Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

18,836 questions
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Let $x,y \in \mathbb{R}$, s.t. $y>0$. Then $|x|<y \iff -y < x < y$. I'm confused about how to join the cases in this proof into a single interval. By definition, $|x|= x$ if $x \geq0$ ...
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Renewed unsolvable inequality for forward differences, falling factorials and harmonic numbers

Ok, maybe the inequality that I propose is not unsolvable but following the instructions of the website I only tried to make the question more challenging. The inequality is the following one: for any ...
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How to prove the following norm inequality

If $x$, $y$ and $a$ are vectors in $\mathbb{R}$, is the following inequality true? $||y - a|| - ||x-a|| \ge (\frac{x-a}{||x-a||})^T(y-x)$ I cannot come up with a counterexample, but I also do not ...
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elementary inequality involving exp and ln

Suppose $x$ and $y$ are real numbers satisfying $x\geq 0$ and $y\geq 1$. Is the following inequality true? $xy \leq e^x + y \ln (y)$ If so, is there a reference or proof?
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Inequality involving $e$

$\forall n,k\in \mathbb{Z^+}$, prove that $\binom{n}{k} < \frac{1}{e}\big(\frac{en}{k}\big)^k$ where $e$ is the euler's constant. I tried my best to solve it and thought of expanding both sides but ...
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How prove this inequality $\frac{2}{(a+b)(4-ab)}+\frac{2}{(b+c)(4-bc)}+\frac{2}{(a+c)(4-ac)}\ge 1$

let $a,b,c>0$,and such $a+b+c=3$, show that $$\dfrac{2}{(a+b)(4-ab)}+\dfrac{2}{(b+c)(4-bc)}+\dfrac{2}{(a+c)(4-ac)}\ge 1$$ I think this inequality use this $$ab\le\dfrac{(a+b)^2}{4}$$
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Showing that for $x \ge 7$, $x\# \ge x^2+x$

Let $x\#$ be the primorial of $x$. I am trying to show that if $x \ge 7$: $$x\# \ge x^2+x$$ Is there a straight forward argument? Here's what I came up with: (1) From Bertrand's Postulate, for ...
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Triangle Inequality and absolute value

I'm curious if the triangle inequality (and reverse triangle inequality) still hold if we only take the absolute value of one term. For example: $$||a| - b| \le |a - b|$$ If $b \ge 0$, then $|b|$ is ...
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lower bound for $\vert a+b\vert^\alpha$

Let $a,b$ be two positives real numbers and $\alpha >1$, by convexity we know that $$\vert a+b\vert^\alpha\le 2^{\alpha-1}(\vert a\vert^\alpha+\vert b\vert^\alpha).$$ But is it possible to have a ...
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Infinity norm is actually a norm : triangle inequality

I have to prove the following assertion : Let $V$ be a finit dimentional vector space with dimension $n$ over the field $K$ which is the field of real numbers or complex numbers. Let the map defined ...
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Given $p_1^{a_1}+ \dots + p_r^{a_r} \leq n$, show that : $r \leq 2\sqrt{\frac{n}{\log(n)}}(1+o(1))$

I post here because I really don't succeed to prove this : Given $p_1^{a_1}+ \dots + p_r^{a_r} \leq n$, $p_i$ distinct prime numbers and $a_i \in \mathbb{N}$, $a_i \geq 1$, $r \in \mathbb{N}$, ...
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Discrete math - The ceiling of a real number x, denoted by$⌈𝑥⌉$, is the unique integer that satisfies the inequality

I have a discrete math question below with a solution written by my teacher. I'm really lost as to what answers I'm trying to find exactly. I don't understand how the teacher got 1 and -1 for the ...
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How to prove/verify the following inequality consisting of polynomial and exponential holds?

How to verify the inequation $C^{H}e^{-\lambda^2 C} \leq \delta$ holds for $C = \frac{1}{\lambda^{2}} (2H log \frac{H}{\lambda^2} + log\frac{1}{\delta})$? Namely, we treat $H \geq 1$, $\lambda > 0$,...
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Inequality w.r.t determinant of a non-negative definite matrix.

I am reading a paper where the author mentioned the following property without proof. Neither can I prove it nor can I find the proof in various textbooks. For any non-negative definite (i.e. ...
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An integral inequality on a disc about twice derivatives.

Denote $f:\mathbb{R^2}\rightarrow\mathbb{R}$ has a continuous twice derivative on the disc $D=\{(x,y):x^2+y^2\leq1\}$, which satisfy $f(\partial D)=\{0\}$, where $\partial D=\{(x,y):x^2+y^2=1\}$, ...
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Is the number of unique values in a matrix bounded by the product of the corresponding statistic for rows and columns?

Suppose we have an $r\times c$ matrix of natural numbers, $M$. Suppose also that the number of unique values in any given row is at most $n$, and the number of unique values in any given column is at ...
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Proof by mathematical induction with the problem $40(2n)! ≥ 30^n$

I want to start by saying that I have far less trouble handling a non-inequality induction problem. I really don't understand the steps to take to get to the desired end product with these inequality ...
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On equivalences of trigonometric inequalities

Let $a$ be a real positive number. \begin{align} \text{(I)}& & a &> \frac{\sin(y_1(a))}{y_1(a)} & &\text{where $y_1(a)$ is the unique root of}& y &= a \cot(y) ~~\text{...
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If ln(2cosh(x)) > x, then what is ln(cosh(x)) >? [on hold]

Not sure how to find this inequality, any help would be appreciated
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Properties of “power inequalities”? [on hold]

Wikipedia defines power inequalities, as inequalities that use exponentiation, and lists a bunch of examples, but gives no names that can be Googled or proofs or citations except for one. Are there ...
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Problem in solving inequality by wavy curve method

Please help me in solving this question: Find the solution set of $x$ for which the expression $$\frac {x (3^x -1)(x+1)^2}{(x-3)(x-2)^4}$$ is positive. My problem is that if I use the wavy ...
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Prove that x.y<1 when 0<x<1 0<y<1 [on hold]

i need a proof of this inequality: 0<x<1 0<y<1 and x,y ∈ (0.1) also x,y ∈ R x.y<1 only using of axioms. Thanks!
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Very curious identity and an inequality $a^{ab}f(a^{ab})+b^{ab}f(b^{ab})\leq a^{a^2}f(a^{a^2})+b^{b^2}f(b^{b^2})$

Related to my post Proving $a^{ab}+b^{ab}\leq a^{a^2}+b^{b^2}$ and an identity . I have found the following identity generalised : Let $x$ a real numbers with $x\neq 0$ then we have and $f(x)$ a ...
if a, b, c>0, then prove: $\frac{a+b}{\sqrt{a^{2}+a b+b^{2}+b c}}+\frac{b+c}{\sqrt{b^{2}+b c+c^{2}+c a}}+\frac{c+a}{\sqrt{c^{2}+c a+a^{2}+a b}} \geq 2+\sqrt{\frac{a b+b c+c a}{a^{2}+b^{2}+c^{2}}}$ I ...
Using induction to prove a sequence $a_1=1$ and $a_{n+1}=3-\frac{1}{a_n}$ is increasing
I have a sequence defined by $a_1=1$ and $a_{n+1}=3-\frac{1}{a_n}$. I need to use induction to show that the sequence is increasing and $a_n<3$ for all $n$. Also to deduce that $a_n$ is ...