Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

0
votes
0answers
12 views

Lower bounds on sum of squared sub-gaussians

Letting $\left\{X_{i}\right\}_{i=1}^{n}$ be an i.i.d. sequence of zero-mean sub-Gaussian variables with parameter $\sigma,$ define $Z_{n} :=\frac{1}{n} \sum_{i=1}^{n} X_{i}^{2} .$ Prove that $$ \...
0
votes
1answer
18 views

Proving an inequality which contains exponentials

Let $x, y \in [0, 1]$ and $t > 0$. I would like to show that $$e^{\frac{-x+y-1}{t}} + 1 \ge e^{\frac{-(1-x)(1-y)}{t}} + 2 e^{\frac{-x}{t}}.$$ First of all I decided to rewrite terms above using ...
-3
votes
2answers
34 views

A problem with Exponential Inequality [on hold]

How can we prove that $1+t ≤ e^t, \ \ t \in \mathbb{R}$ $NOTE:$ Proof should be provided without the use of calculus.
-1
votes
0answers
10 views

If linear inequality system S(x,y) is feasible, can we prove S(f(x),g(y)) feasible? Given f, g are linear functions.

What I'm trying to do is to decide if a system is feasible after multiple linear transformations. If the above statement is valid then I just need to solve the initial system to get the answer.
-2
votes
1answer
45 views

Minimum value of $\sqrt[3]{x}yz$ is

If $x+y+z=14.$ Then minimum value of $\sqrt[3]{x}yz$, where $x,y,z>0$ Try: let $x=t^3.$ Then $y+z=14-t^3$ So we have $$tyz\leq\frac{1}{4}t\bigg(y+z\bigg)^2=\frac{t}{4}(14-t^3)^2$$ Could some help ...
0
votes
0answers
9 views

Obtaining bound for multiplicative weight algorithm

In this paper in Theorem 1, it says $$\Phi_i^t \ge w_i^t$$ and that the desired bound follows from the expressions derived for $\Phi^t$ and $w_i^t$ and that $$-\ln (1 - \varepsilon) \le \varepsilon + \...
1
vote
2answers
37 views

The smallest value of the expression: $(a_1-a_2)^2+(a_2-a_3)^2+(a_3-a_4)^2+(a_4-a_1)^2$ lies in which of the following intervals?

Question: Let $a_1,a_2,a_3,a_4\in\Bbb R$, such that $a_1+a_2+a_3+a_4 = 0$ and $a_1^2+a_2^2+a_3^2+a_4^2 = 1$. Then the smallest value of the expression, $$(a_1-a_2)^2+(a_2-a_3)^2+(a_3-a_4)^2+(a_4-a_1)^...
1
vote
3answers
33 views

Absolute Value Rational Inequalities Help Please

I have been read plenty of questions on questions like this, but I still dont quite get it. For example, this question: $$ \left| \frac{2x+1}{x-3} \right| \ge 2 $$ How would I go about solving this?...
0
votes
1answer
34 views

Misconception about proof by cases

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 $x \geq0$ ...
0
votes
1answer
30 views

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 ...
0
votes
2answers
42 views

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?
0
votes
3answers
27 views

Solving Logarithmic Inequality

Solve $\log$ $\left(5^{\frac{1}{x}}+5^{3}\right)<\log$ $6$ $+$ $\log$ $ $$5^{\left(1+\frac{1}{2x}\right)}$ It is possible to simplify the inequality using the quotient rule property of logarithms, ...
1
vote
1answer
49 views

Proving $F(k)=(16X^2- 24X+18)k^2-11kX+1\geqq0$

Given that $1\leqq X\leqq k$, prove that$$F(k)=(16X^2-24X+18)k^2-11X+1\geqq0.$$ Original problem: Given that $a$, $b$, $c$ are three non−negative numbers and $a+b+c=3$, prove that$$(2+a^2)(2+b^2)(2+c^...
5
votes
1answer
59 views

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 ...
0
votes
1answer
37 views

There is no function in $L^1$ whose Fourier transform is 1/log(x)

I need to prove that there is no odd function on $L^1$ whose Fourier transform is a continuous function $g:\mathbb{R}\to\mathbb{R}$ such that $g(\xi)=1/\log(\xi)$ for $\xi\geq 2$. I am suggested to ...
1
vote
1answer
51 views

Can I split this inequality like this?

Recently I had solved this number theory problem but after I solved it I was a bit uncertain whether my approach was correct so I approached AOPS. The problem is : Prove that $[x] + [y] + [x + y] \...
1
vote
2answers
54 views

How to show that $ab+bc+ca\le \frac34$

Let $a,b$ and $c$ be positive real numbers such that $(a+b)(b+c)(c+a) = 1$ , hen show that $$ab+bc+ca\le \frac34$$ I believe I need to use AM-GM inequality and use the fact $(a+b)(b+c)(c+a) = 1$ ...
0
votes
0answers
26 views

Multivariate Kolmogorov distance bounded by Wasserstein distance

I'm trying to find a bound for the multivariate Kolmogorov distance in terms of the Wasserstein distance. Denoting by $F$ and $G$ two cumulative distribution functions (cdf) on $\mathbb{R}^n$ the ...
0
votes
3answers
90 views

Prove inequality $1+ a^2 + b^2+ c^2+ 4\,abc \geq a + b+ c+ ab+ bc+ ca$

Given that $a,\,b,\,c$ are $3$ non$-$negatve numbers$,$ prove$:$ $$1+ a^{\,2}+ b^{\,2}+ c^{\,2}+ 4\,abc\geq a+ b+ c+ ab+ bc+ ca$$ Let$:$ $X= a+ b+ c$$,$ we have to prove$:$ $$\left ( \frac{1}{X^{\,3}}-...
3
votes
5answers
190 views

Prove that : $\frac{ab^2(b+c)}{c^2}+\frac{bc^2(a+c)}{a^2}+\frac{ca^2(b+a)}{b^2}≥2(ab+ac+bc)$ where $a,b,c>0$

Show that $$\frac{b^2(b+c)a}{c^2}+\frac{c^2(a+c)b}{a^2}+\frac{a^2(b+a)c}{b^2}≥2(ab+ac+bc)$$ where $a,b,c>0$ Can AM-GM work here? I need someone help me or hinting me please. Thanks!
1
vote
2answers
58 views

to prove $x^2 + y^2+1\ge xy + y + x$

$$x^2 + y^2+1\ge xy + y + x$$ $x$ and $ y$ belong to all real numbers my attempt $(u-2)^2\ge0\Rightarrow \frac{u^2}{4}+1\ge u $ let $u=x+y\Rightarrow \frac{(x+y)^2}{4}+1\ge x+y$ $\Rightarrow (x+y)...
0
votes
1answer
40 views

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 ...
0
votes
1answer
77 views

Find : $\lim_{n\to\infty}\sum_{k=0}^n\frac{\binom{n}{k}}{n^k(k+1)}$ [on hold]

I'm try to find this lim $\lim_{n\to\infty}\sum_{k=0}^n\frac{\binom{n}{k}}{n^{k}(k+1)}$ Is this limits can be done by integral !? Or inequality Someone help me hints me Thanks!
1
vote
2answers
35 views

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 ...
0
votes
1answer
45 views

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 ...
0
votes
0answers
21 views

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 $H \geq 1$, $\lambda > 0$,...
1
vote
3answers
41 views

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 ...
3
votes
1answer
42 views

How to prove $\lim_{n\to\infty}\sup A(t)\le\frac{a}{b}$?

Suppose $A(t)>0(t\ge 0)$, $a, b>0$, let $$ A'(t)\le aA-bA^2. $$ Prove $\lim_{n\to\infty}\sup A(t)\le\frac{a}{b}$. Using Taylor formula $$ A(0)=A(t)-tA'(t)+o(t)\ge (1-ta)A(t) +tbA^2(t)+o(t). ...
0
votes
1answer
38 views

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. ...
-1
votes
1answer
56 views

Inequality involving $e$: $\binom{n}{k} < \frac{1}{e}\big(\frac{en}{k}\big)^k$

$\forall n,k\in \mathbb{Z^+}$, prove that $\binom{n}{k} < \frac{1}{e}\big(\frac{en}{k}\big)^k$ where $e$ is the [Euler's number](https://en.wikipedia.org/wiki/E_(mathematical_constant). I tried my ...
3
votes
2answers
61 views

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} $, ...
-1
votes
2answers
53 views

Prove that $\frac{a^3 b}{(a+b)^4} \le \frac{27}{256}$

For $a$ and $b$ positive , prove that $$\frac{a^3 b}{(a+b)^4} \le \frac{27}{256}$$ I tried using weighted means and arrived at the result: $$\frac{a^3 b}{(3a+b)^4} \le \frac{1}{256}$$ I would be ...
0
votes
0answers
10 views

Exponents laws for Inequalities

I'm solving a problem involving inequalities and I arrived at a stage where I got $a^{\alpha+\beta}\leq b$ and I want to show $a<b$ where $a,b\geq0$, $a\leq b$, and $\alpha,\beta \epsilon (0,1)$....
0
votes
0answers
16 views

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 $\partial D=\{(x,y):x^2+y^2=1\}$, ...
3
votes
2answers
69 views

Prove by mathematical induction $4^n > n+1$

Prove the following by mathematical induction: $4^n > n+1$, for all integers $n ≥ 1$ Step 1: $n=1$: LHS $= 4^{(1)} = 4 $ RHS $= (1) + 1 = 2$ LHS > RHS. ∴ $P(1)$ is true. Step 2: Assume $P(k)$...
0
votes
1answer
16 views

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 ...
-2
votes
0answers
18 views

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
0
votes
0answers
16 views

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 ...
2
votes
1answer
49 views

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 ...
-1
votes
0answers
48 views

Prove that x.y<1 when 0<x<1 0<y<1 [closed]

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!
0
votes
1answer
22 views

Resolving a second degree inequation using its factored form

I'm trying to solve an exercise and I'm having troubles to get it right. We have two functions $f(x)$ and $g(x)$ defined as follow : $f(x) = 2x-3$ $g(x) = -x²+x-3$ Question : We have to demonstrate ...
-1
votes
2answers
67 views

Important: Proof of this inequality (?)

I'm really screwed since I don't know how to prove this inequality. I've tried creating an $\epsilon$ surrounding and everything but nothing seems to work. The task is: Prove $$\left(\frac{K+1}{N}...
0
votes
2answers
36 views

Proving an inequality involving absolute value; how do I justify using a conjunction (and) instead of a disjunction (or)?

I'm putting together the following the proof, and I have a question about one of the final steps. Definition of absolute value: $\forall x \in \mathbb{R}, (x \geq 0 \Rightarrow |x| = x) \...
-1
votes
0answers
50 views

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 ...
0
votes
0answers
46 views

Prove a challenging pseudo-cyclic inequality using basic inequalities [duplicate]

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 ...
1
vote
2answers
65 views

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 ...
0
votes
3answers
40 views

Is the statement true?$|z+1|\ge|z|-1\ \forall z\in\Bbb{C}$

I have tried to prove it in the following manner- $||z|-1|=||z|-|1||\le|z-1|\ \forall z\in\Bbb{C}$ (by Triangle inequality) Now, can I write $|z-1|\le|z+1|$ in $\Bbb{C}$? If yes then the proof is ...
1
vote
1answer
58 views

How to prove Ky Fan Inequality using Forward-Backward Induction?

The classical version of the inequality is: $${\frac {{\bigl (}\prod _{{i=1}}^{n}x_{i}{\bigr )}^{{1/n}}}{{\bigl (}\prod _{{i=1}}^{n}(1-x_{i}){\bigr )}^{{1/n}}}}\leq {\frac {{\frac 1n}\sum _{{i=1}}^{...
2
votes
1answer
50 views

inequality about $I=\int_{2012}^{3012}\sqrt[3]{x}\,dx$

Consider $$ \begin{align} & L=\sqrt[3]{2012}+\sqrt[3]{2013}+\ldots +\sqrt[3]{3011} \\ & R=\sqrt[3]{2013}+\sqrt[3]{2014}+\ldots +\sqrt[3]{3012} \\ \end{align}\ $$ and $$ I=\int_{2012}...
0
votes
1answer
37 views

Maximum of $\int_0^1(x^p|f(x)|^q-x^q|f(x)|^p)dx$

Let $p>q>0$ and $C=\{f:[0,1] \to \mathbb{R} \mid f \text{ is continuous} \}$. Determine $$\max_{f \in C}\int_0^1(x^p|f(x)|^q-x^q|f(x)|^p)dx$$ and the functions for which this maximum occurs. ...