Questions on proving and manipulating inequalities.

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0
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1answer
34 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
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2answers
72 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
62 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
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0answers
29 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
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0answers
35 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
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0answers
30 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) ...
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5answers
252 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
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2answers
29 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
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1answer
24 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 ...
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0answers
27 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 ...
2
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1answer
22 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
29 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
52 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 ...
3
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2answers
128 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
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4answers
241 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
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1answer
48 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
16 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
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1answer
46 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
18 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
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0answers
10 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 ...
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2answers
46 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
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1answer
25 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
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1answer
29 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 ...
1
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1answer
30 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
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1answer
65 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
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1answer
61 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
42 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
17 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 ...
1
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2answers
28 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...
3
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1answer
65 views

A combinatoric inequality

How can I show that for every $0 < t < 1$, $$ \frac{n (n - 1) \cdots (n - k + 1)}{(t + n - 1) (t + n - 2) \cdots (t + n - k)} \leq 1 + \frac{k}{t} $$ where $n \in \{1, 2, \dots\}$ and $k \in ...
6
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2answers
296 views

A question about the proof of Schwarz inequality

There are many proofs of the Cauchy-Schwarz inequality, here's one of them: Consider the following quadratic polynomial: $$f(x)=\left(\sum_{i=1}^{n} a_i^2 \right)x^2-2\left(\sum_{i=1}^{n} a_ib_i ...
1
vote
1answer
49 views

Inequality with Logarithms!

I need some help solving this inequality for a question involving the number of bounces, $n$, of ball such that the max. height of the ball is less than 5cm. This is the equation I have gathered from ...
0
votes
1answer
22 views

Angles inequality in acute triangle [duplicate]

Let $\alpha$, $\beta$, $\gamma$ be angles of acute triangle. How to prove that $(\tan(\frac{\alpha}{2}))^2 + (\tan(\frac{\beta}{2}))^2 + (\tan(\frac{\gamma}{2}))^2 \ge 1$? Does left side of equation ...
5
votes
2answers
81 views

How prove Reversing the Arithmetic mean – Geometric mean inequality?

Let $x_1,x_2,\cdots,x_n$ $(n\geq2)$ be a non-decreasing monotonous sequence of positive numbers such that $x_1,\frac{x_2}{2},\cdots,\frac{x_n}{n}$ be a non-increasing monotonous sequence .Prove that ...
5
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1answer
56 views

Prove this $\sum_{1\le i<j\le n}\left((x_{j}-x_{i})-(x_{j}-x_{i})^2\right)\le\frac{n^2-1}{12}$

let $x_{1},x_{2},\cdots,x_{n}\in [0,1]$ show that $$f=\sum_{1\le i<j\le n}\left((x_{j}-x_{i})-(x_{j}-x_{i})^2\right)\le\dfrac{n^2-1}{12}$$ My approach is the following: since ...
0
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1answer
18 views

An upper bound on the rate of convergence of a series with a variable starting index

Let $a, b, k_0$ be fixed real numbers, of which $a$ and $b$ are strictly positive. Is there some non-negative real number $c$ such that, for large enough $n$'s, $$ \sum_{k = k_0 + \sqrt{\frac{a}{b} ...
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2answers
43 views

How to start proof of triangular inequality? [duplicate]

$$\left| {\left| a \right| - \left| b \right|} \right| \le \left| {a \pm b} \right| \le \left| a \right| + \left| b \right| $$
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3answers
91 views

Is it true that $|a^{\alpha} - b^{\alpha}| \leq |a-b|^{\alpha}$?

I am currently reading some papers that seem to use the fact that $$|a^{\alpha} - b^{\alpha}| \leq |a-b|^{\alpha},$$ for $-1< \alpha < 0$ and $a,b$ in the upper half plane of $\mathbb{C}$. Is ...
1
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2answers
44 views

Prove by induction on $n$ that when $x \gt 0$, $ (1+x)^n \ge 1+nx+\frac{n(n-1)}{2}x^2 \text{ for all positive integers } n. $

Here's the problem: Prove by induction on $n$ that when $x \gt 0$ $$ (1+x)^n \ge 1+nx+\frac{n(n-1)}{2}x^2 \text{ for all positive integers } n. $$ So, clearly the base case is true. Here's how far ...
1
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1answer
43 views

Solving Log equation using master theorem

I`m studying Master Theorem, and I got stuck in the case 3. The example is : T(n) = 3T(n/4) + nlogn. I have no idea how my teacher got the final value, c = 3/4, based on the equation below : 3*[n/4 ...
2
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2answers
32 views

Exercise from An Introduction to Inequalities

I am reading An Introduction to Inequalities by Beckenbach and Bellman and on chapter 4.3 there is this exercise. It's regarding AM-GM inequality. How can I prove it? I can't figure it out. $$ ...
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3answers
74 views

Average of square roots's sum vs. square root of an average

I was watching a video on youtube about how colors work in computers, and found this statement: "The average of two square roots is less than the square root of an average" The link to the ...
1
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0answers
45 views

Nonnegative solution of a linear system

Given three collections of parameters $\epsilon_1 > ... > \epsilon_N$, $(a_1,...,a_{N-1})$ and $(b_1,...,b_N)$ that satisfy the following conditions: (i) $\forall i, a_i \geq 0, ...
0
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0answers
32 views

Assume that $1a_1+2a_2+\cdots+20a_{20}=1$, where the $a_j$ a [duplicate]

Assume that $$1a_1+2a_2+\cdots+20a_{20}=1, $$ where the $a_j$ are real numbers and that these values minimize $$1a_1^2+2a_2^2+\cdots+20a_{20}^2.$$ Find $a_{12}$.
2
votes
6answers
189 views

A quick way to prove the inequality $\frac{\sqrt x+\sqrt y}{2}\le \sqrt{\frac{x+y}{2}}$

Can anyone suggest a quick way to prove this inequality? $$\frac{\sqrt x+\sqrt y}{2}\le \sqrt{\frac{x+y}{2}}$$
1
vote
1answer
12 views

How should this statement be understood?

How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content? I am trying to solve ...
1
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0answers
33 views

Solve for a variable in an inequality

In the following inequality $$n - m \left(1 - \left(1 - \frac{1}{m}\right)^n\right) \le r$$ where $n<m$ are integers and $r$ is also a small integer, can one solve for $m$? Thanks D.
0
votes
1answer
61 views

An upper bound for $a_1 (\sum_{i=2}^n a_i^{n-1})$ in terms of $a_1^n + \sum_{i=2}^n a_i^n$

Assume that $n \in \mathbb{N}, n \geq 2$ and $a_i \in \mathbb{R}, a_i > 0, \forall i=1,...,n$ Show that $$ \frac{a_1 (\sum_{i=2}^n a_i^{n-1})}{ a_1^n + \sum_{i=2}^n a_i^n} \leq 1 -\frac{1}{n}$$ ...
2
votes
1answer
18 views

A Simple Question on Comparing Functions

Assume two functions: $U_A = -(a_1 - \hat{a}_2)^2$ and $U_R = -(a_0 - \hat{a}_2)^2$ Given $a)$ $a_0 \geq \hat{a}_2$ what are the possible conditions that satisfy $U_A > U_R$, $U_A < U_R$, and ...
0
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0answers
40 views

How to prove the probability inequality?

Given four mutually independent bounded random variables (RVs), denoted as $x_1,x_2,x_3,x_4$ and we have inequalities that \begin{equation} \mathrm{Pr}(x_1<0)<\mathrm{Pr}(x_3<0) \\ ...