Questions on proving and manipulating inequalities.

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0
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1answer
24 views

Inequality $\sqrt{2x^2+3} < x-a $

$\sqrt{2x^2+3} < x-a $ (I) My try: $2x^2+3 < x^2-2ax+a^2\\ x^2+2ax+a^2 < 2a^2 -3 \\ x+a < \sqrt{2a^2-3} \; \; \lor \; \; -x-ay\sqrt{2a^2-3} \\ x < \sqrt{2a^2-3} -a \; \; \lor \; \; x ...
4
votes
2answers
68 views

Third-degree cosine inequality for obtuse triangle

Suppose $\triangle ABC$ is an obtuse triangle with side lengths $a=BC, b=CA, c=AB$. I want to show that $$a^3\cos A+b^3\cos B+c^3\cos C<abc.$$ My idea is to use the cosine rule. I have $\cos ...
1
vote
1answer
187 views

Prove using Jensen's Inequality

Let $\alpha_1, \alpha_2, . . . , \alpha_n$ be the interior angles of a convex (but not necessarily regular) n-gon. Prove, that for all integers $n\geq3$: $$\cos \alpha_1 + \cos \alpha_2 + \cdots + ...
2
votes
1answer
22 views

Find a minium value of a function

Given x,y,z are positive real numbers such that $$ x^2+y^2+6z^2=4z(x+y). $$ Find the minimum value of the following function $$ P=\frac{x^3}{y(x+z)^2}+\frac{y^3}{x(y+z)^2}+\frac{\sqrt{x^2+y^2}}{z} $$
2
votes
2answers
125 views

How find this inequality minimum $\sum_{i=1}^{n}a^2_{i}-2\sum_{i=1}^{n-1}a_{i}a_{i+1}$

Assume that $n$ is give positive integer numbers,and let $a_{1},a_{2},\cdots,a_{n}\ge 0$,such $$a_{1}+a_{2}+\cdots+a_{n}=1$$ Find this minimum value ...
14
votes
2answers
205 views

Prove $\log_5{30}<\log_8{81}$

It's easy to prove this by calculator or computer, and I wonder can we prove that $$\log_5{30}<\log_8{81}\tag 1$$ by pencil and paper ? Thanks in advance ! Edit: $(1)$ can be written as ...
1
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1answer
70 views

Prove that $\dfrac{|x+y|}{1+|x+y|}\leq\dfrac{|x|}{1+|x|}+\dfrac{|y|}{1+|y|}$ for any $x,y$

Use mean value theorem to prove the inequality: $$\dfrac{|x+y|}{1+|x+y|}\leq\dfrac{|x|}{1+|x|}+\dfrac{|y|}{1+|y|}\quad \forall x,y\in\mathbb{R}$$ I have no idea which function I should consider to ...
3
votes
1answer
172 views

Arithmetic and Geometric Mean Inequality

Use the AM - GM inequality (no other method is acceptable), to prove that for all positive integers $n$: $$\left(1 +\dfrac{1}{n}\right)^n \leq \left(1 + \dfrac{1}{n+1}\right)^{n+1}$$ I see that it ...
1
vote
2answers
377 views

How to prove Schwarz inequality for Hermitian forms?

I'm trying to do something like the proof of the Schwarz inequality for inner product. If $h(y,y)\neq 0$, then we can take $\alpha=-h(x,y)/h(y,y)$ and calculate $h(x+\alpha y,x+\alpha y)$ which is ...
9
votes
1answer
191 views

$a;b;c\in \mathbb{R}^+$. Prove : $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{a+b+c}{\sqrt{a^2+b^2+c^2}} \geq 3+\sqrt{3}$

$a;b;c\in \mathbb{R}^+$. Prove : $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{a+b+c}{\sqrt{a^2+b^2+c^2}} \geq 3+\sqrt{3}$ Thanks :) I have proved that : $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq ...
6
votes
6answers
303 views

which one is larger $\sqrt[n]{x+\delta}-\sqrt[n]{x}$ or $\sqrt[n]{x}-\sqrt[n]{x-\delta}$?

Which is larger? $\sqrt[n]{x+\delta}-\sqrt[n]{x}$ or $\sqrt[n]{x}-\sqrt[n]{x-\delta}$? Algebraic justilation does not help.
1
vote
1answer
121 views

How prove this inequality $\sum_{k=1}^{n}\frac{1}{k!}-\frac{3}{2n}<\left(1+\frac{1}{n}\right)^n<\sum_{k=1}^{n}\frac{1}{k!}$

show that $$\sum_{k=1}^{n}\dfrac{1}{k!}-\dfrac{3}{2n}<\left(1+\dfrac{1}{n}\right)^n<\sum_{k=0}^{n}\dfrac{1}{k!}(n\ge 3)$$ Mu try: I konw $$\sum_{k=0}^{\infty}\dfrac{1}{k!}=e$$ and I can prove ...
1
vote
1answer
90 views

How to prove the following inequality of logarithm?

Let $x,y,z\in\mathbb{C}.$ Suppose $$z=\frac{1}{2}(xy\pm\sqrt{x^2y^2-4(x^2+y^2)} ).$$ Show that $$log^+|z|\leq log^+|x|+log^+|y|+log 2.$$ Where $log^+\phi=max\{0,log\phi\}.$ Here we are also ...
1
vote
1answer
36 views

Solving an inequality

Name the property that justifies each statement. If $-3x < 24$, then $x > -8$.
1
vote
0answers
33 views

Modeling 4 people going to same place over 3 different places for at least 5 days

I'm trying to model a linear programming task with the condition 4 people going to the same place among 3 different places for at least 5 days. I have the variables for the time spend each person in ...
1
vote
1answer
48 views

Mistake in simplification of large polynomial inequality?

We are to solve for $p$, and the inequality to simplify is $$10p^3(1-p)^2 + 5p^4(1-p) + p^5 - 3p^2(1-p) - p^3 > 0$$ On the next line of the textbook, the author simplifies this expression to ...
0
votes
1answer
124 views

Proof of the nonexistence of an identity $\phi$ involving convolution

The Banach space $L^1(\mathbb{R}^n)$ is an algebra with a product (convolution) which is both commutative and associative. But this algebra does not have a multiplicative identity. An attempt to show ...
0
votes
0answers
39 views

Does $\left|\left(\int_{\Omega}u^p\right)^{1/p}-\left(\int_{\Omega}v^p\right)^{1/p}\right|\leq C\left(\int_{\Omega}|u-v|^p\right)^{1/p}$ hold?

Does $$\left|\left(\int_{\Omega}u^p\right)^{1/p}-\left(\int_{\Omega}v^p\right)^{1/p}\right|\leq C\left(\int_{\Omega}|u-v|^p\right)^{1/p}$$ hold? If not, what about when $v$ is a constant? Here ...
0
votes
3answers
70 views

Inequation solving. $\frac{(x+2)²}{x+1}<4$

I am trying to solve this inequality, but I always get the wrong score. This is how I did it. $$ \frac{(x+2)^2}{x+1}<4\\ (x+2)^2< 4(x+1)\\ (x+2)^2 < 4x+4\\ x^2+4x+4 < 4x+4\\ x^2+4x+4-4x-4 ...
3
votes
1answer
81 views

Interesting inequality $\|F\|_p\le \frac{\pi}{\sin(\pi/p)}\|f\|_p$ over $L^p$

Consider the function $$F(x)=\int_0^\infty \frac{f(y)}{x+y} \, dy, \quad0<x<\infty$$ Prove that if $1<p<\infty$, $$\|F\|_p\le \frac{\pi}{\sin(\pi/p)}\|f\|_p$$ and show that the constant ...
7
votes
2answers
199 views

Simple Divisor Summation Inequality (with Moebius function)

Show that $$\left| \sum_{k=1}^{n} \frac {\mu(k)}{k} \right| \le 1 $$ where $\mu$ is Moebius function and n is a positive integer. The hard thing here is that the sum is not directly ...
0
votes
2answers
65 views

Find an index n such that inequality is true

I need to find an index $n$ such that: $|e^{x} - S_{n}(x)| \leq \frac{|e^{x}|}{10^{4}}$ (1), where $S_{n} = \sum\limits_{i=0}^{n} \frac{x^{k}}{k!}$ is the n-th Partial Sum of $e^x$. Let $x$ be a ...
1
vote
3answers
82 views

problem on solving equations with three variables

Find all positive real numbers $x,y,z$ which satisfy the following equations simultaneously. $x^3+y^3+z^3=x+y+z$ $x^2+y^2+z^2=xyz$
2
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1answer
73 views

a mathematical problem on inequalities

If $a,b,c,d$ are positive real numbers such that $a+b+c+d=1$,show that $$ \frac{a^3}{b + c} + \frac{b^3}{c + d} + \frac{c^3}{d + a} + \frac{d^3}{a + b} > \frac 1 8$$
1
vote
1answer
106 views

If $ax^2 + 2bx +c$ and $px^2 + 2qx + r$ are both $\geq 0$ prove $apx^2 + bqx + cr \geq 0$ as well

$a, b, c, p, q, r \in \mathbb{R}$ such that for every real $x$: $$\begin{equation} ax^2 + 2bx + c \geq 0 \end{equation}$$ and $$px^2 + 2qx + r \geq 0$$ Prove that $$apx^2 + bqx + cr \geq 0$$ ...
1
vote
3answers
128 views

How prove this inequality $\sum_{i=1}^{n}\frac{\ln{i}}{i^4}<\frac{1}{14}$

I need to show that $$\sum_{i=2}^{n}\dfrac{\ln{i}}{i^4}<\dfrac{1}{14}$$ This problem from high school competition, so usage of integrals and infinite serries is forbidden. My try: let $x\in ...
0
votes
1answer
38 views

Equality question

Hi I'm a bit confused with this? $\frac{1}{x} < 0 \iff x\frac{1}{x} < x\times 0 =0 \iff 1 < 0$ This was another question that I saw which was $\frac{1}{x} < 0$ but when I multiplied by ...
1
vote
2answers
273 views

How prove this inequality $\frac{x}{x+yz}+\frac{y}{y+zx}+\frac{z}{z+xy}\ge \frac{3}{2}$

let $x,y,z>0$,and such $x^n+y^n+z^n=3(n\ge 1),n\in N^*$, show that: $$\dfrac{x}{x+yz}+\dfrac{y}{y+zx}+\dfrac{z}{z+xy}\ge \dfrac{3}{2}$$ My try: if $n=1$ , since $x+y+z=3$,then use Cauchy-Schwarz ...
1
vote
1answer
190 views

If $M$ is positive definite, then $\operatorname{det}{(M)}\leq \prod_i m_{ii}$

In the Wikipedia article on positive definite matrices they claim that if $M$ is positive definite, then the determinant of $M$ is bounded by the product of its diagonal entries. How might we show ...
5
votes
2answers
63 views

How do I solve this simple inequality algebraically?

How do I solve this inequality: $\frac{1}{x} < 0 $ Its deceptively tricky. I've spent some time thinking about it, but came up with nothing. The answer is obviously $x < 0$, but how do I ...
0
votes
1answer
36 views

Average of m weights equals the average of expected value

I'm reading a paper that says $max\{w_1,\sum_jM^j/m\} = max\{w_1,\sum_i w_i/m\}$. There are $n$ weights $w_1 \dots w_n$ and $w_1$ is the max of the weights. $M^j$ is defined as $\sum_i p^j_iw_i$. ...
2
votes
1answer
199 views

Using Bernoulli's Inequality to prove an inequality

Using Bernoulli's Inequality show that $$\left(1+\frac{1}{k+1}\right)\left(1-\frac{1}{(k+1)^2}\right)^k\geq 1$$for all $k\in\mathbb{Z^+}$. My initial thought was to start be noting that ...
1
vote
2answers
176 views

Prove the inequality $ \sqrt{a^2 + b^2} \geq \frac{|a-b|}{\sqrt{2}}$

Please help me to prove the inequality $$ \sqrt{a^2 + b^2} \geq \frac{|a-b|}{\sqrt{2}}. $$
4
votes
3answers
164 views

How prove $\sum\frac{1}{2(x+1)^2+1}\ge\frac{1}{3}$

let $x,y,z>0$ and such $xyz=1$ show that $$\dfrac{1}{2(x+1)^2+1}+\dfrac{1}{2(y+1)^2+1}+\dfrac{1}{2(z+1)^2+1}\ge\dfrac{1}{3}$$ My try: I will find a value of the $k$ such ...
0
votes
1answer
64 views

How do I use Cauchy-Cchwarz inequality?

I wouldn't have posted if I hadn't searched in every site about Cauchy-Schwarz or bcs or Bunyakovsky inequality. But the only thing I can find is the statement and a proof. In many answers in this SE ...
-1
votes
2answers
103 views

How prove this $\sqrt{x^4+7x^3+x^2+7x}+3\sqrt{3x}+x^2-10x\ge 0$

let $x\ge 0$,show that $$\sqrt{x^4+7x^3+x^2+7x}+3\sqrt{3x}+x^2-10x\ge 0$$ My try: let $a=\sqrt{x^4+7x^3+x^2+7x},b=3\sqrt{3x}+x^2-10x$ so $$\Longleftrightarrow a+b\ge 0$$ if $b\ge 0$ then $a+b\ge 0$ ...
1
vote
1answer
213 views

Why is normalization of inequalities possible?

I have seen, in many proofs for inequalities, the author does something called normalization. I believe this is only possible for homogeneous inequalities. I saw this in a proof of Nesbitt's ...
3
votes
3answers
159 views

How prove this inequality $\frac{x^2y}{z}+\frac{y^2z}{x}+\frac{z^2x}{y}\ge x^2+y^2+z^2$

let $x\ge y\ge z\ge 0$,show that $$\dfrac{x^2y}{z}+\dfrac{y^2z}{x}+\dfrac{z^2x}{y}\ge x^2+y^2+z^2$$ my try: $$\Longleftrightarrow x^3y^2+y^3z^2+z^3x^2\ge xyz(x^2+y^2+z^2)$$
3
votes
1answer
107 views

Lower bound on a number theoretic function

Let $n$ be a positive odd integer, let $$n_j = \Bigl\{\frac{n}{2^{j+1}}\Bigr\}\,,$$ where $\{x\}$ denotes the fractional part of $x$, and finally let $k = \lceil \log_2 n\rceil$. Consider the ...
3
votes
1answer
476 views

Evans PDE p.308 Exercise 16 (2nd ed)

Here is the statement of the problem (Evans PDE 2nd Ed., p.308, exercise 16) Show that for $n \geq 3$ there exists a constant $C$ so that $$ \int_{\mathbb {R}^n} \frac{u^2}{\vert x ...
6
votes
3answers
165 views

How would I prove $|x + y| \le |x| + |y|$?

How would I write a detailed structured proof for: for all real numbers $x$ and $y$, $|x + y| \le |x| + |y|$ I'm planning on breaking it up into four cases, where both $x,y < 0$, $x \ge 0$ ...
3
votes
2answers
57 views

$C$ such that $\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^4|a_{ij}|^2$

More generally, can we find $C_n>0$ such that $$\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^2|a_{ij}|^4$$ for all $\{a_k\}_{k\in \mathbb{Z}^n} \in ...
2
votes
1answer
96 views

An inequality involving expectation

Let $f,g$ be two pdfs, and suppose $X$ is a random variable that has pdf $f$. Is it necessarily true that $E[f(X)] \ge E[g(X)]$? Although I doubt this will help, but I got this problem from studying ...
2
votes
0answers
70 views

What is $iav-\log(v)$? Any series expansion or inequality for it?

I am investigating the integral of this question here where \begin{equation} \frac{\exp(i a v)}v=\frac{\exp(i a v)}{\exp(\log(v))}=\exp(iav-log(v)) \end{equation} where I am interested in the ...
3
votes
1answer
67 views

Find the complex $z$ such $\max{(|1+z|,|1+z^2|)}$ is minimum

find the complex $z$,such $$\max{(|1+z|,|1+z^2|)}$$ is minimum My try: let $z=a+bi$,then $$|1+z|=\sqrt{(a+1)^2+b^2}$$ $$|1+z^2|=|1+a^2+2abi-b^2|=\sqrt{(1+a^2-b^2)^2+4a^2b^2}$$ Then I can't,Thank ...
2
votes
1answer
97 views

How prove $\frac{2}{3}<\frac{3x^6+15x^2+2}{2x^6+15x^4+3}\le\frac{3}{2}$

Let $x\in(0,1]$, show that $$\dfrac{2}{3}<\dfrac{3x^6+15x^2+2}{2x^6+15x^4+3}\le\dfrac{3}{2}$$ My try: since $$\begin{align}\dfrac{3x^6+15x^2+2}{2x^6+15x^4+3} ...
5
votes
1answer
112 views

How prove this inequality $a^2+b^2+c^2+8(ab+bc+ac)+3-10(a+b+c)\ge 0$

let $a,b,c\ge 0$,and such $abc=1$,show that $$a^2+b^2+c^2+8(ab+bc+ac)+3-10(a+b+c)\ge 0$$ My solution: Without loss of generality,assume that $a=\max{(a,b,c)}$, since $abc=1$,we have $a\ge ...
0
votes
2answers
43 views

Matrix norm inequality involving max and stacked matrices

In a paper I found the following inequality for matrices $A$ and $B$: $\max\left\{||A||, ||B||\right\} \le \left\| \begin{align}A \\ B\end{align} \right\|_2 $ I suspect that this is a well-known ...
1
vote
1answer
135 views

Using Sobolev-Nirenberg-Gagliardo

I am currently studying a proof of a General Sobolev Inequality. I have the following question: Consider the Sobolev Space $W^{k,p}(U)$. With the added assumption that $k > \frac{n}{p}$. Let $l = ...
4
votes
1answer
115 views

Generalization of an inequality $0\lt e^6-{\pi}^4-{\pi}^5\lt 0.00002$

Question : Is the following true? For any $n\in\mathbb N$, there exists a triple $(k,l,m)\ (k,l,m\in\mathbb N)$ such that $$0\lt e^k-{\pi}^l-{\pi}^m\lt{10}^{-n}.$$ Motivation : A friend ...