Tagged Questions

Questions on proving and manipulating inequalities.

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0
votes
0answers
18 views

How to provee SinA/A +SinB/B +SinC/C <(9*(3)^.5)/2

How can I prove: $$ \frac{\sin \alpha}{\alpha} +\frac{\sin \beta}{\beta}+\frac{\sin \gamma}{\gamma} < \frac{9\sqrt{3}}{2}$$ Only for an acute angle triangle. $\alpha,\beta,\gamma$ are angles of ...
1
vote
1answer
13 views

Question about Mobius function.

Let $N \in \mathbb{N}.$ I would know if is it true that $$-\underset{k\mid N}{\sum}\mu\left(k\right)\log\left(k\right)>0.$$I know that $$-\mu\left(k\right)\log\left(k\right)=\underset{r\mid ...
0
votes
6answers
58 views

How can I prove $(a+b+c)!>a!b!c!$

In fact, I couldn't prove the inequality because I don't know which method is used for this.
1
vote
1answer
11 views

How to justify that $d_{\mid.\mid} ^{\alpha} (x,y) = \mid x-y\mid^{\alpha}$ (with $0<\alpha <1$) is a distance on $\mathbb{R}$

I know directly that : $\forall x,y \in \mathbb{R}$, $d_{\mid.\mid} ^{\alpha} (x,y)=0$ $\Leftrightarrow$ $x=y$ $\forall x,y \in \mathbb{R}$, $d_{\mid.\mid} ^{\alpha} (y,x) = d_{\mid.\mid} ^{\alpha} ...
7
votes
3answers
68 views

Inequality $(1+x_1)(1+x_2)\ldots(1+x_n)\left(\dfrac{1}{x_1}+\dfrac{1}{x_2}+\cdots+\dfrac{1}{x_n}\right)\geq 2n^2.$

Let $n\geq 2$, and $x_1,x_2,\ldots,x_n>0$. Show that $$(1+x_1)(1+x_2)\ldots(1+x_n)\left(\dfrac{1}{x_1}+\dfrac{1}{x_2}+\cdots+\dfrac{1}{x_n}\right)\geq 2n^2.$$ For $n=2$, this reduces to ...
0
votes
1answer
24 views

proving limits with Epsilon-Delta definition

Say I've got $\lim_{x\to a} f(x)= A$ and $\lim_{x\to a} g(x)= B$ How do I do the following: prove that if A < B, then there is the existence of a $\delta$ such that when $0 < |x-a| < ...
0
votes
0answers
6 views

Hilbert transform of the product of functions

Let $H[g]$ denotes a Hilbert transform of function $g$. What would be the constant $C$ in the following inequality: $$ \|H[(\cos n)(\cos{1/(2n))}f](x)\|_{L_2}\leq C\|f\|_2? $$
3
votes
1answer
32 views

Maximum of $\sin A\sin B\cos C+\sin B\sin C\cos A+\sin C\sin A\cos B$ in triangle

What is the maximum value of $$\sin A\sin B\cos C+\sin B\sin C\cos A+\sin C\sin A\cos B,$$ where $A,B,C$ are angles in a triangle? We can rewrite as $$-\sin A\sin B\sin(A+B)+\sin B\sin(A+B)\cos ...
0
votes
1answer
31 views

How could I proof that there can not be equality in Chebyshev's inequality?

For $k>0$. I have gotten the expresion $F(\mu+\sigma k)-F(\mu-\sigma k) = 1-1/k^2$ for all $k>0$. I can not see why this equality is not possible for any continuous RV, what does this mean for ...
0
votes
0answers
18 views

Sobolev norm inequality.

I would like to prove or to disprove the following statement. Let $u$ and $v$ be functions in $H^{s}(S^1)$, the for every $s'\leq s$ $$\|uv\|_s\leq (\|u\|_{s}\|v\|_{s'}+\|v\|_{s}\|u\|_{s'}).$$ I ...
2
votes
3answers
39 views

floor ceiling proof

Hi I would like to proof without using induction that: $$ \left\lceil\frac{n}{m}\right\rceil \leq \frac{n+m-1}{m} $$ I tried: $$ \left\lceil\frac{n}{m}\right\rceil \leq ...
1
vote
1answer
28 views

How to deal with a sum in the denominator

In a book, I saw the following inequality: $$e^{-\frac{\lambda^2}{2 (\lambda + \Delta)}}\leq e^{- \min\{\lambda, \frac{\lambda^2}{\Delta}\}/4},$$ for some (specific) $\lambda, \Delta \geq 0$ Does ...
2
votes
3answers
29 views

Conjectured Inequality

I noticed that for positive integers $(a,b,c)$, it happens that $ab+bc+ca$ is always greater than $abc+1$. Is this true, and if it is, how would I prove it?
0
votes
1answer
19 views

Riemann Integrables

Let $f: [a,b] \rightarrow \mathbb{R}$ and $g: [a,b] \rightarrow \mathbb{R}$ Prove the following. $f \le M \implies \int^b_a f(x)dx \le M(b-a)$ $f+g \in \mathbb{R[a,b]}$ and $\int^b_a [f(x)+g(x)]dx ...
0
votes
1answer
30 views

Integral of the log is less than the integral of the log of the average value

This is an interesting property that I came across while reading an old proof on this website. The poster didn't really explain it, so I thought I might ask. We suppose $u$ is a positive measure on ...
0
votes
0answers
45 views

Showing the inequality $\sqrt{a^2+\frac 1c}+\sqrt{b^2+\frac 1a}+\sqrt{c^2+\frac 1b}\ge (a+b+c)\sqrt{1+\frac 1{abc}}$

How to show that $$\sqrt{a^2+\frac 1c}+\sqrt{b^2+\frac 1a}+\sqrt{c^2+\frac 1b}\ge (a+b+c)\sqrt{1+\frac 1{abc}}$$ for $a,b,c>0$? I have used Cauchy-Schwarz inequality. I need some idea on this ...
1
vote
2answers
42 views

How are some equations and inequalities called identities (how do they have infinite solutions?)?

I've seen identities, which means they have infinite solutions.$\longleftarrow$This is incorrect; see the comments below. Examples are$$4x+6=2(2x+3)$$$$9q-6\lt9q+3$$$$12u\le3(4u)$$$${x\over ...
0
votes
2answers
33 views

Why do some equations or inequalities have no solution?

I've seen some equations and inequalities that have no solution. Examples of these are$$3m+4=3m-9$$$$128y-10\lt128y-25$$$$10t+45\ge2(5t+23)$$The third example evaluates to$$10t+45\ge10t+46$$using the ...
0
votes
1answer
34 views

Using induction to prove that $ \prod_{i=1}^{n} (1+a_{i}) \geq 1 + \sum_{i=1}^{n}a_{i} $ [on hold]

I started a course in my university and I am having trouble with answering this question: Prove using Mathematical induction, for every real, non-negative 'n' number $$(a_{i}\geq 0)$$ the ...
6
votes
1answer
81 views

A conjecture concerning primes and algebra

A monoid morphism $\psi:\mathbb Z_+\!\!\rightarrow\mathbb Z_+$ is defined by an arbitrary function $f:\mathbb Z_+\!\!\rightarrow\mathbb Z_+$ and defines a group homomorphism $\varphi:\mathbb ...
3
votes
1answer
24 views

Reason behind solution in this inequality with absolute values

Solve the inequality $|3x-2|-|x+2|>x$ When $|x+2|<0$: $-(3x-2)+(x+2)>x\iff x <\frac{4}{3}$ When $|x+2|>0\land |3x-2|<0$: $-(3x-2)-(x+2)>x\iff x < 0$ When $|3x-2|>0$: ...
0
votes
1answer
26 views

prove $\limsup_{n\to\infty}\frac{\sqrt{2({{x_1}^2+{x_2}^2+…+{x_n}^2})}}{n}\leq\limsup_{n\to\infty}\frac{x_n}{\sqrt{n}}$

Let $x_1,x_2,x_3$,... be a sequence of nonnegative real numbers. Prove that $\limsup_{n\to\infty}\frac{\sqrt{2({{x_1}^2+{x_2}^2+...+{x_n}^2})}}{n}\leq\limsup_{n\to\infty}\frac{x_n}{\sqrt{n}}$ I ...
1
vote
3answers
52 views

How to get the inequality?

Prove that for any positive integer n, $$2\sqrt{n + 1} − 2 ≤ 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ · · · +\frac 1 {\sqrt n} ≤ 2\sqrt n − 1$$
1
vote
1answer
49 views

Real Analysis Riemann Integration - Strict Monotonicity for Integrals

If $f,g$ are Riemann integrable on $[a,b]$, and $f(x) < g(x)$ for all $x \in [a,b]$, prove that $$ \int_a^b f(x) \,dx < \int_a^b g(x) \,dx$$ This is a strict inequality. I know how to prove the ...
0
votes
3answers
43 views

Show that $a \lt \frac{a + b}{2} \lt b$ for $a\lt b $ and $a, b \in \mathbb{R}$

How can I prove this statement true? I have tried saying starting like this: $a = 0; \qquad b>0.$ But I don't know where to proceed from here
1
vote
1answer
30 views

Tricky Substitution to get AM-GM inequality

So, I'm reading the literature to find different proofs of the AM-GM inequality, the following proof quite hit me, and I don't seem to understand at all. The proof is as follows: For any positive ...
2
votes
1answer
46 views

What's bigger, the sum of powers or the power of the sum?

Do we know if $(\sum\limits_{i=1}^n a_i)^k \geq \sum\limits_{i=1}^n a_i^k$ for any $k\geq1$?
4
votes
2answers
43 views

The minimum value of $\frac{(x+\frac{1}{x})^6-(x^6+\frac{1}{x^6})-2}{(x+\frac{1}{x})^3+x^3+\frac{1}{x^3}}$

Problem : The minimum value of $$\frac{(x+\frac{1}{x})^6-(x^6+\frac{1}{x^6})-2}{(x+\frac{1}{x})^3+x^3+\frac{1}{x^3}}$$ Can I use this in numerator and denominator : The minimum value of $a ...
1
vote
3answers
41 views

Finding the range of equation. Any tricks?

I m working on the following problem For real numbers $a,b$, if $a+ab+b=3$, then find the range of $m=a-ab+b$. Is there any inequalities here to use?
3
votes
2answers
55 views

Applications of the limit nth root of n.

I have the following question, and is that: Given that I've already proven that $$ \lim_{n \rightarrow \infty} \sqrt[n]{n}= 1 $$ Let $a_n = \sqrt[n]{n}$. I want to prove that $ a_n > a_{n+1} $, my ...
0
votes
2answers
69 views

Is it true that $|x^a - y^a| \leq |(x-y)^a|$ on $[0,1]$, where $a\le 1$?

It looks to me like for a function $f(x) = x^a$ on the domain $[0,1]$ where $a \leq 1$, and $x,y$ are points in the domain, $|x^a - y^a| \leq |(x-y)^a|$ I would like to use this in a proof and so if ...
0
votes
0answers
15 views

A geometric inequality involving the sum of distances of an interior point in the triangle $\bigtriangleup ABC$ to its vertices.

Let $\bigtriangleup ABC$ be a triangle in the plane and suppose that $P$ is an interior point of $\bigtriangleup ABC$. Now, I recall seeing somewhere that $$ s < PA + PB + PC < 2s,$$ where $s$ ...
1
vote
2answers
59 views

$ (1+\sin{x})^{\cos{x}} + (1+\cos{x})^{\sin{x}} > 3x $

How do I show that, for $ 0 < x < \dfrac{\pi}{4} $ (first quadrant), the inequality $ (1+\sin{x})^{\cos{x}} + (1+\cos{x})^{\sin{x}} > 3x $ is valid? I've tried Bernoulli's, but it took me to ...
0
votes
4answers
28 views

Prove $(n!)/n^n \leq 1/2^{k}$, where $k$ is the floor of $n/2$.

I suppose the natural way to prove this is by induction. When I follow the rather natural steps $$\frac{(n+1)!}{(n+1)^{n+1}} = \frac{n!}{(n+1)^{n}} \leq \frac{n!}{(n)^{n}}$$ in order to apply the ...
0
votes
0answers
13 views

$f(x)\leq c_1\max(e^{c_2\cdot x^2},e^{-c_2\cdot s \cdot x})$

Let $f:(0,\infty)\rightarrow \mathbb{R}$ be such that $$f(x)\leq e^{C \cdot t^2\cdot s^2-t\cdot x\cdot s},$$for all $t\in[0,1]$ and for some positive $s,C$. I have to prove the bound $$f(x)\leq ...
0
votes
2answers
19 views

If the graph of the function f(x) = $2x^3+ax^2+bx , a,b \in N$ cut the x -axis at three distinct points,

If the graph of the function f(x) = $2x^3+ax^2+bx , a,b \in N$ cut the x -axis at three distinct points, then find the maximum values of a+b. Please suggest as I am not getting any clue on this how ...
2
votes
3answers
39 views

If a,b,c $>0$ and a+b+c=1, then find the maximum / minimum value of the following

If a,b,c $>0$ and a+b+c=1, then find the maximum / minimum value of the following : (a) abc (b) $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ (c) $(1+\frac{1}{a})(1+\frac{1}{b})(1+\frac{1}{c})$ ...
0
votes
1answer
37 views

Is it correct to approach this with Holder Inequality? What am I doing wrong?

I know that $ \forall n\in N, a_0 + a_1 + ... + a_n = 1$, with $a_0, a_1, ... a_n > 0$ and $f(t) = a_0t^n+a_1t^{n-1}+...+a_n, \forall t\in R$ I have to prove that for every $x > 0$ $$ ...
0
votes
1answer
28 views

If $x\in \mathbb{R}^n$ and is a unit vector, why is $\sum\limits_{j,k=1}^n |x_j||x_k| < n^2$?

This is an excerpt of a larger proof: Other pertinent information: $A$ is a positive definite $n \times n$ matrix The set $C$ is the unit sphere I don't get the last inequality: $\gamma \sum ...
-1
votes
1answer
27 views

what is the area of the polygon with given constraints?

What is the area of the polygon formed by all points $(x, y)$ in the plane satisfying the inequality $ ||x| – 2 | + | |y| – 2 | ≤ 4 $ ?
3
votes
0answers
28 views

Trigonometric sum evaluation

Let $q$ a prime number and $1 \leq a<q$ a positive integer. We know from Ramanujan identity that $$\underset{h=1,\left(h,q\right)=1}{\overset{q}{\sum}}e^{2\pi ...
3
votes
5answers
82 views

How prove this $xyzw>0$

let $x,y,z,w\neq 0$ are real numbers,and such $$x+y+z+w=\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}+\dfrac{1}{w}=0$$ show that $$xyzw>0$$ My idea: let ...
6
votes
4answers
162 views

How prove $\pi^2>2^\pi$

show that $$\pi^2>2^\pi$$ I use computer found $$\pi^2-2^\pi\approx 1.044\cdots,$$ can see this I know $$\Longleftrightarrow \dfrac{\ln{\pi}}{\pi}>\dfrac{\ln{2}}{2}$$ so let ...
2
votes
4answers
30 views

Prove $\left|\sum_{i=1}^n x_i y_i \right| \le \dfrac{1}{a} \sum_{i=1}^n {x_i}^2 + \dfrac{a}{4}\sum_{i=1}^n {y_i}^2$

If $X,Y$ are vectors in $\mathbb{R}^n$ and $a>0$ show that: $$\left|\sum_{i=1}^n x_i y_i \right| \le \dfrac{1}{a} \sum_{i=1}^n {x_i}^2 + \dfrac{a}{4}\sum_{i=1}^n {y_i}^2 (*)$$ I started with ...
1
vote
1answer
26 views

Why this inequality is true?

Why this following two inequalities are true? Where $c$ is a constant. $P(X_n+Y_n\le x) \ge P(X_n\le x-c-\epsilon) -P(Y_n>c+\epsilon)$ $P(X_n+Y_n\le x) \le P(X_n\le x-c+\epsilon) ...
0
votes
1answer
27 views

For an entire function $|f(z)| \le \sup_{\xi \in \partial D} |f(\xi)|$ for all $z \in D$

I was presented with a fact that for any entire function $f$ and any open disk $D$ it is true that $\forall z \in D$ $$ |f(z)| \le \sup_{\xi \in \partial D} |f(\xi)|. $$ But why is it true? Maybe it ...
0
votes
2answers
37 views

Can multiple inequality symbols be used in an inequality?

About these "inequalities" (I don't know if I'm being sarcastic): $$1 \lt 4 \gt 3$$$$6 \lt 8 \lt 15 \gt -1$$$$(6\cdot9)\gt(5\cdot10)\gt(2\cdot7)\lt(3\cdot5)\gt(-1\cdot3)$$ can multiple inequality ...
1
vote
1answer
40 views

$\mathbb{E}(e^{tS_{n}})\leq e^{Ct^{2}\sigma^{2}}\quad ?$

Let $S_n=X_1 + \cdots+ X_n$ be a sum of independent random variables such that each $X_i$ has mean zero, variance $\sigma_i ^2$ and lies in $[-1,1]$. Denote with $S_n$ the sum of these random ...
1
vote
3answers
20 views

Proving inequalities using induction all natural numbers that's greater than or equal to 5

using mathematical induction, prove that $n\le5: 4n<2^n$ base case: $4(5) < 2^5$ $20 < 32$ Correct I need help with the inductive process
1
vote
1answer
53 views

Prove that $\sum_{i=1}^na_i\sum_{i=1}^na^{-1}_i\ge n^2$ and $\sum_{i=1}^na_i^2\ge\frac1n$ [on hold]

For $a_i>0$, $i=1, \dots,n$ prove the inequalities $a)$ $$\sum_{i=1}^na_i\sum_{i=1}^na^{-1}_i\ge n^2$$ $b)$ $$\sum_{i=1}^na_i^2\ge\frac1n,\quad \text{if additionally}\sum^n_{i=1}a_i=1$$ ...