Questions on proving, manipulating and applying inequalities.

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2
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1answer
56 views

An inequality using Sobolev norms

Let $\| \cdot \|_{H^s(\mathbb R)}$ be the usual Sobolev norm in $\mathbb R$ and $r>0$. If we have $$ \|f\|_{L^\infty(\mathbb R)} \le \| f\|_{H^k(\mathbb R)} $$ for all $k>r$, then the inequality ...
0
votes
2answers
88 views

Solve $|x-2| \leq 2|x|$

This is an in-class example we were given in calculus class, I am having some difficulty understanding one of the instructor's steps. The following is my attempt of the question: Since this is an ...
5
votes
2answers
168 views

Inequality $\frac{\sqrt a+\sqrt b+\sqrt c}{2}\ge\frac{1}{\sqrt a}+\frac{1}{\sqrt b}+\frac{1}{\sqrt c}$ with weird condition

I want to prove the following inequality: $$\frac{\sqrt a+\sqrt b+\sqrt c}{2}\ge\frac{1}{\sqrt a}+\frac{1}{\sqrt b}+\frac{1}{\sqrt c}$$ Where $a,b,c$ are positive reals and with the horrible ...
0
votes
1answer
63 views

Inequality with square roots

I need help with another inequality. $\sqrt{2x+4}-2\sqrt{2-x}>\dfrac{12x-8}{\sqrt{9x^2+16}}$. Squaring leads nowhere as you get a polynomial of degree 4 with no rational roots.
0
votes
2answers
149 views

Writing a proof of an inequality between fractions

I have no idea how to do this. Suppose $x,y,z,n$ are positive integers. Given that $\frac{x}{y} < \frac{z}{n}$, prove that $$\frac{x}{y} < \frac{x+z}{y+n} $$
1
vote
1answer
56 views

Find the greatest value of this expression

Let $x$, $y$, $z$ be nonnegative real numbers with $x + y + z = 3$. Find the greatest value of the expression \begin{equation} P = \sqrt{(x+1)(y^2 + 2)(z^3 + 3)} + \sqrt{(y+1)(z^2 + 2)(x^3 + 3)} + ...
3
votes
3answers
111 views

Prove that $x+\frac{1}{2x}-\frac{1}{8x^3}<\sqrt{x^2+1}<x+\frac{1}{2x}$ for all positive real numbers x

This is a math problem from the German Math Olympiad, but in this case I do not know where to start, probably because I do not have enough intuition regarding inequalities. However, I tried to apply ...
0
votes
1answer
58 views

Inequalities and their solutions

a. Since: $9|x + 9| + 6 > 5$ $9|x + 9| > -1 $ $|x + 9| > -\frac19$ which is true for every real $x$ value, so it has infinitely many solutions. $(−\infty, \infty)$ How would I graph ...
1
vote
1answer
45 views

Inequalities with function $e^{x^2+e^{x^2}}$

Let $f(x)=e^{x^2+e^{x^2}}$ for $x\in\mathbb{R}$. How to prove that for any $a,b>0$, $a\neq b$ the following inequalites hold $$(b-a)f(\frac{a+b}{2}) < \int_a^b f(x)\ dx < ...
6
votes
1answer
1k views

Inequality involving square roots

I need help with this inequality: $\sqrt x +\sqrt{x+7} + 2\sqrt{x^2+7x} <35-2x$ It doesn't seem solvable. All roots of the corresponding equation are irrational.
2
votes
0answers
38 views

How to solve ordinary differential inequations with vector variables?

Given $a\in\mathcal{R}_+^d$ and $s\in\mathcal{R}^d$,we wanna a function f(.) which maps s to a vector $f=\begin{bmatrix}f(s_1),\cdots,f(s_d)\end{bmatrix}^T$ and satisify the following inequation. ...
1
vote
2answers
39 views

Turning $2\le x$ into $\sqrt{1+\frac{4}{x^6}}\le \sqrt{5}$?

I am supposed to turn $2\le x$ into $\sqrt{1+\frac{4}{x^6}}\le \sqrt{5}$, and I have no idea on how to approach this. I'll post my steps, even though I don't think they'll be of much help. $$2\le x ...
3
votes
1answer
64 views

Does such a function exist always?

Suppose that $f(x)$ is some smooth function on $[0,1]$ with $f(x) \geq c > 0$. Can we always find a function $g(x)$ smooth satisfying $g'(x) \not= 0$ for all $x \in [0,1]$ and $f'(x)g'(x) + ...
3
votes
1answer
121 views

Is this proof that $\lfloor x \rfloor \geq n \left\lfloor \frac{x}{n} \right\rfloor$ correct?

In this text the fractional part of a real $x$ shall be denoted $\{x\}$, such that $x = \lfloor x \rfloor + \{x\}$. Theorem: $$ \forall x \in \mathbb{R}_{\geq 0} \forall n \in \mathbb{N}_{\geq 1} : ...
3
votes
2answers
64 views

How to show that $\|a+b+c\|^2\leq 3\|a\|^2+3\|b\|^2+3\|c\|^2$

Show that $$\|a+b+c\|^2\leq 3\|a\|^2+3\|b\|^2+3\|c\|^2$$ where $a,b,c$ are in some Hilbert space $(H,\langle\cdot,\cdot \rangle)$? I see that we have $$\|a+b\|^2\leq2\|a\|^2 +2 \|b\|^2$$ due to the ...
2
votes
1answer
107 views

Upperbound for integral of a function times a cosine

Let $f$ be a function such that $0<c_1\leqslant f(x)\leqslant c_2$ for all $x$, and $g$ be a positive function. Assuming that we know the integral $\int_0^\infty g(x)\cos{x}\,\mathrm{d}x$, is it ...
3
votes
3answers
100 views

Prove or Disprove Inequality By Induction

Prove or Disprove $\sum_{i=0}^n(2i)^3 \le (8n)^3 $ If true, prove using induction. If false, give the smallest value of n that is a counter example and the values for the left and right hand sides ...
0
votes
5answers
118 views

Prove or give a counterexample: For all $x > 0$, $x^2 + 1 < (x+1)^2 \le 2(x^2 + 1)$

I am working on the following problem from Lay's Analysis with an Introduction to Proof: Prove or give a counter example: For all $x > 0$ we have $x^2 + 1 < (x+1)^2 \le 2(x^2 + 1)$ Now, ...
2
votes
2answers
75 views

How to prove that $0 < 1$ from the order axioms of $\mathbb R$?

My homework question: From the order axioms for $\mathbb{R}$, show that $0 < 1$. [Hint: From the field axioms, $0 \not=1$. By the trichotomy property, either $0<1$ or $4<0$. Assuming $1 ...
0
votes
2answers
43 views

Inequalities with logarithms and limits

For my analysis homework, I am to show that $\lim_{n \to \infty} \frac{3^n}{n!} = 0$ using the epsilon definition. My approach is to invoke the squeeze theorem and show that the above sequence is less ...
1
vote
3answers
83 views

$(x+y)^c\le x^c+y^c$ for $0<c\le1$ [duplicate]

The statement I'm trying to prove is: $(x+y)^c\le x^c+y^c$ whenever $0\le x,y$ and $0\le c\le1$. This comes up in the proof that $|x|_*^c$ is an absolute value whenever $0<c\le1$ and $|x|_*$ ...
3
votes
1answer
89 views

Proving that if $n>2$ then $n!>n^{n/2}$ using induction. [duplicate]

How to prove that if $n>2$ then $n!>n^{n/2}$ using induction?
0
votes
5answers
104 views

Proving that if $a < b$ and $c < 0$, then $bc<ac$

Let $a$, $b$, and $c$ be real numbers with $a < b$ and $c < 0$. Prove that $bc < ac$. Proof: Let $a$, $b$, and $c$ be real numbers with $a < b$ and $c < 0$. Since $a < b$, ...
1
vote
1answer
76 views

How $\frac{\cos \alpha_1}{\sin \alpha}+\frac{\cos \beta_1}{\sin \beta}+\frac{\cos \gamma_1}{\sin \gamma}\leq\cot \alpha+\cot \beta+\cot \gamma$

Let are any two triangles with angles $\alpha, \beta, \gamma$ and $\alpha_1, \beta_1, \gamma_1$. How prove that $$\frac{\cos \alpha_1}{\sin \alpha} + \frac{\cos \beta_1}{\sin \beta}+ \frac{\cos ...
0
votes
2answers
73 views

Proof of the inequality: $\sum_{k=1}^N\frac{1}{k^N}\gt\frac{N}{\Gamma(N+1)}$.

The following inequality seems to hold: $$\forall N\gt2,\displaystyle\sum_{k=1}^N\dfrac{1}{k^N}\gt\dfrac{N}{\Gamma(N+1)}$$ Is it possible to prove it analytically? Thanks.
4
votes
1answer
147 views

How show this inequality $P_{1}\cdot P_{n}\le (m-1)(n-1)^2$

Question: $n$ students attend a test of $m$ problems where $m, n \ge 2$. The scoring rule for each problem is: If $x$ students answer a problem incorrectly, then a correct answer worth ...
3
votes
5answers
66 views

Solve the following inequality

I have the following inequality $\frac {2x^2}{x+2} < x-2$. I tried to solve it the with the following steps. step 1 $\frac {2x^2}{x+2} < x-2$ step 2 $\frac {2x^2}{x+2} - (x-2) < 0$ step 3 ...
0
votes
1answer
52 views

How can I find the greatest value of this expression?

This is my problem. Find the greatest of the expression $$P = x^3 + 2y^3 +2x-y-1,$$ knowing that $$x^2 + y^2 -x-y=0. $$ I tried. We have $$x^2 + y^2 -x-y=0 \Leftrightarrow \left (x-\dfrac{1}{2}\right ...
1
vote
4answers
77 views

Solve $n^4>3^n$

$n^4>3^n$ I'm trying to solve this inequality problem, but everything I can find online is either how to solve log inequality problems or exponent inequality problems. I think this may be a ...
3
votes
6answers
99 views

Proving a trigonometric inequality $(1-\sin a)x^2 -2x\cos a + 1+ \sin a \ge 0$

$(1-\sin (a))x^2 -2x\cos(a) + 1+ \sin( a) \ge 0$, where $a,x$ are any two real constants. Any suggestions on how to prove this ? I tried playing with it, but nothing useful came out.
1
vote
0answers
29 views

An elementary algebraic inequality [duplicate]

How do I go about proving that $$\frac{x^2}{yz+2}+\frac{y^2}{xz+2}+\frac{z^2}{xy +2}\ge \frac{x+y+z}{3} $$ for $1\le x,y,z\le 2$ ? This is straightforward when $x=y=z$, but I don't see where to go ...
1
vote
1answer
151 views

The supremum norm is submultiplicative

Is the following proof correct: Let $X$ be compact a compact Hausdorff space and $C(X)$ the continuous functions $f: X \to \mathbb{C}$ on X. We can equip $C(X)$ with the (edit: sorry, semi-)norm ...
4
votes
0answers
357 views

My favorite proof of the generalized AM-GM inequality: where it came from?

I have already posted (most of) the present question as a (misplaced) answer to a question about understanding a particular proof of the AM-GM inequality. I sincerely hope I am not breaking the code ...
1
vote
1answer
70 views

$L_2$ error between a non-negative monotone function and its mean?

I have been recently trying to prove a lemma which seems true in every single example I have tried, yet that I didn't manage to prove so far unless making extra (not desirable) assumptions. A ...
3
votes
1answer
77 views

Are some of the Real number axioms redundant?

We are taking a course in Real Analysis with the text: "Elementary Analysis, the theory of Calculus", by Kenneth Ross. I have also been reading a little bit of Beckenbach and Bellman's book, ...
1
vote
4answers
86 views

How many integer solutions to$ x+y+z \le 6$ where $x,y,z$ are non-negative?

I understand the steps towards working this question out if $x + y + z = 6$, but what are the steps for an inequality?
1
vote
2answers
59 views

Proof an inequality by mathematical induction

I have a problem that I have to solve using mathematical induction but I'm stuck from a part. The problem is: Proof that $\large n<2^n$ is true for $\large n \in \mathbb{N}\ $ So, I did that ...
3
votes
1answer
82 views

Inequality with prime numbers

I found exercise in my book for number theory that I can't resolve. How do you show that $$p_n < e^{1+n}$$ where $p_n$ is $n$-th prime number?
5
votes
5answers
248 views

Need to prove inequality $\sum\limits_{k=0}^n \frac{1}{(n+k)} \ge \frac{2}{3}$

Prove that for $n \geq 1$: $$\sum\limits_{k=0}^n \frac{1}{(n+k)} \ge \frac{2}{3}$$ I have tried math induction but that didn't work. Although I'm pretty sure that the solving can be done by induction ...
4
votes
1answer
68 views

If $x_1+x_2+\cdots x_8=1$, then show that $x_1x_2+x_2x_3+x_3x_4+x_4x_1+x_5x_6+x_6x_7+x_7x_8+x_8x_5+x_1x_5+x_2x_6+x_3x_7+x_4x_8\le 1/4$.

If $x_1+x_2+\cdots x_8=1$ and $x_1,\ldots x_8$ are all non-negative numbers, then show that $x_1x_2+x_2x_3+x_3x_4+x_4x_1+x_5x_6+x_6x_7+x_7x_8+x_8x_5+x_1x_5+x_2x_6+x_3x_7+x_4x_8\le 1/4$. I'd like to ...
1
vote
1answer
67 views

The expression $1 + x^2 +(-T_px+y)^2 +z^2$ is bounded below by a constant multiple of $(1+x^2+y^2+z^2)$

Suppose $T_p > 0$. Is there a simply way to show that $1 + x^2 +(-T_px+y)^2 +z^2 \geq C (1+x^2+y^2+z^2)$, for all $(x,y,z) \in \mathbb R^3$, where $C>0$.
4
votes
1answer
346 views

Prove of Nesbitt's inequality in 6 variables

I was just reading "Pham kim hung secrets in inequalities,Volume 1" book and there was an interesting problem on it's Cauchy-Schwarz and Holder section that caught my eye. Prove that for all ...
1
vote
1answer
39 views

Absolute values and inequalities

So I've been trying to solve this one for a few hours and am now out of ideas on how to approach this problem. Here are the inequalities: $$\text{show that if}$$ $$z,w \in \Bbb C$$ $$|z| < ...
1
vote
1answer
105 views

How find this maximum $\sqrt{x^2+x^4}+\sqrt{(x-\frac{3}{2})^2+(x^2-\frac{9}{4})^2}$

Question: Let $0<x<\dfrac{3}{2}$, then find the maximum of $$\sqrt{x^2+x^4}+\sqrt{\left(x-\dfrac{3}{2}\right)^2+\left(x^2-\dfrac{9}{4}\right)^2}$$ I Found when $x=\dfrac{1}{2}$ it's ...
0
votes
0answers
43 views

Necessary and sufficient condition of $f$ satisfying $f(E[X]) - E[f(X)] > 0$

I am trying to looking into function $f$ which satisfies the inequality $f(E[X]) - E[f(X)] > 0$ where $E[X]$ is the expectation of positive random variable $X$. My questions is, what is the ...
1
vote
1answer
121 views

How prove this inequality with $x+y+z=1$

let $x,y,z>0$,and such $$x+y+z=1$$ show that:$$\dfrac{(1+xy+yz+xz)(1+3x^2+3y^2+3z^2)}{9(x+y)(y+z)(x+z)} \ge \left(\dfrac{x\sqrt{1+x}}{\sqrt[4]{3+9x^2}}+\dfrac{y\sqrt{1+y}}{\sqrt[4]{3+9y^2}} ...
1
vote
1answer
50 views

Proving inequality $\frac{1}{2}e^x\left(2+e^x\right) > \left(1+e^x\right)\ln(1+e^x)$

Do you have any ideas on how to show the following inequality? $$\frac{1}{2}e^x\left(2+e^x\right) > \left(1+e^x\right)\ln(1+e^x)$$ It's not about the convexity of any of those functions. ...
0
votes
1answer
154 views

How to prove that $\frac{x^2}{yz+2}+\frac{y^2}{zx+2}+\frac{z^2}{xy+2}\geq \frac{x+y+z}{3}$ holds for any $(x,y,z)\in[1,2]^3$

Prove that for $x,y,z\in [1,2]$ the following inequality holds: $$\frac{x^2}{yz+2}+\frac{y^2}{zx+2}+\frac{z^2}{xy+2}\geq \frac{x+y+z}{3}.$$ I tried to apply the Cauchy-Schwarz inequality or the power ...
2
votes
3answers
65 views

Inequality: $2(p^2+q^2+r^2)+2(pq+qr+rp)\ge pqr$

I need to determine the range of $p,q,r$ such that $2(p^2+q^2+r^2)+2(pq+qr+rp)\ge pqr$. I am not given any other information except that $p,q,r\in \mathbb{R}$. I haven't solved a problem like this ...
1
vote
1answer
34 views

An inequality related to Pythagorean theorem: if $A^{2} + B^{2} = C^{2}$, then $A+B>C$

If $A^{2} + B^{2} = C^{2}$, prove $A+B>C$ for all $A>0$ and $B>0$ Intuitively it seems to apply to all positive real numbers(since the hypotenuse of a right triangle is shorter than the sum ...