Questions on proving and manipulating inequalities.

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8
votes
1answer
237 views

How to prove there exists $n_{1}a_{n_{0}}+n_{2}a_{n_{1}}+\cdots+n_{k}a_{n_{k-1}}<3(a_{1}+a_{2}+\cdots+a_{N})$

Let $a_{1},a_{2},\cdots,a_{N}$ be nonnegative reals, not all $0$. Prove that there exists a sequence $$1=n_{0}<n_{1}<\cdots<n_{k}=N+1$$ of integers such that ...
3
votes
0answers
48 views

Find $\Big\{ (a,b)\ \Big|\ \big|a\big|+\big|b\big|\ge 2/\sqrt{3}\ \text{ and }\forall x \in\mathbb{R}\ \big|a\sin x + b\sin 2x\big|\le 1\Big\}$ [on hold]

Find all (real) numbers $a $ and $b$ such that $|a| + |b| \ge 2/\sqrt{3} $ and for any $x$ the inequality $|a\sin x + b \sin 2x | \le 1$ holds. In other words, find the set $Q$ defined as ...
5
votes
3answers
113 views

Given $a,b,c,d>0$ and $a^2+b^2+c^2+d^2=1$, prove $a+b+c+d\ge a^3+b^3+c^3+d^3+ab+ac+ad+bc+bd+cd$

Given $a,b,c,d>0$ and $a^2+b^2+c^2+d^2=1$, prove $$a+b+c+d\ge a^3+b^3+c^3+d^3+ab+ac+ad+bc+bd+cd$$ The inequality can be written in the condensed form ...
2
votes
2answers
80 views

Prove QM-AM inequality

$$\dfrac{x_1^2+ x_2^2 + \cdots + x_n^2}n \geq \left(\dfrac{x_1+x_2+\cdots+x_n}n\right)^2$$ I don't think AM, GM can be used here. And simple expansion doesn't help too. What should I do?
28
votes
5answers
864 views

Which is bigger: $(\pi+1)^{\pi+1}$ or $\pi^{\pi+2}$?

I've been struggling for a while with the following problem: Which is bigger: $(\pi+1)^{\pi+1}$ or $\pi^{\pi+2}$? Needless to say software aid is not allowed. All manual calculations should be ...
5
votes
2answers
73 views

Proving that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)}\ge\frac32$ using derivatives

Let $a,b,c\in\mathbb{R}^+$ and $abc=1$. Prove that $$\frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)}\ge\frac32$$ This isn't hard problem. I have already solved it in following way: Let ...
1
vote
1answer
330 views

Equality in the Cauchy-Bunyakowsky-Schwarz inequality for a semi-inner product

I am stuck with an exercise that I found in a textbook by Conway. First, I would like to clarify what is meant by a semi-inner product. Definition. Suppose that $\mathscr X$ is a vector space over ...
1
vote
2answers
32 views

Inequality $\left|z+w\right|\geq\left||z|-|w|\right|$ when $|w|\leq A|z|$.

Let $z,w\in\mathbb C$ and $|w|\leq A|z|$ for $A>0$. I want estimate from below $|z+w|$. I proceeded as follows. Since $$\left|z+w\right|\geq\left||z|-|w|\right|$$ and $-|w|\geq -A|z|$, I write ...
2
votes
5answers
108 views

Determine whether $f(x)$ is increasing or decreasing

Let $f(x) = -x + (x^3/3!) + \sin(x)$ How do I determine if $f(x)$ is increasing or decreasing? I have already found the derivative of this function which is: $f'(x) = -1 + (x^2/2) + \cos(x)$ And I ...
-3
votes
1answer
37 views

The integer part of $x+1$ is the integer part of $x$ plus $1$ [closed]

How do you solve the proof: If $x$ is a real number, then: $[x+1] = [x] + 1$. For my proof, I tried to describe the interior of the argument inside the parentheses, but I was unsuccessful. Please ...
1
vote
2answers
38 views

Solve for real value of $x$: $|x^2 -2x -3| > |x^2 +7x -13|$

Here I have a question: Solve for real value of $x$: $$|x^2 -2x -3| > |x^2 +7x -13|$$ I got the answer as $x = (-\infty, \frac{1}{4}(-5-3\sqrt{17}))$ and ...
2
votes
0answers
19 views

$\sum_{i=1}^n\sum_{j=1}^n|c_i+c_j|a_ia_j\geq\sum_{i=1}^n\sum_{j=1}^n|c_i-c_j|a_ia_j$

Let $c_i\in\mathbb R$, $a_i\geq0$ with $\sum_{i=1}^n a_i=1$, prove $$\sum_{i=1}^n\sum_{j=1}^n|c_i+c_j|a_ia_j\geq\sum_{i=1}^n\sum_{j=1}^n|c_i-c_j|a_ia_j$$ This inequality comes from there, when $X$ is ...
0
votes
1answer
34 views

About inequality $\sum_{k=1}^n |a_k|^2 \lessgtr \sum_{k\neq s} |a_k| |a_s|$

Let $a_k$ a sequence of complex number. We have $$\left(\sum_{k=1}^n |a_k|\right)^2 \geq \sum_{k=1}^n |a_k|^2$$ It is a basic fact because $$\left(\sum_{k=1}^n |a_k|\right)^2 = \sum_{k=1}^n |a_k|^2 + ...
1
vote
2answers
41 views

prove using Lagrange multipliers that for $x,y>0,\space n\in \mathbb N,\space (\frac{x+y}2)^n \leq \frac{x^n+y^n}2 $

I have been asked to prove using Lagrange multipliers that for \begin{equation*} \space (\frac{x+y}2)^n \leq \frac{x^n+y^n}2,~x,y>0,~n\in \mathbb {N} \end{equation*} I am familiar with the ...
2
votes
1answer
34 views

Find an example that the following equality doesn't apply

I need a sequence $(f_n)_{n\in\mathbb{N}}, f_n:X\to\mathbb{R}^\star$ which is measurable, $f_n \ngeq 0$ and doens't accept the following equality: $$\int_X\sum_{n=1}^\infty f_n \, d\mu = ...
0
votes
3answers
48 views

The inequality $k(n-1)<n^2-2n$ for all odd $n$ and $k<n$

How one can prove the following statement: $k(n-1)<n^2-2n$ for all odd $n$ and $k<n$ Tried so far: induction on $n$, graphing, and rewriting $n^2−2n$ as $(n−1)^2−1$.
1
vote
1answer
49 views

Struggling to prove inequality

I've been given to following inequality to prove: (The hint given was not to evaluate the integral) \begin{equation*} \frac{1}{4} \leq \int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{sin(x)}{x}dx\leq ...
3
votes
2answers
68 views

uniform bound for sine integral function

Prove that for any $0<a<b$, $$ \left|\int_a^b\frac{\sin x}{x}\,dx\right|\le4 $$ Here is my approach. I used integration by parts to prove that LHS is bounded by $3$ when $a\ge 1$. I will be done ...
1
vote
2answers
145 views

Complex inequality $||u|^{p-1}u - |v|^{p-1}v|\leq c_p |u-v|(|u|^{p-1}+|v|^{p-1})$

How does one show for complex numbers u and v, and for p>1 that \begin{equation*} ||u|^{p-1}u - |v|^{p-1}v|\leq c_p |u-v|(|u|^{p-1}+|v|^{p-1}), \end{equation*} where $c_p$ is some constant dependent ...
0
votes
0answers
22 views

A smart way to bound this function and get rid of covariance matrix

I have the following function which I am trying to bound as follows $$A({\bf h},\Sigma)= \log(1+ {\bf h}^* \Sigma {\bf h} )$$ $$\Sigma = \begin{bmatrix} 1-|\rho_1|^2 & \rho_3 - \rho_1 \rho_2^* ...
0
votes
3answers
36 views

Proving inequality involving real numbers [closed]

$x, y$ are real positive numbers. Let $m$ be the smallest number among $x, y + \frac{1}{x}, \frac{1}{y}$. How to prove that $m \le \sqrt{2}$? I really don't know how to start.
1
vote
0answers
21 views

Completing Karlin's proof of variation diminishing transformation theorem

In S Karlin's book total positivity there's a theorem that says if $K(x,y)$ is $TP_r$ (totally positive with degree $r$) and the sign change count of function $h$, $S(h) = n\leq r-1$, then ...
1
vote
1answer
32 views

Maximum singular value of a matrix valued function

Let $f$ be an analytic matrix-valued function, $\Lambda(A)$ be the spectrum of $A$ and $\sigma_1(A)$ the maximum singular value of $A$. It is known that $$\Lambda(f(A)) = f(\Lambda(A)) := ...
6
votes
4answers
941 views

Proving $\sqrt{x + y} \le \sqrt{x} + \sqrt{y}$

How do I prove $\sqrt{x + y} \le \sqrt{x} + \sqrt{y}$? for $x, y$ positive? This should be easy, but I'm not seeing how. A hint would be appreciated.
0
votes
1answer
25 views

How to show this is decreasing

I'd like to show $$\sum_{i=1}^n \frac{1}{i((n+1)-i)} $$ is decreasing for n>1, which is Cauchy product of $$\sum_{i=1}^n \frac{1}{i}$$ Numerical computation until n=50 shows it's decreasing but I ...
13
votes
6answers
204 views

show that $\frac{1}{F_{1}}+\frac{2}{F_{2}}+\cdots+\frac{n}{F_{n}}<13$

Let $F_{n}$ is Fibonacci number,ie.($F_{n}=F_{n-1}+F_{n-2},F_{1}=F_{2}=1$) show that $$\dfrac{1}{F_{1}}+\dfrac{2}{F_{2}}+\cdots+\dfrac{n}{F_{n}}<13$$ if we use Closed-form expression ...
0
votes
1answer
30 views

Chebiyshev Inequality

In proving the Chebyshev inequality in Probability theory an important step is to observe that: $P((|x-E(x)|≥a))=P(|x-E(x)|^2≥a^2)$. It is assumed that X has a moment of order 2. Can somebody help ...
-1
votes
0answers
33 views

How to solve this inequality using AM-GM? [duplicate]

Let $a,b,c>0$ and $a+b+c=1$. Prove $$\sqrt{\frac{ab}{c+ab}}+\sqrt{\frac{bc}{a+bc}}+\sqrt{\frac{ca}{b+ca}}\le\frac32$$
1
vote
1answer
28 views

Upperbound a logarithmic expression that has a covariance matrix

Let $\Sigma$ be a $2\times 2$ covariance matrix and ${\bf h}$ a vector of complex values entries. $$A= \log(1+ {\bf h}^* \Sigma {\bf h} )$$ $$\Sigma = \begin{bmatrix} 1-|\rho_1|^2 & \rho_3 - ...
0
votes
1answer
32 views

Hilbert's inequality for $\left|\sum_{n,m}a_n \bar a_m\right|$.

We know that, an Hilbert's inequality states $$\left|\sum_{n\neq m}\frac{a_n \bar{a}_m}{n-m}\right|\leq\pi \sum_n |a_n|^2$$ Give $a_n, b_n$ two sequences of complex numbers. Then write an inequality ...
1
vote
1answer
47 views

Proving $(n+1)c^{1/(n+1)} - nc^{1/n} \le 1$ from first principles

Is it possible to prove that \begin{align*} (n+1)c^{1/(n+1)} - nc^{1/n} \le 1 \qquad c \in \mathbb{R}_+, n \in \mathbb{N} \end{align*} using only elementary techniques? (No calculus, no appeasement to ...
1
vote
1answer
53 views

Convex function inequality for Euclidean norm: $\|(f(x_1),\cdots,f(x_n))\|_2\leq f(\|x\|_2)$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a positive, convex, continuous function such that $f(0)=0$. (If you wish you can also suppose $f$ to be monotone increasing.) I would like to prove or to ...
2
votes
2answers
165 views

Minimum Value of expression

Given that $x$, $y$ and $z$ are positive real numbers satisfying $xyz=32$, find the minimum value of: $$x^2+4xy+4y^2+2z^2$$ Perhaps AM-GM and manipulation but I'm not quite sure how? Source BMO.
0
votes
0answers
22 views

Determine the order of $\exp Q(z)$ when $Q$ is a polynomial of degree $q$.

I am looking to determine the order of $f(z) = \exp Q(z)$ when $Q$ is a polynomial of degree $q$. I think the order is $q$, but I am struggling to prove it. The definition of order is: An entire ...
1
vote
1answer
335 views

$\frac{a}{\sqrt{a+b}} + \frac{b}{\sqrt{b+c}} + \frac{c}{\sqrt{c+a}} > \sqrt{a+b+c}$ is true for positive a,b,c

How do you prove that for all positive $a,b,c$ this formula holds true: \begin{equation*} \frac{a}{\sqrt{a+b}} + \frac{b}{\sqrt{b+c}} + \frac{c}{\sqrt{c+a}} > \sqrt{a+b+c}? \end{equation*} Any ...
1
vote
1answer
47 views

Inequalities using AM-GM

Use the AM-GM inequality to prove $(5xy + 6y)^3$ ≥ $1215xy^3$ for all real numbers x, y > 0. Not sure if I was on the right track but so for my understanding is: since there is a power of 3 on the ...
3
votes
2answers
45 views

Proving $x\ln(\frac{x}{a})+y\ln(\frac{y}{b})\geq (x+y)\ln(\frac{x+y}{a+b})$

Let a,b,x,y be positive reals. Prove $x\ln(\frac{x}{a})+y\ln(\frac{y}{b})\geq (x+y)\ln(\frac{x+y}{a+b})$ I don't have any olympic background, so I may be missing some standard trick. The ...
4
votes
1answer
47 views

Struggling with inequality involving a bunch of binomial coefficients

I want to find a lower bound on $n$, i.e. isolate $n$, or more realisticly, approximate $n$ that satisfies the following : $$ {n \choose k}\left( 1 - \frac{{n \choose \frac{n-1}{2} - k}}{{n \choose ...
3
votes
2answers
317 views

Proving that $f(x) = \vert x \vert^{\alpha}$ is Holder continuous, inequality help

The definition of $\alpha$-Holder continuity for a function $f(x)$ at the point $x_0$ is tha there exist constant $L$ such that for all $x \in D$ \begin{equation} \vert f(x) - f(x_0) \vert \leq L ...
3
votes
1answer
135 views

The complex numbers inequality $(|a+tb|^p-|a|^p)/t \leq |a+b|^p-|a|^p$

The following inequality is from the proof that the $L^p$ norm is Gâteaux differentiable for $ 1 < p<\infty$ (from "Analysis" by Lieb and Loss). Let $a$, $b\in\mathbb{C}$ and $-1\leq t\leq 1$, ...
1
vote
0answers
28 views

$ \| v\| \leq Ce^x\| w\| \quad \left(x\rightarrow 0 \right) $ if $\| w\|=0$

Let $v\in \mathbb{R}^J$ and $J\in \mathbb{N}$. I have the follow inequality: $$ \| v\| \leq \| w\| + x \quad \left(x\rightarrow 0 \right) $$ If $\|w\|\not=0$ we can find a $C$ positive constant with ...
6
votes
1answer
787 views

An inequality like Riemann sum involving $\sqrt{1-x^2}$

How can I prove that for every positive integer $n$ we have \begin{equation*} \frac{n\pi}{4}−\frac{1}{\sqrt{8n}}<\frac{1}{2}+\sum_{k=1}^{n−1}\sqrt{1−\frac{k^2}{n^2}}? \end{equation*}
3
votes
2answers
260 views

Absolute Value Equation

Please help me with this! $$x^3+|x| = 0$$ Now one solution is clearly $0.$ We have to find the other solution (i.e, $-1$) $$Solution:$$ CASE $1$: If $x<0,~|x| = -x$, we can write $x^3+|x| = 0$ as ...
3
votes
1answer
61 views

Inequality of elementary symmetric polynomials

Let $\lambda=(\lambda_1,\lambda_2,\lambda_3,\lambda_4)$ with $\lambda_i>0$ for $i=1,2,3,4$. Let $$\sigma_k(\lambda)=\sum_{1\leq ...
0
votes
0answers
10 views

If $f_{1}(x)<f_{2}(x)$, is it true that $ \min_B \max_{Bx=0} f_{1}(x)<\min_B \max_{Bx=0} f_{2}(x)$?

One "obvious" question but I hope I can get some explanations... If $f_{1}(x)<f_{2}(x)$, is it true that $ \min_B \max_{Bx=0} f_{1}(x)<\min_B \max_{Bx=0} f_{2}(x)$? $B$ is an arbitrary matrix ...
1
vote
4answers
30 views

Inequality for sides and height of right angle triangle

Someone recently posed the question to me for the above, is c+h or a+b greater, without originally the x and y lengths. I used this method: (mainly pythagorus) $a^2+b^2=c^2=(x+y)^2=x^2+y^2+2xy$ ...
0
votes
1answer
15 views

$y^tv<0,z^tv<0 \text{ unsolveable} \Leftrightarrow \exists \lambda\geq 0: y=-\lambda z$

I am trying to show: Let $y,z\in\mathbb{R}^n$ and $z\neq 0$. Then $y^tv<0,z^tv<0 \text{ unsolveable} \Leftrightarrow \exists \lambda\geq 0: y=-\lambda z$. '$\Leftarrow$' is trivial. ...
1
vote
1answer
31 views

Prove that $\left|\sum_{r\neq s}u_r\overline u_s\csc\pi(x_r-x_s)\right|^2\leq\sum_{r}\left|\sum_s\overline u_s\csc\pi(x_r-x_s)\right|^2$

On a paper that I'm studying, it is written, without another: "By Cauchy's inequality $$\left|\sum_{r\neq s}u_r\overline u_s\csc\pi(x_r-x_s)\right|^2\leq\sum_{r}\left|\sum_s\overline ...
2
votes
2answers
27 views

Find the minimum value of k $(k \in I)$ for which the equation $e^x =kx^2$ has exactly three real solution.

Problem : Find the minimum value of k $(k \in I)$ for which the equation $e^x =kx^2$ has exactly three real solution. My approach : We apply log on both sides $x=2\ln(k x^2)$ $\Rightarrow ...
1
vote
3answers
53 views

Is the following inequality true $(a^3-b^6)^3+(3abc)^3 \leq (a^3-b^6+3cb^3)^3$?

Let $a,b,c$ be all positive integers greater than $1$. If $$a>b^2$$ and $$a^3-b^6> 3c$$ Is this the following inequality true?: $$(a^3-b^6)^3+(3abc)^3 \leq (a^3-b^6+3cb^3)^3$$ I tried to ...