounded, $x(t)$ is bounded in a positive region. What are some possible ways to obtain an upper bound on $...
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0answers
19 views

Is there a Chernoff Bound version for ANY random variable? [on hold]

Is there a Chernoff Bound verson for ANY random variable? Or is it just through the Markov Inequailty? Like there's is a version of Chernoff bounds for sum of binomial random variables. So is there a ...
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0answers
51 views

Maximum of : $\Big(\frac{a}{a^{11}+1}\Big)^n+\Big(\frac{b}{b^{11}+1}\Big)^n+\Big(\frac{c}{c^{11}+1}\Big)^n$

I'm interested by the following problem : Let $a,b,c$ be real positive numbers such that $abc=1$ with $n>0$ a natural number then find the maximum of : $$\Big(\frac{a}{a^{11}+1}\Big)^n+\Big(\...
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1answer
78 views

How to prove by mathematical induction that $(y-x)x^n \leq \frac{y^{n+1}-x^{n+1}}{n+1} \leq (y-x)y^n$? [on hold]

Prove by induction that: $$(y-x)x^n \leq \frac{y^{n+1}-x^{n+1}}{n+1} \leq (y-x)y^n\ . $$ As a hint, the professor told us to use the following expression that we had previously proven: $$\sum_{i=...
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2answers
37 views

If $\sum_{n=1}^{\infty} a_n>0$,can one deduce that $\sum_{n=1}^{\infty} a_n<|\sum_{n=1}^{\infty} n^2a_n|$

If $\sum_{n=1}^{\infty} a_n>0$,can one deduce that $$\sum_{n=1}^{\infty} a_n< \left |\sum_{n=1}^{\infty} n^2a_n \right |$$ Of course, both series are convergent.
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Hoeffding's inequality : where did root of log t/N came from? [on hold]

I want to know where did the expression given below come from in Hoeffding's inequality $$ \sqrt{\frac{2\log{t}}{N_t(a)}} $$
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1answer
78 views

If $a+b+c=1$ and a,b,c >0 prove $\dfrac{b^2}{a+b^2}+\dfrac{c^2}{b+c^2}+\dfrac{a^2}{c+a^2} \geqslant \dfrac{3}{4}$ [duplicate]

If $a+b+c=1$ and a,b,c>0 prove $\dfrac{b^2}{a+b^2}+\dfrac{c^2}{b+c^2}+\dfrac{a^2}{c+a^2} \geqslant \dfrac{3}{4}$. I tried with CS Engel form,homogenization but ina anyway i can't prove inequality. Can ...
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42 views

Conditional Expectation Inequality for bounded moment

I was going through a proof and encountered the following: Let $X$ be a random variable with $\mathbf{E}[X] = \mu$ and $\mathbf{E}[(X-\mu)^2] = \sigma^2$ and let $\mathbf{E}[(X-\mu)^4] \leq C_4 \...
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2answers
44 views

Values of $x$ satisfying $\lfloor{x^2\rfloor}=\left(\lfloor{x\rfloor}\right)^2$

Prove that values of $x$ satisfying $\lfloor{x^2\rfloor}=\left(\lfloor{x\rfloor}\right)^2$ is $(0,\sqrt{2})\cup \mathbb{Z}$ My try: Its trivial that every integer satisfies the given equation. Now ...
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1answer
35 views

Prove $\sum_{i=1}^n\frac{1}{\sqrt{i}}\le\frac{1}{\sqrt{n!}}\prod_{i=2}^n(\sqrt{i-1})+2\sum_{i=2}^n\frac{1}{\sqrt{i}}$ using Weierstrass inequality

Use Weierstrass's inequality to prove that $$\sum_{i=1}^n \frac{1}{\sqrt{i}}\le \frac{1}{\sqrt{n!}} \prod_{i = 2}^n (\sqrt{i-1}) + 2 \left(\sum_{i = 2}^n \frac{1}{\sqrt{i}}\right).$$ Using ...
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1answer
48 views

Show this inequality $\frac{n}{a_1 - a_0} + \frac{n - 1}{a_2 - a_1} + \cdots + \frac{1}{a_n - a_{n-1}} \ge \sum_{k=1}^n \frac{k^2}{a_k}$

For $a_1, \ldots , a_n \in \mathbb{R}, a_1 < a_2 < \cdots <a_n$ and $a_i \ne 0$, show that $\dfrac{n}{a_1 - a_0} + \dfrac{n - 1}{a_2 - a_1} + \cdots + \dfrac{1}{a_n - a_{n-1}} \ge \sum_{k=1}^...
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2answers
44 views

Understanding the example of metric

I would like to get some help about the next problem: I'm trying to understand the following example about metric in my book: In the set $\mathbb{R}^n$, just as in any non-empty set, metric can be ...
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3answers
48 views

Doubt in the solution provided to an inequality question

I have the following question with me: The numbers $x_1, x_2, . . . , x_n$ obey $−1 \leq x_1, x_2, . . . , x_n \leq 1$ and $$x_1^3 + x_2^3 + ... + x_n^3 = 0 $$ Prove that $$x_1 + x_2 + · · · + x_n ≤ \...
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3answers
97 views

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

Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=4$. Prove that $$\frac{1}{ab}+\frac{1}{cd} \geq \frac{a^2+b^2+c^2+d^2}{2}$$ I write $$a^2+b^2+c^2+d^2=16-2\left(ab+cd+\left(a+b\right)\left(c+...
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0answers
28 views

Landau inequality for several variables

For $f \in C^n(\mathbb{R})$ and $0 < \alpha < n$, Landau-Kolmogorov inequlity is geven by $$ \|f^{(\alpha)}\| \leq K(n,\alpha)\| f\|^{1-\alpha/n}\|f^{(n)}\|^{\alpha/n}, 0 < \alpha < n,$$ ...
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1answer
40 views

Maximizing Area of a quadrilateral inside of a square

The square ABCD has point M located on side AB and point N on side CD. Lines CM and BN intersect at point U. Lines DM and AN intersect at point V. Determine where points M and N should be placed to ...
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0answers
33 views

Big $O$ of integral and estimate

Suppose $f: (0,+\infty) \to \mathbb{R}$ is s.t. \begin{equation}\tag{1} f(s) = \frac{1}{4\pi s} + \mathcal{O} \biggl (\frac{1}{\sqrt{s}} \biggr ) \quad \quad\text{ as } s \to 0^+ \end{equation} and $f ...
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2answers
51 views

How can we show that $e^{-2\lambda t}\lambda^2\le\frac1{e^2t^2}$ for all $\lambda,t\ge0$? [duplicate]

How can we show that $$e^{-2\lambda t}\lambda^2\le\frac1{e^2t^2}\tag1$$ for all $\lambda,t\ge0$? Applying $\ln$ to both sides yields that $(1)$ should be equivalent to $$t\lambda\le e^{t\lambda-1}\...
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3answers
63 views

About the fact that $\dfrac{a^2}{b} + \dfrac{b^2}{a} + 7(a + b) \ge 8\sqrt{2(a^2 + b^2)}$.

There's a math competition I participated yesterday (19/3/2019). In these kinds of competitions, there will always be at least one problem about inequalities. Now this year's problem about ...
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2answers
108 views

Difficult inequality with 4 variables

I am struggling with this inequality from the book Advanced Olympiad Inequalities: Algebraic & Geometric Olympiad Inequalities, any idea please? Thanks. Question: Let $a,b,c,d>0$ such that $a^...
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2answers
70 views

Prove that $\sum_{cyc}\dfrac{a}{a + b^4 + c^4} \le 1$ where $abc = 1$.

If $a$, $b$ anc $c$ are three positives such that $abc = 1$ then prove that $$\large \sum_{cyc}\dfrac{a}{a + b^4 + c^4} \le 1$$ Here's what I did. $$\large \sum_{cyc}\dfrac{a}{a + b^4 + c^4}$$ $$\...
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2answers
60 views

Is there a non-trivial lower bound for $|a_1+a_2+ \cdots + a_n|$?

$|a_1+a_2+ \cdots + a_n|$ comes up a lot when working with polynomials and power series. For the sake of getting an answer, $a_1, ..., a_n$ can be whatever you want; real numbers, complex numbers, ...
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0answers
23 views

Xor properties in inequalities

Lately I was trying to figure out some mathematical properties of XOR. So for equalities with XOR we can do $\oplus\ x$ on both sides of it and still get the right equality. Example: $a \oplus b = c$ ...
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23 views

Implicit function derivative: Simplification

Simplify the following system of inequities: $$\frac{d^2y}{dx^2}<\frac{dy}{xdx}$$ $$\frac{d^2x}{dy^2}<\frac{dx}{ydy}$$ Known that $F(x,y)=0$. $F$ is monotonic. I got something like this: $$\...
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1answer
44 views

How can we show that $\left|\frac{e^{-\lambda t}-1}t\right|\le|\lambda|$ for all $t\in\mathbb R\setminus\left\{0\right\}$ and $\lambda\in\mathbb R$?

How can we show that $$\left|\frac{e^{-\lambda t}-1}t\right|\le|\lambda|$$ for all $t\in\mathbb R\setminus\left\{0\right\}$ and $\lambda\in\mathbb R$? Clearly, $e^{-\lambda t}\in[0,1]$ for all $t,\...
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3answers
47 views

How to prove an inequality by mathematical induction?

How to prove the following inequality by mathematical induction? $$x + \frac 1x \ge 2, x \gt 0$$ I am aware of this. First, I have to prove $P(1)$; then $P(n+1)$. I am stuck at $P(n+1)$ because I ...
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2answers
40 views

Prove of $\Vert A \Vert_\infty$ submultiplicativity

How can I prove that $\Vert AB \Vert_\infty\le\Vert A \Vert_\infty.\Vert B \Vert_\infty$ ? What I have already done: $\max_{1\le i \le n}(\sum_{j=1}^n|\sum_{k = 1}^n A_{ik}.B_{kj}|)$ $\le \max_{1\...
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1answer
54 views

Prove that $0.4 ≤ \int_0^1 f(x) dx ≤ 0.5$ for $f(x) = x^{\cos x + \sin x} $. [duplicate]

Consider the function $$f(x) = x^{\cos x + \sin x} $$ defined for $x \ge 0.$ Prove that $$0.4 ≤ \int_0^1 f(x) dx ≤ 0.5.$$ I tried using max-min inequality but it didn't work.
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5answers
68 views

Minimum value of $\cos x+\sin x$ for $0 \le x \le 1$

What will be the minimum value of $$\cos x+\sin x$$ for $0\le x \le 1$? The answer is $1$. I tried finding it's minima, but there is none for critical point. Which other approach shall I try?
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0answers
261 views
+100

Bounding a polynomial from below

Let $\sigma >0$ be fixed. For even $k \in \mathbb{N} \cup \{0\}$, we consider the polynomial \begin{equation} \varphi_k(x) = \sum_{j=0}^{k} (-1)^j {k \choose j} b_j \, x^{2j} \quad x \in (-1,1), \...
1
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1answer
85 views

Prove an inequality with positives $a$, $b$ and $c$.

If $a$, $b$ and $c$ are positives such that $(a + b + c)\left(\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c}\right) = x$ ($x \ge 9$) then prove that $$\large(a^2 + b^2 + c^2)\left(\dfrac{1}{a^2} + \dfrac{...
2
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1answer
38 views

Proving inequality relation

I would like to get some help with the next problem: I'm trying to prove that $$\sum_{i = 1}^n (x_i - y_i)^2 \le \sum_{i = 1}^n (x_i - z_i)^2 + \sum_{i = 1}^n (z_i - y_i)^2\;\;\;\;\;(1).$$ I ...
0
votes
3answers
30 views

Why is a < 0 the only solution to the following inequality?

I have been given the following equation, semi-derived from the quadratic equation: $\frac{+\sqrt{b^2-a}}{a}<\frac{-\sqrt{b^2-a}}{a}$ I need to prove that ${a}<0$ is a possible real solution ...
0
votes
2answers
41 views

Prove $\forall x,y \in \mathbb{R} \ $ $x^2+y^2 \geq x^2-y^2$ [on hold]

I know this may sound obvious, but I was wondering if both $x, y$ are real numbers, then why is it that $$x^2+y^2\geq x^2-y^2.$$
6
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0answers
98 views

Proof of an interesting inequality

I think this question was asked here before, but I am unable to find it at the moment. Apologies if this is due to my ineptitude. Anyway, the question is as follows: let $n>1$ be an integer number ...
1
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0answers
44 views

Given four real numbers $a_1<a_2<a_3<a_4$, rearrange them in such an order $a_{i_1}$, $a_{i_2}$, $a_{i_3}$, $a_{i_4}$ that the sum

Given four real numbers $a_1 \lt a_2 \lt a_3 \lt a_4$, rearrange them in such an order $a_{i_1} , a_{i_2} , a_{i_3} , a_{i_4} $ that the sum $$S = (a_{i_1}-a_{i_2})^2 + (a_{i_2}-a_{i_3})^2 + (a_{i_3}-...
0
votes
0answers
18 views

Solve $S = \frac{P^K}{P^K + \left(1 - P^K\right) (1 - P)^K}$ for $K$

I have an inequality $$ S \leq \frac{P^K} { P^K + (1 - P^K) (1 - P)^K} $$ with $P \in [0..1]$ and $S \in [0..1]$. I need to solve it for $K$.
0
votes
4answers
56 views

$[a+b]\geq[a]+[b]$ for all a,b belongs to Real number [duplicate]

At first I think of triangle inequalities but it is totally different. Then I consider that is this bracket symbolize anything in maths or these were just normal square brackets. I am really not ...
0
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1answer
25 views

How to use Chebyshev's inequality to find a lower bound?

I have that $\text{var}(X) = \frac13v^2$ and I want to find the lower bound for $P(|X| \le v)$ I tried doing the following: $P(|X| \le v) = 1 - P(|X| \ge v)$ so using Chebyshev's inequality I have ...