Questions on proving and manipulating inequalities.

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7
votes
2answers
322 views

How prove this inequality $\sum_{cyc}\frac{a^2}{b(a^2-ab+b^2)}\ge\frac{9}{a+b+c}$

let $a,b,c>0$, show that $$\dfrac{a^2}{b(a^2-ab+b^2)}+\dfrac{b^2}{c(b^2-bc+c^2)}+\dfrac{c^2}{a(c^2-ca+a^2)}\ge\dfrac{9}{a+b+c}$$ My try: since this inequality is homogeneous ,without loss of ...
12
votes
8answers
770 views

prove $\frac{1}{ n+1}+\frac{1}{ n+2}+\cdots+\frac{1}{2n}<\frac{25}{36}$ by mathematical induction

How to prove $$\frac{1}{ n+1}+\frac{1}{ n+2}+\cdots+\frac{1}{2n}<\frac{25}{36}$$ by Mathematical induction,n$\ge $1
10
votes
2answers
341 views

determinant inequality $ \det(A^2+B^2+(A-B)^2)\ge 3\det(AB-BA) $

A and B are two $2\times2$ reals matrices. then $$ \det \Big(A^2+B^2+(A-B)^2\Big)\ge 3\det(AB-BA) $$ well, it is seems interesting, but it is really hard to get started Thank you very much!
5
votes
1answer
151 views

How prove this number theory inequality $\left(\dfrac{1}{N}\sum_{n=1}^{N}(\omega{(n)})^k\right)^{\frac{1}{k}}\le k+\sum_{q\le N}\frac{1}{q}$

show that: for any positive numbers $k$ and $N$, have $$\left(\dfrac{1}{N}\sum_{n=1}^{N}(\omega{(n)})^k\right)^{\frac{1}{k}}\le k+\sum_{q\le N}\dfrac{1}{q}$$ where $\displaystyle\sum_{q\le N}$ is ...
15
votes
3answers
705 views

Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$

Let $a,b,c$ are non-negative numbers, such that $a+b+c = 3$. Prove that $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ Here's my idea: $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ ...
15
votes
3answers
1k views

Inequality. $\frac{1}{16}(a+b+c+d)^3 \geq abc+bcd+cda+dab$

I want to prove the following inequality : $$\frac{1}{16}(a+b+c+d)^3 \geq abc+bcd+cda+dab, $$ $a,b,c,d \in \mathbb{R}_{+} .$ In my book, at the answers chapter the author uses AM $\geq$ GM, ...
11
votes
2answers
1k views

Relationship between diameter and radius of a point set

Consider a set of $n$ points in $\mathbb{R}^k$. The diameter of this set is the maximum distance between two of its points; its radius is the radius of the smallest (closed) k-ball that contains all ...
4
votes
6answers
143 views

Summation inductional proof: $\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<2$

Having the following inequality $$\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<2$$ To prove it for all natural numbers is it enough to show that: ...
4
votes
1answer
294 views

Prove that: $\cos(x) -1 < -\frac{x^2}{2} + \frac{x^4}{24}$

Prove that: $$\cos(x) -1 < -\frac{x^2}{2} + \frac{x^4}{24}$$ for $x \ne 0$ I need to prove this using Cauchy's mean value theorem. What I did: $f(x) = \cos(x) -1$ $$g(x) = ...
3
votes
1answer
126 views

Show that $\frac1{\sqrt{(n+\frac12) \pi}} \le\frac{1\cdot 3\cdot 5 … (2n-1)}{2\cdot 4\cdot 6 … (2n)} \le \frac1{\sqrt{n \pi}} $

Show that, if $n$ is a positive integer, $$\frac1{\sqrt{(n+\frac12) \pi}} \le\frac{1\cdot 3\cdot 5 ... (2n-1)}{2\cdot 4\cdot 6 ... (2n)} \le \frac1{\sqrt{n \pi}} . $$ This result is in a current ...
3
votes
3answers
166 views

Inequality: $ab^2+bc^2+ca^2 \le 4$, when $a+b+c=3$.

Let $a,b,c $ are non-negative real numbers, and $a+b+c=3$. How to prove inequality $$ ab^2+bc^2+ca^2\le 4.\tag{*} $$ In other words, if $a,b,c$ are non-negative real numbers, then how to prove ...
13
votes
2answers
350 views

A conjecture concerning primes and algebra

A monoid morphism $\psi:\mathbb Z_+\!\!\rightarrow\mathbb Z_+$ is defined by an arbitrary function $f:\mathbb Z_+\!\!\rightarrow\mathbb Z_+$ and defines a group homomorphism $\varphi:\mathbb ...
12
votes
6answers
431 views

Asymptotic behaviour of a multiple integral on the unit hypercube

A few days ago I found an interesting limit on the "problems blackboard" of my University: $$\lim_{n\to +\infty}\int_{(0,1)^n}\frac{\sum_{j=1}^n x_j^2}{\sum_{j=1}^n x_j}d\mu = 1.$$ The correct claim, ...
12
votes
2answers
319 views

Prove that $\dfrac{a}{b^2+5}+ \dfrac{b}{c^2+5} + \dfrac{c}{a^2+5} \le \dfrac 12$

Let $a,b,c>0$ and $a^3+b^3+c^3=3$. Prove that $$\dfrac{a}{b^2+5}+ \dfrac{b}{c^2+5} + \dfrac{c}{a^2+5} \le \dfrac 12$$ I have an ugly solution for this solution.
10
votes
3answers
364 views

Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality $\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$?

Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality $$\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$$?
6
votes
1answer
157 views

How prove $\frac{(k+1)^{k+1}}{k^k}\sum_{t=k+1}^{n}\frac{1}{t^2}<e$

Let $k,n\in \mathbb{N},n\ge k$, prove that $$\dfrac{(k+1)^{k+1}}{k^k}\sum_{t=k+1}^{n}\dfrac{1}{t^2}<e.$$ I got the impression that this inequality is very sharp. My idea: ...
5
votes
2answers
62 views

Prove the inequality: $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$

Prove the inequality: $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$ Where $a,b,c$ are sides of a triangle. It is clear that $c+a-b$ is positive but how to use it?
4
votes
2answers
157 views

Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$

I'm trying to solve these induction exercises proposed by the department of mathematics of Oxford University. I don't know how to give a valid proof for the third one which says the following: ...
2
votes
1answer
148 views

Rudin: Problem Chp3.11 and need advice.

I am working on the following problems and I have a couple of questions. Suppose $a_n>0, s_n = \sum_{i = 1}^{n}$ and $\Sigma a_n$ diverges. RTP (a) $\Sigma \frac{a_n}{1+a_n}$ ...
1
vote
1answer
71 views

Find minimum of $P=\frac{\sqrt{3(2x^2+2x+1)}}{3}+\frac{1}{\sqrt{2x^2+(3-\sqrt{3})x +3}}+\frac{1}{\sqrt{2x^2+(3+\sqrt{3})x +3}}$

For $x\in\mathbb{R}$ find minimum of $P$. $P=\dfrac{\sqrt{3(2x^2+2x+1)}}{3}+\dfrac{1}{\sqrt{2x^2+(3-\sqrt{3})x +3}}+\dfrac{1}{\sqrt{2x^2+(3+\sqrt{3})x +3}}$ Source : Viet Nam national test for high ...
18
votes
3answers
538 views

An Inequality Involving Bell Numbers: $B_n^2 \leq B_{n-1}B_{n+1}$

The following inequality came up while trying to resolve a conjecture about a certain class of partitions (the context is not particularly enlightening): $$ B_n^2 \leq B_{n-1}B_{n+1} $$ for $n \geq ...
10
votes
3answers
428 views

proving :$\frac{ab}{a^2+3b^2}+\frac{cb}{b^2+3c^2}+\frac{ac}{c^2+3a^2}\le\frac{3}{4}$.

Let $a,b,c>0$ how to prove that : $$\frac{ab}{a^2+3b^2}+\frac{cb}{b^2+3c^2}+\frac{ac}{c^2+3a^2}\le\frac{3}{4}$$ I find that $$\ ...
9
votes
2answers
619 views

Proof that $t-1-\log t \geq 0$ for $t > 0$

Using basic calculus, I can prove that $f(t)=t-1-\log t \geq 0$ for $t > 0$ by setting the first derivative to zero \begin{align} \frac{df}{dt} = 1 - 1/t = 0 \end{align} And so I have a critical ...
8
votes
1answer
260 views

If $f$ is a positive, monotone decreasing function, prove that $\int_0^1xf(x)^2dx \int_0^1f(x)dx\le \int_0^1f(x)^2dx \int_0^1xf(x)dx$

If $f$ is a positive, monotone decreasing function, prove that $\int_0^1xf(x)^2dx \int_0^1f(x)dx\le \int_0^1f(x)^2dx \int_0^1xf(x)dx$
8
votes
4answers
278 views

How many sequence of integers ($j_1 , j_2 , . . . , j_k$) are there such that $0 ≤ j_1 ≤ j_2 ≤ . . . ≤ j_k ≤ n$?

I need to solve the problem, How many sequence of integers ($j_1 , j_2 , . . . , j_k$) are there such that $0 ≤ j_1 ≤ j_2 ≤ . . . ≤ j_k ≤ n$? I've been given a hint, (Hint: Reduce the ...
7
votes
5answers
706 views

Arithmetic mean is less than geometric mean (Spivak Calculus 3rd Chapter 2 Problem 22)

If $a_1, \ldots, a_n \ge 0$, the arithmetic mean $$A_n={a_1 + \cdots + a_n \over n}$$ and the geometric mean $$G_n = \sqrt[n]{a_1 \cdots a_n}$$ satisfy $G_n \le A_n$. As a first step to prove this ...
7
votes
1answer
171 views

Prove inequality: $\sum \frac{a^4}{a^3+b^3} \ge \frac{a+b+c}{2}$

Prove inequality with $a,b,c >0$ $$\frac{a^4}{a^3+b^3} + \frac{b^4}{b^3+c^3}+\frac{c^4}{c^3+a^3} \ge \frac{a+b+c}{2}$$ I tried the inequality: $\sum \frac {a^4+b^4}{a^3+b^3} \ge \sum ...
6
votes
3answers
94 views

Prove $(x+r_1) \cdots (x+r_n) \geq (x+(r_1 \cdots r_n)^{1/n})^{n}$.

I can show that for $x > 0$ and $r_{i} > 0$ we have $$ \left(\, x + r_{1}\,\right)\ldots\left(\, x + r_{n}\,\right)\ \geq\ \left[\, x + \left(\, r_{1}\ldots r_{n}\,\right)^{1/n}\,\right]^{n}.$$ ...
6
votes
1answer
115 views

Solving an inequality : $n \geq 3$ , $n^{n} \lt (n!)^{2}$.

I proved this inequality in the following way: Lemma: $r \in \Bbb N, r \geq 3$. We have $r^r \gt (r+1)^{r-1}$. Proof: We apply the AM-GM inequality to the $r$ positive integers where there are ...
5
votes
1answer
94 views

Prove that $\|a\|+\|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane.

Prove that $\|a\| + \|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane. Gentle hints only, please! I know that attempting to decompose R.H.S. into $$\alpha a + \beta b + ...
5
votes
3answers
793 views

Derivatives of the Riemann zeta function at $s=0$

It's a curious fact that for $n>0$, $\zeta^{(n)}(0)\approx -n!$. Apostol gave a table for $\frac{\zeta^{(n)}(0)}{n!}$, among other results on $\zeta^{(n)}(0)$ . the sequence : $$\delta_{n}=\left | ...
5
votes
2answers
201 views

Inequality involving sides of a triangle

How can one show that for triangles of sides $a,b,c$ that $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} < 2$$ My proof is long winded, which is why I am posting the problem here. Step 1: let ...
4
votes
4answers
219 views

Working with proofs help?

I'm trying to study for my midterm and doing some random practise questions to work with proofs. However I'm stuck on, as the only way I know how to prove it is through plugging in numbers, however as ...
4
votes
1answer
1k views

Liapunov's Inequality for $L_p$ spaces

Let $1 \leq p,q < \infty$ and $0 \leq \lambda \leq 1$. If $r = \lambda p + (1 - \lambda)q$ and $f \in L_p \cap L_q $, then $$||f||_r^r \leq ||f||_p^{\lambda p} ||f||_q^{(1 - \lambda)q} \tag{*}$$ ...
3
votes
2answers
60 views

Prove QM-AM inequality

$$\dfrac{x_1^2+ x_2^2 + \cdots + x_n^2}n \geq \left(\dfrac{x_1+x_2+\cdots+x_n}n\right)^2$$ I don't think AM, GM can be used here. And simple expansion doesn't help too. What should I do?
3
votes
2answers
178 views

Find maximum of $P$

Let $$P = \frac{{{x^2}}}{{{x^2} + yz + x + 1}} + \frac{{y + z}}{{x + y + z + 1}} - \frac{{1 + yz}}{9}.$$ Find maximum of $P$ where $x, y,z$ are nonnegative real numbers such that ${x^2} + {y^2} + ...
3
votes
1answer
147 views

little inequality conjecture

proof or disproof for $n\geq2$ even and $x>0$ $$\sum\limits_{i=0}^{n}x^i\geq \frac{(1+2\sum\limits_{i=1}^{ \frac{n}{2} }x^i)^2}{ \frac{n}{2} (x+1)+1}$$ I came up with this little inequality while ...
3
votes
2answers
161 views

Prove that $\frac{\int_0^1xf^2(x) \mathrm{d}x}{\int_0^1 xf(x) \mathrm{d}x}\le\frac{\int_0^1 f^2(x) \mathrm{d}x}{\int_0^1 f(x) \mathrm{d}x}$ [duplicate]

Let $f:[0,1]\rightarrow\mathbb{R_+}$ be a monotone decreasing function. We want to prove that $$\frac{\int_0^1x(f(x))^2 \,\mathrm{d}x}{\int_0^1 xf(x) \,\mathrm{d}x}\le\frac{\int_0^1 (f(x))^2 ...
2
votes
1answer
120 views

If $m,n\in \mathbb N$ and $n>m$, prove that $\text{lcm}(m,n)+\text{lcm}(m+1,n+1)>\frac{2mn}{\sqrt{n-m}}$.

Where $\text{lcm}$ is the least common multiple. I've changed it to: $$\frac{mn}{\gcd(m,n)}+\frac{(m+1)(n+1)}{\gcd(m+1,n+1)}>\frac{2mn}{\sqrt{n-m}}$$ Can't see how to continue. Is there a way to ...
2
votes
2answers
302 views

Mean Value Theorem: Real Analysis

I need to show that $\dfrac{2}{\pi}<\dfrac{\sin(x)}{x}<1$ for $0<x<\dfrac{\pi}{2}$. I know I need to use the mean value theorem, would I just say that since $f$ is continuous in the ...
2
votes
2answers
556 views

If $a^2 + b^2 + c^2 = 2$, find the maximum of $\prod(a^5+b^5)$

Given that $a, b, c > 0$ and $a^2 + b^2 + c^2 = 2$, what is the maximum value of $(a^5 + b^5)(a^5 + c^5)(b^5 + c^5)$? Normally when I encounter a problem like this, I seem to be able to push ...
1
vote
2answers
108 views

Assume that $ 1a_1+2a_2+\cdots+na_n=1$, where the $a_j$ are real numbers.

Assume that $$ 1a_1+2a_2+\cdots+na_n=1, $$ where the $a_j$ are real numbers. As a function of $n$, what is the minimum value of $$1a_1^2+2a_2^2+\cdots+na_n^2?$$
1
vote
2answers
110 views

Comparison theorem for ODE

Here is something I'm trying to prove: Conjecture: Suppose $f'(x) \leq \phi(f(x), x)$ and $f(a)=\alpha$. Suppose $g'(x)=\phi(g(x),x)$ and $g(a)\geq \alpha$. Then $f(x)\leq g(x)\,\,\forall x$. ...
1
vote
1answer
63 views

Does $\neg(x > y)$ imply that $y \geq x$?

Given any arbitrary binary relation $\geq$ defined on some set $S$, we define a new binary relation $>$ on $S$ by: $$ x > y \quad\text{iff}\quad (x \geq y) \wedge \neg(y \geq x) $$ In accordance ...
1
vote
1answer
807 views

Prove Friedrichs' inequality

I'm trying to show that the theorem (Friedrichs' inequality) in my book: Assume that $\Omega$ be a bounded domain of Euclidean space $\Bbb R^n$. Suppose that $u: \Omega \to \Bbb R$ lies in the ...
1
vote
4answers
466 views

Problems with Inequalities

It seems like I am facing some confusion while handling with inequalities,I was doing some tasks where it is asked to find the interval of the variable,after some steps I deduced the the necessary ...
8
votes
3answers
154 views

Proving the inequality $4\ge a^2b+b^2c+c^2a+abc$

So, a,b,c are non-negative real numbers for which holds that $a+b+c=3$. Prove the following inequality: $$4\ge a^2b+b^2c+c^2a+abc$$ For now I have only tried to write the inequality as ...
8
votes
1answer
396 views

Showing that $\log(\log(N+1)) \leq 1+\sum\limits_{p \leq N} \frac{1}{p}$

I can't see how you get this. I want to show that $$\log(\log(N+1)) \leq \sum_{p \leq N} \frac{1}{p}+1$$ Can't see how it follows from this. So you show that $$0 \lt -\log(1-x)-x \lt ...
7
votes
2answers
262 views

How prove this Stronger AM-GM inequality $\frac{n^2-1}{6}\min_{1\le i<j\le n}\left(\sqrt{a_{i}}-\sqrt{a_{j}}\right)^2\le A_{n}-G_{n}$

let $a_{i}>0,i=1,2,\cdots,n,n\ge 3$,show that $$\dfrac{n^2-1}{6}\min_{1\le i<j\le n}\left(\sqrt{a_{i}}-\sqrt{a_{j}}\right)^2\le\dfrac{a_{1}+a_{2}+\cdots+a_{n}}{n}-\sqrt[n]{a_{1}a_{2}\cdots ...
7
votes
1answer
146 views

Inequality with four positive integers looking for upper bound

Umm. This comes from Diophantine quartic equation in four variables and will finish the most important part if it can be done. Four positive integers $w,x,y,z.$ One equation and two inequalities $$ ...