Questions on proving and manipulating inequalities.

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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
vote
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
votes
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 ...
1
vote
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
votes
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
vote
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
vote
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
vote
1answer
64 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
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
vote
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
votes
1answer
64 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
votes
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
votes
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
votes
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} ...
0
votes
2answers
43 views

How to start proof of triangular inequality? [duplicate]

$$\left| {\left| a \right| - \left| b \right|} \right| \le \left| {a \pm b} \right| \le \left| a \right| + \left| b \right| $$
1
vote
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
vote
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
vote
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
votes
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. $$ ...
1
vote
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
vote
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
votes
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
vote
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
votes
0answers
39 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) \\ ...
2
votes
3answers
50 views

Find $N \gt 0$ so that $ n \ge N \implies \lvert \left(1+\frac3n \right)^2 -1\rvert \lt \frac{1}{10}$.

Here's the problem: Find $N \gt 0$ so that $$ n \ge N \implies \left\lvert \left(1+\frac3n \right)^2 -1\right\rvert \lt \frac{1}{10}$$ So, I'm not quite sure what this is asking me. I found a value ...
0
votes
0answers
15 views

Monotonicity of n-th root by induction

Suppose a,b are real numbers. I'm trying to prove that $\forall n\geq 1$ ( $0<a<b$ entails $0 <a^{1/n}<b^{1/n}$ ) with the method of Induction. P.S : I already know how to ...
0
votes
2answers
45 views

Tricky Logarithmic Inequality Problem

I am having a problem solving this question - If $\log_{\frac{1}{\sqrt{2}} }{\sin{x}}>0$, $x\in [0,4\pi]$,then number of values for chating which are integral multiples of $\pi/4$,is A-6 B-12 ...
3
votes
2answers
31 views

summation inequlity

how to derive the following inequality: $$\sum_{k=1}^n \sqrt{x_k}\sqrt{\frac{1}{x_k}} \leq \sqrt{\sum_{k=1}^n x_k}\sqrt{\sum_{k=1}^n\frac{1}{x_k}} $$ The main question is to prove $n^2 \leq ...
0
votes
1answer
22 views

In the finite difference formulas, how can we pick h to give a certain tolerance?

There's lots of questions on here about finite differences. In particular, picking the 'best' h value. But what if I want to find the biggest 'h' which bounds to a given tolerance? On first glance, I ...
0
votes
1answer
35 views

Triangle inequality on the projective space

Given a unit $n$-sphere $\mathbb{S}^n = \{x \in \mathbb{R}^{n+1} : \langle x,x \rangle = 1\}$, we define the set $\mathbb{P}^n = \{[x] : x \in \mathbb{S}^n\}$, where $[x] = \{-x, x\}$, and a function ...
2
votes
2answers
37 views

Conditional Extremum, need help finding the extreme points in calculation.

Find the conditonal extremums of the following $$u=xyz$$ if $$(1) x^2+y^2+z^2=1,x+y+z=0.$$ First i made the Lagrange function $\phi= xyz+ \lambda(x^2+y^2+z^2-1) + \mu (x+y+z) $, now making the ...
-1
votes
3answers
39 views

Show that $2(2^n-2)^{n-1}>(2^n-1)^{n-1}$ for all $n\geq 2$.

Subject says it all. Show that $2(2^n-2)^{n-1}>(2^n-1)^{n-1}$ for all $n\geq 2$. I am confident that it is true but a proof has remained elusive.
2
votes
0answers
25 views

Asymptotic solution to $m \leqslant e^{\lambda t} (c t^q - \varepsilon)$

What is the smallest $t$ statisfying the inequality: $m \leqslant e^{\lambda t} (c t^q - \varepsilon)$, where $\varepsilon$ is arbitrary small positive number? I believe $t$ must be of the from: $$t = ...
1
vote
3answers
34 views

how to solve inequation involving modulus

how to solve $$ǀx^2 + 3xǀ + x^2-2 ≥ 0 ?$$ I got stuck in the above problem. What would be the classic process to solve these type of problems. Also, if u have some fast processes then please explain ...
2
votes
2answers
47 views

Proof of a limit inequality

Prove that if $a, b \in \mathbb R$ and $a \le x_n \le b$ for all $n \in \mathbb N^+$ and $(x_n)$ converges, then $a \le lim_{n \to \infty} x_n \le b$. Is the following proof valid? For this proof I ...
6
votes
2answers
365 views

Prove an inequality on natural number

Show that if $ a,b\in N$ and $a < b$, then $$\frac{a^a}{(a+1)^{a+1}} > \frac{b^b}{(b+1)^{b+1}}.$$
0
votes
3answers
40 views

How to get values of $a$ from $3 > \log_{5/4}a$?

I cannot come up with the solution to how to get $a$ from $$3 > \log_{5/4}a.$$ If it would be equality, we would get $(5/4)^3 = a$ But what should I do for inequality? Thanks
0
votes
2answers
43 views

Show that $n! < (n/2)^n$ for all large enough $n$ in as elementary a way as possible

Show that $n! < (n/2)^n$ for large enough $n$ in as elementary a way as possible. Using Stirling's formula is not allowed. Of, course, what is true, is that $n! < (n/c)^n$ for any $c < e$ ...
4
votes
7answers
101 views

explaining $|a+b|≤|a|+|b|$ in simple terms

I'm struggling to get to grips with the Triangle Inequalities. The problem is I don't really understand what it means. This is what my lecturer has written in the notes: $$ |a+b|≤|a|+|b|. $$ First of ...