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
14 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
0answers
45 views

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 ...
0
votes
1answer
22 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
33 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?
1
vote
1answer
42 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^...
0
votes
3answers
24 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, ...
5
votes
1answer
194 views
+50

Maximum of $\int_a^b \frac{f(x)}{x}\,\mathrm dx$

Let $b>a>0$ and $M>0$ be fixed. Let $F$ be the set of all functions $f:[a,b]\to[-M,M]$ such that $$\int_a^bf(x)\,\mathrm dx=0.$$Find$$\max_{f\in F}\int_a^b\frac{f(x)}x\,\mathrm dx.$$ I tried ...
0
votes
2answers
48 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
3answers
82 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}}-...
-1
votes
1answer
47 views

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 ...
20
votes
4answers
895 views

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 $$ab\le\dfrac{(a+b)^2}{4}$$
4
votes
1answer
55 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 ...
1
vote
1answer
50 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] \...
4
votes
1answer
101 views

Proving $\frac12 < \int_0^1 \frac1{\sqrt{4-x+x^3}} dx < \frac{ \pi }6$

I have a question where I have to show $$\frac12 < \int_0^1 \frac1{\sqrt{4-x+x^3}} dx < \frac{ \pi }6 \approx 0.52359$$ using the result $$\frac12 < \int_0^{1/2} \frac{1}{\sqrt{1-x^{2n}}} ...
0
votes
1answer
36 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
votes
1answer
53 views

Prove that : $\exists m \in ({\pi/4},{\pi/3})$ such that

I need to prove that $\exists m \in ({\pi/4},{\pi/3})$ such that $$\int_\frac{\pi}{4}^{\pi/3}\frac{1}{x\tan x}dx≤\frac{\pi}{12}\left(\frac{\ln(m)}{m^2}-\frac{\ln(m)}{\sin^2 (m)}+\frac{1}{m\tan m}\...
0
votes
0answers
70 views

Why is sum of square root (given product) maximized with equal summands?

Prove $\sum_{t=1}^{T}\sqrt{x_t-1}$, where $x_t\ge 1$ and $\prod_{t=1}^{T}x_t\le T$, gets its maximum when $x_t=T^{1/T}$. Preliminary idea: The square root is a concave function. Sum of concave ...
0
votes
0answers
18 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 ...
1
vote
1answer
69 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!
0
votes
0answers
21 views

KL divergence contraction coefficient - basic question

I'm studying the Blahut Arimoto algorithm using these notes and towards the end of section 6, an interesting quantity arises. The author does not talk about how to compute it so I was hoping I could ...
2
votes
1answer
136 views

Prove an inequality using complex analysis

If $f:\mathbb{D}\rightarrow\mathbb{D}$ is holomorphic then prove that $$\frac{|f(0)|-|z|}{1 + |f(0)||z|} \leq|f(z)| \leq\frac{|f(0)| + |z|}{1 - |f(0)||z|} $$ I have been wracking my brain for hours ...
1
vote
2answers
53 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
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 ...
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 ...
3
votes
2answers
59 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
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 ...
0
votes
0answers
20 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 $H \geq 1$, $\lambda > 0$,...
0
votes
1answer
33 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. ...
3
votes
1answer
41 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). ...
3
votes
2answers
68 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
2answers
34 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
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 ...
1
vote
1answer
81 views

Minimizing $2$-norm subject to non-convex constraints

Let $|Ax|$ be the element-wise absolute value of $A x$, i.e., $|Ax|_i = |A(i,:)x|$. The inequalities are element-wise inequalities, i.e., $|A(:,i)x| \geq b(i)$. Also, let $\|x\|$ denote the $2$-norm ...
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 $\alpha,\beta \epsilon (0,1)$....
1
vote
1answer
57 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}}^{...
0
votes
0answers
15 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 $\partial D=\{(x,y):x^2+y^2=1\}$, ...
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
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
vote
0answers
38 views

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{...
-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 ...
0
votes
3answers
2k views

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 ...
-1
votes
0answers
47 views

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!
-1
votes
2answers
66 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
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 ...
4
votes
0answers
138 views
+50

Proving $a^{ab}+b^{ab}\leq a^{a^2}+b^{b^2}$ and an identity .

I was working on this Prove that $a^{ab}+b^{bc}+c^{cd}+d^{da} \geq \pi$ when I have discovered the following identity : $$\Bigg|\Big(\frac{1}{2}\Big)^{\frac{x}{8}}\pm\Big(\frac{1}{4}\Big)^{\frac{x}{8}}...
-1
votes
0answers
49 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 ...