Questions on proving and manipulating inequalities.

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0
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2answers
46 views

What can we say about $x$?

If we have three reals (not positive reals, just reals) $x,y,z$ such that $x^2+3y^2+z^2+(x+y-z)^2=2$, what can we say about $x,y,z$? Is it possible to find the minimum of $x$? I don't know where to ...
1
vote
1answer
43 views

Which term of the binomial expansion of $\left(1+\sqrt{2}\right)^{50}$ is the greatest?

Which term of the binomial expansion of $\left(1+\sqrt{2}\right)^{50}$ is the greatest? How can I find it, without comparing all 51 values? Is there a quicker way to do it? (The solution says ...
1
vote
0answers
22 views

Inequality with complex number

Let $z,z'\in\mathbb{C}$. I want to prove that $$\vert\vert z\vert^{\alpha}z-\vert z'\vert^{\alpha}z'\vert\leq C (\vert z\vert^{\alpha}+\vert z'\vert^{\alpha})\vert z-z'\vert$$ where $\alpha$ is an ...
10
votes
0answers
37 views

Valid proof of Young's Inequality?

Part of an exercise to prove Holder's inequality in Rudin involves proving Young's Inequality... That is, given $\frac{1}{p}+\frac{1}{q} = 1$, prove $$ab \leq \frac{a^p}{p} + \frac{b^q}{q}.$$ ...
1
vote
1answer
23 views

Proving statements about ceiling and floor functions.

Prove or disprove the statements below. (a) For all positive real numbers x and y, $\lfloor x \cdot y\rfloor ≤ \lfloor x\rfloor \cdot \lfloor y\rfloor $. (b) For all positive real numbers x and y, ...
-1
votes
0answers
20 views

$E[f(X)g(X)] \geq E[f(X)]E[g(X)]$

Why is $E[f(X)g(X)] \geq E[f(X)]E[g(X)]$? Would Jensen's inequality be helpful?
-2
votes
2answers
68 views

Impossible to prove that is unbounded!

How we can demonstrate that $$\int _1^e\:\left(1+\log\left(x\right)\right)^ndx$$ is unbounded as $n\to \infty$, without using Bernoulli's Inequality?
2
votes
1answer
33 views

How do I integrate the inequality $ \frac{f(\frac{1}{2}+h)+f(\frac{1}{2}-h)}{2} \leqslant f(\frac{1}{2})$ over the range $h\in[0,1/2]$?

I would like to know the formal steps and theory. I was told that, by integrating this inequality, I can achieve one of the definitions of a concave function in the interval [0,1]. Thanks for your ...
1
vote
0answers
17 views

Prove a lower bound $\left|\int_{-\infty}^\infty k(t) g(t) e^{i\theta t}dt\right|\geq C \int_{-\infty}^\infty |f(t)|dt$

Let $k(t)$ be any function absolutely integrable over $(-\infty,\infty)$ and let $$g(t)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(u) e^{itu}du$$ Consider $$\int_{-\infty}^\infty k(t) g(t) e^{i\theta t}dt$$ can ...
1
vote
2answers
54 views

Prove that $\sqrt{2011} + \sqrt{2013} + \sqrt{2015} + \sqrt{2017} < 4 \sqrt{2014}$

How to prove that $\sqrt{2011} + \sqrt{2013} + \sqrt{2015} + \sqrt{2017} < 4\sqrt{2014}$ without using calculator?
-2
votes
1answer
36 views

Direct comparison test for ( Improper ) Integrals [on hold]

How we can prove with direct comparison test for ( Improper ) Integrals that is bounded: $\int _1^n\:e^{-x^3}dx$ ?
1
vote
2answers
62 views

Inequality very difficult to show

1) $\int _0^1\:\frac{x^n}{x^n+1}dx\ge \int _0^1\:\frac{x^{n+1}}{x^{n+1}+1}dx$ but I dont want to use $I_{n+1}-I_n$ 2) How we can prove with direct comparison test for ( Improper ) Integrals that is ...
1
vote
3answers
46 views

Does $a \leq b + c $ imply $a^2 \leq (b+c)^2 + (b-c)^2$?

Givens $$ a \leq b + c $$ or $$ a^2 \leq b^2 + c^2+2bc $$ Can we prove that?: $$ a^2 \leq (b+c)^2 + (b-c)^2 = 2b^2 + 2c^2 $$
1
vote
1answer
31 views

Very difficult to prove a convergent with Weierstrass

How we can prove that is monotone and bounded: $I_n=\int _1^n\:e^{-x^3}dx\:$ , Have any ideea how we can solve? and explain all to understand, I am a student... P.S: for all guys on this site, you ...
-2
votes
4answers
71 views

What's the easiest proof for $\left|\frac{\sin x}x\right|\le\frac 12$ for all $|x|\ge 2$? [on hold]

As asked in the title: What's the easiest proof for $$\left|\frac{\sin x}x\right|\le\frac 12\;\;\;\text{for all }|x|\ge 2$$
0
votes
0answers
12 views

Fourier series for convex plane curves.

The following problem is from Stein's Fourier analysis. This problem explores another relationship between the geometry of a curve and Fourier series. The diameter of a closed curve $\Gamma$ ...
5
votes
4answers
63 views

$\sum_{k=1}^n \log k \ge \int_1^n \log x \, dx$

Why is $$\sum_{k=1}^n \log k \ge \int_1^n \log x \, dx$$ is there an intuitive or graphical way to think about it?
0
votes
0answers
15 views

How to write Holder's inequality for random vectors?

For $1 < p,q < \infty$ satisfying the constraint $1/p + 1/q =1$ and for $X, Y$ random variables such that $\mathbb E [\vert X \vert ^p ], \mathbb{E} [\vert X \vert ^q ] < \infty $ we have ...
1
vote
2answers
63 views

How to prove that the $\lceil x y \rceil \le \lceil x\rceil\lceil y\rceil$ for real numbers $x, y$?

I know intuitively that this is true, but whenever I try and break apart that intuition to see where it's coming from, I essentially end up re-writing the assumption I'm trying to prove. I've tried ...
-5
votes
1answer
49 views

Estimating partial sums $\sum_{n = 1}^m \frac{1}{\sqrt{n}}$

Apostol's Calculus, exercise number I 4.7 13. Prove that if $n \geq 1$, then $$ 2(\sqrt{n+1} - \sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n} - \sqrt{n-1}) $$ and use this to prove that if ...
2
votes
0answers
27 views

On the existence of a certain sequence of positive numbers II

I wish to find a sequence of strictly positive real numbers $(a_1, a_2, \dots)$, such that $$ \sum_{k = 1}^\infty \frac{a_k}{k} < \infty $$ and such that for all $m, n \in \{1, 2, \dots\}$ with $m ...
-1
votes
0answers
25 views

About Riemann's hypothesis and a comment by 'almagest'.

When I was user 128932 I asked if a and b are relatively prime and are both exceptions to Robin's inequality then so is ab. The user 'almagest' confirmed this I think. So if N is sufficiently large ...
-1
votes
0answers
26 views

Proving Inequality $\sum_{m=1}^{M}\beta_m\ln(x_m)\geq 0 $

We have $\beta_m=\dfrac{x_m^{M-2}}{\prod_{j=1,j \neq m}^{M} (x_m-x_j)}$ Hence, it is easy to show that $\sum_{m=1}^{M}\beta_m = 0$ However, I am unable to show that $\sum_{m=1}^{M}\beta_m\ln(x_m) ...
5
votes
5answers
225 views

How to write an expression in an equivalent form without absolute values?

The question I have in front of me is the very first problem in Trench's Introduction to Real Analysis: Write the following expression in equivalent form not involving absolute values: $a+b+|a-b|$ ...
1
vote
2answers
25 views

Inequality with absolution value for complex number

How to show that inequality: $|1-\bar{\alpha} z| \ge |z-\alpha|$ $z$ and $\alpha$ are complex number, $\alpha$ is constans and $|z|<1$, $| \alpha| < 1$ I can proof that by using substition ...
0
votes
2answers
53 views

Let $a_1, \ldots, a_n$ be distinct positive integers. Show that $\sum a_n/n^2\ge\sum 1/n$ [on hold]

Let $a_1, \ldots, a_n$ be distinct positive integers. Show that $$\frac{a_1}{1^2} + \frac{a_2}{2^2} + \cdots + \frac{a_n}{n^2} \geq \frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{n}.$$
0
votes
1answer
23 views

Why does the unit vector of form $x_i=\frac{-1}{\sqrt{n}}$ minimize sum of $x_i$?

Cauchy-Schwarz implies that for $||\vec{x}||=1, \vec{y}=(1,\ldots,1)\in\mathbb R^n,\sum_{i=1}^{n} x_i = \pm\sqrt{n}$ if $\vec{x}=\pm{k}\vec{y}$. This implies that ...
0
votes
0answers
26 views

Simple number theory inequality

$$-b < -r \leq 0\text{ and } 0 \leq r' < b \implies -b < r'-r < b$$ how is that implication possible? I'm going over the proof for the division theorem mainly the uniqueness part ...
1
vote
1answer
17 views

Show $\sum_{n\leq k\leq 2n}2^{-2k}\log(k)\leq C\, 2^{-2n}\log(n)$

I'd like to prove $$\sum_{n\leq k\leq 2n}2^{-2k}\log(k)\leq C \, 2^{-2n}\log(n),$$ where $C>0$ is a constant. Can someone give me a hint.
2
votes
1answer
27 views

Combinatorial sum inequality

Prove the following inequality: $$ \forall k\in\left\{4n+5:n\in\mathbb{N}\right\},\qquad\sum_{m=0}^{\frac{k-1}{2}}{\left( -1 \right) }^{m}\binom{k}{2m}2^{2m}\neq 1. $$ I'm particularly interested ...
2
votes
1answer
46 views

Let $a,b,c$ be the nonnegative real numbers such that $a+b+c=1$. Prove that $\sqrt{a+\frac{(b-c)^2}4}+\sqrt b+\sqrt c\le\sqrt3$

Let $a,b,c$ be the nonnegative real numbers such that $a+b+c=1$. Prove that $$\sqrt{a+\frac{(b-c)^2}4}+\sqrt b+\sqrt c\le\sqrt3$$ I first wrote $a$ as $1-b-c$ and substituted it in main ...
-5
votes
2answers
49 views

Proof of a mathematical inequality [on hold]

I am looking for the solution(by induction method) of the inequality given below $1/2^2 + 1/3^2 + ... + 1/n^2 < 1$ for $n \geq 2$ Looking for help
3
votes
2answers
124 views

Inequality and Induction

I needed to prove that $\prod_{i=1}^n\frac{2i-1}{2i}$ $<$ $\frac{1}{\sqrt{2n+1}}$, $\forall n \geq 1$ . I've atempted by induction. I proved the case for $n=1$ and assumed it holds ...
2
votes
4answers
216 views

Integral inequality 5

How can I prove that: $$8\le \int _3^4\frac{x^2}{x-2}dx\le 9$$ My teacher advised me to find the asymptotes, why? what helps me if I find the asymptotes?
4
votes
1answer
45 views

Is $x(x!)^{1/x}$ an increasing function of $x$, for $x > 0$?

Is $x(x!)^{1/x}$ an increasing function of $x$, for $x > 0$? Here $x!$ is the factorial of $x$. Sure, I do know differential calculus, but my problem is that I do not know how to compute for the ...
0
votes
1answer
15 views

Prove a lower bound on $\left|\int_{-\infty}^{+\infty}k(t) f(t) e^{\lambda_n ti}dt\right|$.

Let $k(t)$ be any function absolutely integrable over $(-\infty,+\infty)$, let $$K(u)=\int_{-\infty}^{+\infty}k(t) e^{-uti}dt$$ and let $$f(t)=\sum_n a_n e^{-\lambda_n t i}, \ \ \ \lambda_n\in\mathbb ...
1
vote
1answer
37 views

Holder's inequality with expectation norm

One defines the ``p-expectation norm" of a function $f$ as $\vert f \vert_p = (\mathbb E ( f^p) )^{ \frac{1}{p}}$. What is the intuition for this norm? Now why are the following true? $\vert ...
0
votes
2answers
17 views

Don't understand adding a system of compound inequalities

I'm reading a proof of the Division Theorem and one line that comes up is Since 0 ≤ r1 < b and 0 ≤ r2 < b , we have −b < r1 − r2 < b. I do not ...
0
votes
0answers
8 views

Upper bound $f(t,d)$ such that $\sum_{j=1}^{d} cos(2 \pi j \; t) \leq f(t,d)$. [duplicate]

I have a sum of a series of trig functions as follows: $\sum_{j=1}^{d} cos(2 \pi j \; t)$ where t is just a constant. Here, we can assume $t$ is a small number and $t \neq 0$. what is the upper ...
1
vote
2answers
44 views

Inequality with condition $x^2+y^2+z^2=1$.

let $x,y,z>0$ such that $x^2+y^2+z^2=1$. Find the minimum of $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$$ Is the answer $3\sqrt{3}$ by any chance?
0
votes
1answer
23 views

Inequality $(n+1)^{-s} \leq (2n)^{-s}$ true for all $s\leq1$ and natural $n$?

On the line $S_{2n}-S_n$ I don't understand how the first inequality was established for $s \leq 1$. I see how it works for $0 \leq s \leq 1$ but not s < 0. Any clues?
1
vote
1answer
28 views

Show that $r_k^n/n \le \binom{kn}{n} < r_k^n$ where $r_k = \dfrac{k^k}{(k-1)^{k-1}}$

Show that for $n \ge 2$, $\dfrac{r_k^n}{n+1} \le \binom{kn}{n} < r_k^n$ where $r_k = \frac{k^k}{(k-1)^{k-1}}$. This is a generalization of How to prove through induction which asks for a proof ...
3
votes
0answers
56 views

Is my proof of the Schoenfeld's inequality correct?

Full preprint here. Theorem 4.1. For any $x\ge 2$ we have $$ \begin{equation} \theta(x)-x<\frac{1}{8\pi}\sqrt x\log^2 x. \;\;\;\;\;\;\;\;\;\;\;(4.1) \end{equation} $$ Proof: It's known that ...
1
vote
1answer
28 views

Question about inequality in linear algebra

$V$ is inner product space. $u, v \in V$ are two orthogonal vectors. Prove that $\|v-u\| \geq \|v\|$. Because $\|v-u\|, \|v\| \geq 0$ it's enough to prove that $||v-u||^2 \geq \|v\|^2$. ...
1
vote
1answer
49 views

trace inequalities: linear algebra

If S is any $n \times n$ real, symmetric, invertible matrix and D is any $n \times n$ diagonal matrix such that $0\prec D \prec I$ then does there exist a constant $\gamma$ such that: ...
3
votes
1answer
59 views

On the existence of a certain sequence of positive numbers

I wish to find a sequence of strictly positive real numbers $(a_1, a_2, \dots)$, such that $$ \sum_{k = 1}^\infty \frac{a_k}{k} < \infty $$ and such that for all $m, n \in \{1, 2, \dots\}$ with $m ...
1
vote
2answers
32 views

Upper bound $f(t,d)$ such that $\sum_{j=1}^{d} cos(2 \pi j \; t) \leq f(t,d)$?

I have a sum of a series of trig function as follows: $\sum_{j=1}^{d} cos(2 \pi j \; t)$ where t is just a constant. I am looking for the upper bound $f(t,d)$ such that $\sum_{j=1}^{d} cos(2 \pi j ...
2
votes
1answer
16 views

Poincaré constant of a cover

If $U\subset \mathbb{R}^n$ is a bounded, open, connected set and $U \subseteq \bigcup_{i=1}^N{U_i}$ (with, say, $U_i$ open bounded and connected), and $C_V$ denotes the $L^2(V)$-Poincarè constant of ...
-4
votes
1answer
43 views

Gronwall inequality [closed]

Let $f$ and $g$ continuous on $[a,b]$ and non-negative functions. Let $C>0$. Suppose that $f(x)\leq C+\int_a^x fg$. Demonstrate the Gronwall inequality: $$f(x)g(x)\leq C(e^{\int_a^x g})$$ HINT: ...
1
vote
2answers
23 views

Prove the logarithmic inequality

Prove that: $(\log_{24}{48})^2+(\log_{12}{54})^2>4$ I tried to put $t=\log_23$ and get the equation $6t^4+32t^3+22t^2-84t-74>0$. But I can't do anything with it...