Questions on proving, manipulating and applying inequalities.

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1answer
34 views

Simple Inequality for Proving Equivalent Besov Seminorms

For $f\in L^{p}(\mathbb{R}^{n})$, $1\leq p<\infty$, and $h\in\mathbb{R}^{n}$, define the quantity $$I_{p}(h):=\left(\int_{\mathbb{R}^{n}}\left|f(x+h)-f(x)\right|^{p}dx\right)^{1/p}$$ and define ...
22
votes
13answers
755 views

Which is greater, $98^{99} $ or $ 99^{98}$? [duplicate]

Which is greater, $98^{99} $ or $ 99^{98}$? What is the easiest method to do this which can be explained to someone in junior school i.e. without using log tables. I don't think there is an ...
1
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2answers
73 views

How to find the minimum of the $a^2+b^2+c^2+a+b+c+ab+bc$

(1): Let $a,b\in R$, Find this minimum of the value $a^2+b^2+ab+a+b$ I have prove $a^2+b^2+ab+a+b\ge-\dfrac{1}{3}$ (2): Let $a,b,c\in R$, find the minimum of the value $$a^2+b^2+c^2+a+b+c+ab+bc$$ I ...
1
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0answers
31 views

Proof of Gruss inequality

I've been reading articles that use the Gruss inequality for some time now, but I can't seem to find a proof of it anywhere. The only source I could find that actually has the proof is the original ...
1
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1answer
43 views

Prove that if $|a_1\sin x+a_2\sin2x+a_3\sin3x+\cdots+a_n\sin nx|\leq|\sin x|$ for $x\in R,$then $|a_1+2a_2+3a_3+…+na_n|\leq1$

Prove that if $|a_1\sin x+a_2\sin2x+a_3\sin3x+\cdots+a_n\sin nx|\leq|\sin x|$ for $x\in R,$then $|a_1+2a_2+3a_3+\cdots+na_n|\leq1$ When we try to differentiate it on both sides wrt $x$,then modulus ...
1
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1answer
47 views

Proving inequality involving convex combination

What are the steps in proving this following inequality: for $p,q \in [0,1]$: $$ p \log \frac{p}{q}+(1-p) \log \frac{1-p}{1-q} \ge 2(p-q)^2 $$ I'm just looking for an outline of the approach to tackle ...
0
votes
1answer
25 views

Solve this inequality equation with 3 sides?

Could you guys solve them and show me the solutions and all working so that I learn? This is another practice question and I have not been taught (I haven't been taught how to do it with 3 sides). I ...
-2
votes
2answers
45 views

Solve this inequality equation with fraction? [closed]

Could you guys solve this and show me the solutuions and all working so that I learn? Thanks in advance. This is one of the practice questions I was given but I was never thought how to do anything ...
0
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3answers
41 views

For what values of $x$ does $x^2-2x+2<0$ hold?

Can someone help me solve this inequality? $$x^2-2x+2>0$$ I can't factor this, but I know that $x^2-2x=x(x-2)>0$ when $x$ and $(x-2)$ are the same sign and not zero, so in other words when ...
1
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1answer
34 views

Prove that $\sum\sqrt{\frac{a+b}{c}}\ge2\sum\sqrt{\frac{c}{a+b}}$

Let $a,b,c$ be positive numbers. Then we need to prove $\sqrt{\frac{a+b}{c}}+\sqrt{\frac{b+c}{a}}+\sqrt{\frac{c+a}{b}}\ge2\left(\sqrt{\frac{c}{a+b}}+\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}\right).$ ...
-1
votes
1answer
29 views

Two Column Inequality Proof

I'm an absolute beginner in proving so I need help please. Let $a$ and $b$ be both positive numbers. If $a \gt b$, then $a^2\gt b^2$.
2
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2answers
67 views

Proving: $2(n-1)<(n+1)\sqrt{n}$ [closed]

How can I prove the following inequality ($n\in N$)? Any help would be greatly appreciated. $$2(n-1)<(n+1)\sqrt{n}$$
1
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3answers
41 views

Proving: $\frac{2x}{2+x}\le \ln (1+x)\le \frac{x}{2}\frac{2+x}{1+x}, \quad x>0.$

$$\begin{equation}\frac{2x}{2+x}\le \ln (1+x)\le \frac{x}{2}\frac{2+x}{1+x}, \quad x>0\end{equation}$$ I found this inequality in this paper: http://ajmaa.org/RGMIA/papers/v7n2/pade.pdf (Equation ...
1
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2answers
47 views

Is there some $M \geq 0$ such that $|T(x)| \leq M|x|$ for all $x$?

Let $T: \mathbb{R}^{m} \to \mathbb{R}^{n}$ be linear. Is there some $M \geq 0$ such that $|T(x)| \leq M|x|$ for all $x \in \mathbb{R}^{m}$? I am not sure, as I just found a bound not that sharp. Let ...
1
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1answer
41 views

Inequality for a power function

Can anyone tell me under what conditions the following is correct?? Given two real variables $x$ and $y$, \begin{equation} (y-x)^b \le y^b - x^b \end{equation} where y > x and $0\le b \le 1$. ...
-1
votes
0answers
31 views

Triangle inequality $ax ≥ br + cq$

I got stuck on this problem : Given a triangle (△ABC) of sides $a$, $b$ and $c$, let $O$ be a point inside △ABC. Let $D$, $E$ and $F$ be points on sides a, b and respectively c such that $OE ⊥ ...
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2answers
80 views

How to see that $e^{x(1+x/3)} \le (1+x)^{(1+x)}$ for very small x>0?

I am reading a research paper and one of the calculation states that $e^{x(1+x/3)} \le (1+x)^{(1+x)}$ for very small x>0. Is this true? How to prove it? Thank you very much.
1
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0answers
24 views

finding the shortest distance of a hermitian matrix to a set of hermitian matricies with specific eigenvalues 2-norm

The title is more general, and all that I require is to show an inequality that I already have verified using random matrices in matlab. Let $\lambda_1 \leq ... \leq \lambda$ and $\mu_1 \leq ... \leq ...
3
votes
3answers
93 views

Upper and lower bounds for $S(n) = \sum_{i=1}^{2^{n}-1} \frac{1}{i} = 1+\frac{1}{2}+ \cdots +\frac{1}{2^n-1}.$ [duplicate]

For a positive integer $n$ let $S(n) = \sum_{i=1}^{2^{n}-1} \frac 1i = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+ \cdots +\frac{1}{2^n-1}.$ Then which of the following are true. (a) ...
-6
votes
1answer
45 views

Please help me solve $4-(r-2)>3-5$ [closed]

$4-(r-2)>3-5$ I don't really get how to solve this problem. So please explain to me step by step.
2
votes
3answers
40 views

Triangle area inequalities

I've got stuck on this problem : Proof that for every triangle of sides $a$, $b$ and $c$ and area $S$, the following inequalities are true : $4S \le a^2 + b^2$ $4S \le b^2 + c^2$ ...
1
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2answers
57 views

Powers of complex numbers property

I would like to prove the following statement. Let $\lambda_1,\dots,\lambda_s \in \mathbb{C}$ be such that $|\lambda_1| = \dots = |\lambda_s|=1$. Then $\forall \varepsilon \gt 0$ there exist ...
1
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1answer
31 views

Lower and upper bound of the Stirling's approximation

Perhaps everybody has heard of the Stirling's approximation, namely: $$ \Gamma(z)\approx\sqrt{\frac{2\pi}{z}}\left(\frac{z}{e}\right)^z $$ Thus (the very basic example): $$ ...
3
votes
2answers
58 views

find smallest $x>0$ such that $\frac{A}{cx}e^{-cx^2}\le \varepsilon$

I was estimating some error and I got $$\varepsilon(x)\le\frac{A}{cx}e^{-cx^2}$$ $A,c$ are known and positive, $x$ is also positive. The bigger the $x$ smaller the error. But I need to find the ...
0
votes
1answer
39 views

Finding a positive lower bound of hte sequence $\frac{\sqrt[n]{n!}}n$

I am given a sequence {(n'th root of n!)/n}. Can I show that the sequence is bounded below by a real no. which is greater than 0, by not calculating the limit of it....???thank you
2
votes
4answers
45 views

Inequality between altitude and sides in triangle

Let $a,b,c$ be the side lengths and $h_a,h_b,h_c$ the altitudes each connect a vertex to the opposite side and are perpendicular to that side. Then we need to prove ...
0
votes
1answer
38 views

Are these inequalities useless for getting better estimates? If not what is needed?

Are these inequalities useless for getting better estimates? If not what is needed? My motivation for asking this question is to get a glimpse to the mind of masters that can tell if a line of ...
2
votes
2answers
43 views

Inequality with decay of modified Bessel functions of the second kind

I think that the following inequality holds for all $x > 0$ and all $\nu$ above some constant that is somewhere around 0.2: $$ K_\nu(2 x) \le \frac{2^{2 - 2 \nu}}{\Gamma(\nu)} x^\nu K_\nu^2(x) $$ ...
1
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3answers
25 views

average of an inequality

what is the result of $<1 + <1$? ; $<1 + <1 = <2$? or <1 + <1 = <1? Now what is that number divided by 2? Either we have: <2 / 2 = <1 or <1 / 2 = <0.5
1
vote
1answer
68 views

Inequality prove $ e^x \ge 1+x $ [duplicate]

I want to prove that $ e^x \ge 1+x $ for all $ x \in R $ , using Mean Value Theorem it can be proved for $ x \gt 0 $, and equality holds for $ x = 0$, however I can't solve it for $ x \lt 0 $
1
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2answers
59 views

Prove $|a+b+c| \leq |a| + |b| + |c|$ for all $a,b,c \in \mathbb{R}$.

Here is the proof that I am currently working on. Prove $|a+b+c| \leq |a| + |b| + |c|$ for all $a,b,c \in \mathbb{R}$. Hint: Apply the triangle inequality twice. Do not consider eight cases. I ...
-1
votes
3answers
39 views

Prove $|e^{ix}-1-ix-\cdots-\frac{(ix)^k}{k!}| \leq \frac{|x|^{k+1}}{(k+1)!}$

I would like to prove $|e^{ix}-1-ix-\cdots-\frac{(ix)^k}{k!}| \leq \frac{|x|^{k+1}}{(k+1)!}$, which trick can be used?
19
votes
1answer
244 views

Integers $n$ for which the digit sum of $n$ exceeds the digit sum of $n^5$

This question is strongly inspired by The smallest integer whose digit sum is larger than that of its cube? by Bernardo Recamán Santos. The number $n=124499$ has digit sum $1+2+4+4+9+9=29$ while its ...
0
votes
1answer
46 views

Question on a proof of Hilbert's Inequality using Cauchy Schwarz

A simpler version of Hilbert's Inequality states that: For any real numbers $a_1,a_2\cdots,a_n$ the following inequality holds: $\sum_{i=1}^n\sum_{j=1}^n\frac{a_ia_j}{i+j}\leq\pi\sum_{i=1}^na_i^2$. I ...
2
votes
1answer
83 views

Proof with Cauchy-Schwarz

Prove that if $x_i > 0$ for all $i$ then \begin{align*} &(x_1^{19} + x_2^{19} + \cdots + x_n^{19})(x_1^{93} + x_2^{93} + \cdots + x_n^{93}) \\ &\geq (x_1^{20} + x_2^{20} + \cdots + ...
1
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1answer
81 views

Prove that $(a+b)(b+c)(c+a) \ge8$

Given that $a,b,c \epsilon \mathbf R^+$ Also $abc(a+b+c)=3$ Prove that $(a+b)(b+c)(c+a) \ge8$ My attempt- BY AM-GM inequality we have- $\frac{a+b}{2}\ge\sqrt ab$ ... (1) similarly ...
4
votes
1answer
77 views

How to show that $|\exp(z)-1|\le2|z|$ for $|z|\le 1$

How to show that $|\exp(z)-1|\le2|z|$ for $|z|\le 1$ ...
5
votes
1answer
104 views

The smallest integer whose digit sum is larger than that of its cube?

79 is an example of a number whose digital sum is greater than that of its square (6241). Which is the least number, if any, whose digital sum is greater than that of its cube?
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votes
1answer
70 views

Inequalities help! [closed]

Let $a,b,c > 0$ and $a + b + c = 1$. Prove: $$\sqrt{\frac{ab}{c + ab}} + \sqrt{\frac{bc}{a + bc}} + \sqrt{\frac{ac}{b + ac}}\leq \frac32$$
1
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1answer
38 views

Inequality in proof of SLLN

This comes from theorem 5.1.2 of KL Chung's A Course in Probability Theory. Suppose ${X_n}$ are uncorrelated and their second moments have a common bound. Then For each $n \ge 1 $, $D_n:= ...
0
votes
3answers
67 views

prove that $an(n+1)+2n\ge4 \sqrt a(\sqrt1+\sqrt2+\cdots+\sqrt n)$

It is given that $a \in \mathbf R^+$ and $n \in \mathbf Z$ prove that $$an(n+1)+2n\ge4 \sqrt a(\sqrt1+\sqrt2+\cdots+\sqrt n)$$ No idea about how to solve this one.please help. A full answer with a ...
14
votes
3answers
173 views

Show that $\frac{x}{3!}-\frac{x^3}{5!}+\frac{x^5}{7!}-\cdots\leq \frac{1}{\pi}$.

My problem is to show that $$\frac{x}{3!}-\frac{x^3}{5!}+\frac{x^5}{7!}-\cdots\leq \frac{1}{\pi}$$ for all $x\in\Bbb R$. I was thinking of first finding the max and then show that its less ...
0
votes
0answers
41 views

Inequality Steps

I am stuck in the following inequality. $$2\sum_{q=1}^N(q-1)\frac{1}{q^2}<2 \sum_{q=1}^\infty\frac{1}{q^2}$$ Can anyone help to get the inequality? N is a fixed natural number.
1
vote
2answers
37 views

Concentration property of entropy

Let $X$ be a random variable taking its values in $A = \{a_1,\ldots,a_n\}$ such that $Pr[X = a_i] = p_i$ for all $1 \leq i \leq n.$ The entropy of $X$ is defined as $$H(X) = -\sum_{i=1}^n p_i ...
0
votes
5answers
109 views

How to resolve $n>(1+\frac{1}{n})^n$?

I'm trying to prove that $\forall n\geq 3, n^{n+1}>(n+1)^n$. I came that this is true for $n>(1+\frac{1}{n})^n$. WolphramAlpha gives $n>2.293166...$ but I failed to compute it analytically.
9
votes
1answer
60 views

inequality $\max\{a_1,a_2,\cdots,a_n \}\leq {n^2}^{n-1}.$with Egyptian fraction

Let $a_1,a_2,\cdots,a_n $ be positive integer such that$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=1.$ Prove that$$\max\{a_1,a_2,\cdots,a_n \}\leq {n^2}^{n-1}.$$ This Problem from:1
2
votes
1answer
69 views

The proof of theorem 3.19 from baby Rudin

If $s_n\leqslant t_n$ for $n\geqslant N$, where $N$ is fixed, then $$\liminf_{n\to \infty} s_n\leqslant \liminf_{n\to \infty} t_n$$ $$\limsup_{n\to \infty} s_n\leqslant \limsup_{n\to \infty} t_n$$ ...
0
votes
1answer
30 views

Smoothing Inequalities

Can anyone explain to me how the "smoothing argument for inequalities" works? I know that basically it can be used to prove an inequality $f(a_1,a_2,\cdots,a_n)\geq C$ subject to the constraint ...
1
vote
3answers
49 views

How can I prove that if $\lim_{n \to \infty}s_n=s$ then $|s_n-s|< \epsilon$ is equivalent to $s-\epsilon <s_n <s+ \epsilon$

My professor casually mentioned this in class and told us to prove it if we weren't convinced, however, I cannot find how to prove it.
6
votes
2answers
309 views

Is this proof of Cauchy Schwarz inequality circular or valid?

I'm a college freshman learning linear algebra on my own, and I'm in the section on inner products. I noticed a proof of the Cauchy Schwarz inequality for vectors in my book, and it seems to contain ...