Questions on proving and manipulating inequalities.

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8
votes
2answers
109 views

How to show a Determinantal inequality

If $A, B$ and $C$ are $n\times n$ positive semidefinite matrices. How to show that $$\det(A + B) + \det(A + C)\le \det A + \det(A + B + C)?$$
6
votes
3answers
183 views

Prove that $\zeta (4)\le 1.1$

Prove the following inequality $$\zeta (4)\le 1.1$$ I saw on the site some proofs for $\zeta(4)$ that use Fourier or Euler's way for computing its precise value, and that's fine and I can use it. ...
2
votes
1answer
52 views

notion of the minimum of a function over a polytope

Let $P$ be a polytope with $M$ vertices. (The polytope $P$ is the intersection of the hypercube $0≤x _j ≤1$ with the hyperplane $\sum_{j=1}^nx_j=t$, $0\leq t\leq n$). Suppose that the volume of $P$ ...
7
votes
2answers
246 views

for any acute triangle $ABC:\cos^3 (A)+ \cos^3 (B)+\cos^3 (C)+\cos(A)\cdot\cos(B)\cdot\cos(C)\ge\frac{1}{2} $

What is the proof that for any acute triangle $ABC$,then : $$\cos^3 (A)+ \cos^3 (B)+\cos^3 (C)+\cos(A)\cdot\cos(B)\cdot\cos(C)\ge\frac{1}{2} $$
2
votes
3answers
4k views

Taking the square roots in inequalities

I have a question regarding taking square roots in inequalities. I have a problem asking: Suppose $3x^2+bx+7>0$ for every real number x. Show that $|b|<2\sqrt{21}$. In an earlier question it ...
-1
votes
3answers
182 views

proving a inequality about sup [duplicate]

Possible Duplicate: How can I prove $\sup(A+B)=\sup A+\sup B$ if $A+B=\{a+b\mid a\in A, b\in B\}$ I want to prove that $\sup\{a+b\}\le\sup{a}+\sup{b}$ and my approach is that I claim $\sup ...
4
votes
2answers
140 views

Inequality under condition $a+b+c=0$

I don't know how to prove that the following inequality holds (under condition $a+b+c=0$): $$\frac{(2a+1)^2}{2a^2+1}+\frac{(2b+1)^2}{2b^2+1}+\frac{(2c+1)^2}{2c^2+1}\geqq 3$$
6
votes
3answers
131 views

the least possible value for :$ \lfloor \frac{a+b}{c}\rfloor +\lfloor \frac{b+c}{a} \rfloor+\lfloor \frac{c+a}{b} \rfloor $

If we know that for every $a,b,c>0$ ,how we can find the least possible value for : $$ \lfloor \frac{a+b}{c}\rfloor +\lfloor \frac{b+c}{a} \rfloor+\lfloor \frac{c+a}{b} \rfloor $$
1
vote
2answers
106 views

calculus limit inequality

How can I elegantly show that: $(1 + \frac{1}{k})^k \leq 3$ For instance I could use the fact that this is an increasing function and then take $\lim_{ k\to \infty}$ and say that it equals $e$ and ...
1
vote
1answer
144 views

Inequality for the Binomial Theorem

I want to show that $(1+ \frac{1}{k})^k \geq 2$ for say $k \geq 2$. Here is what I have so far: By the Binomial Theorem we know $(1 + x)^k$ for $k \geq 2$ gives us: $1^k + {k\choose 1}1^{k-1}x^1 + ...
1
vote
1answer
155 views

Chernoff bound for Geometric RVs compared to exact tail bound

I keep getting a result I can't interpret. X is a Geometric RV with distribution ($0<\rho<1$) $$ \pi_k = \rho^k(1- \rho) $$ so directly applying Geometric series the tail bound is $$ B_1 = ...
0
votes
1answer
35 views

correctness of inequality

Let $k \geq 6$ and I know $k!$ < $\dfrac{k^k}{2^k}$ I want to show the following: $(k+1)! < \dfrac{(k+1)^{k+1}}{2^{k+1}}$ Now I am going to show my solution, let me know if my reasoning is ...
2
votes
1answer
105 views

why is the Cantor-Lebesgue function increasing

I am trying to derive the fact that the function $F$ defined below is monotonically increasing. The only thing I can use is that the any member of the Cantor set has a ternary expansion involving only ...
3
votes
4answers
428 views

Prove $\log_{\frac{1}{4}} \frac{8}{7}> \log_{\frac{1}{5}} \frac{5}{4}$

Prove $$\log_{\frac{1}{4}} \frac{8}{7}> \log_{\frac{1}{5}} \frac{5}{4}$$ How to prove without a computer?
4
votes
1answer
143 views

Is this AM/GM refinement correct or not?

In Chap 1.22 of their book Mathematical Inequalities, Cerone and Dragomir prove the following interesting inequality. Let $A_n(p,x)$ and $G_n(p,x)$ denote resp. the weighted arithmetic and the ...
1
vote
1answer
134 views

How to solve for the $n$-th Fibonacci number that is greater than or equal to $N$?

The general formula for the $n$-th Fibonacci number is: $$\frac{\phi^n - (1 - \phi)^n}{\sqrt{5}}$$ where $$\phi = \frac{1 + \sqrt{5}}{2}$$ Given $N$, is there a way to solve for $n$ in this ...
1
vote
1answer
47 views

Inequality involving $|u|^{p-1} u$

For $u,v \in L^q(\Omega)$ with $q \ge p \ge 1$, how does one show that: $$ \begin{aligned} \||u|^{p-1}u - |v|^{p-1}v\|_{L^{p/q}} & \le C\,\|(|u|^{p-1} + |v|^{p-1})\,|u-v|\,\|_{L^{p/q}}\\ & ...
1
vote
1answer
150 views

Exact upper bound vs Markov and Chernoff inequalities

This is probably not a very smart question, more of a confusion I guess. For example, if $X \sim Geometric(1-p)$, i.e. $p$ being the probability of failure, the exact tail bound is $$ ...
1
vote
1answer
52 views

Inequality with numbers

It seems its a simple question, but I am confused. Let a be natural number and let b be some number $1\le b\le a$. Find an upper bound for $$ \frac{a^2+2b^2-4ab-a}{a(a-1)}. $$ I've got $$ ...
13
votes
2answers
669 views

How can one prove that $\pi^4 + \pi^5 < e^6$?

A proof of the inequality using properties of $\pi$ and $e$, for example, is what I'm looking for. Not calculator approximations showing the inequality holds.
3
votes
1answer
122 views

Littlewood's Inequality

Working on an exercise from Shorack's Probability for Statisticians, Ex 3.4.3 (Littlewood's inequality). Can't seem to find any material for this. Help is appreciated. Prove:Exercise 4.3 ...
0
votes
1answer
88 views

Showing an inequality with interest rates and compounding

Suppose that your grandmother is receiving Social Security at the start of next year and has to choose between two plans for how to receive her payments. Divide each year into $k$ payment periods ($k$ ...
2
votes
2answers
129 views

Proving $a>b$ implies $P(Y>a|Y>b) > P(Y>a)$

Suppose you have a constants $a>b$, and $P(Y>b)<1$, then how can we show that $$P(Y>a|Y>b) > P(Y>a)?$$ I used the definition of conditional probability to get that $$ ...
2
votes
1answer
39 views

Which theorem can I use to solve the following sequence of random variables

If $\{X_j\}_{j=1}^n$ is a sequence of random variables, which theorem should I use to show that for any $p \ge 1$: $$ \mathbb{E}\left|\frac{1}{n}\sum_{j=1}^n X_j\right|^p \le \frac{1}{n}\sum_{j=1}^n ...
5
votes
1answer
323 views

Minimal set of inequalities

I have a set of $m$ linear inequalities in $R^n$, of the form $$ A x \leq b $$ These are automatically generated from the specification of my problem. Many of them could be removed because they are ...
2
votes
2answers
593 views

How to prove this inequality $\sqrt{\frac{ab+bc+cd+da+ac+bd}{6}}\geq \sqrt[3]{{\frac{abc+bcd+cda+dab}{4}}}$

How to prove this inequality $$\sqrt{\frac{ab+bc+cd+da+ac+bd}{6}}\geq \sqrt[3]{{\frac{abc+bcd+cda+dab}{4}}} ?$$ Thanks
5
votes
0answers
705 views

Proof of $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$

Let $a_n>0$ and $b_n\geq 0$, then $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$ My attempt at a proof is as follows. Let $A_n=\sup\{a_n, a_{n+1},...\}$, $B_n=\sup\{b_n, b_{n+1},...\}$, and ...
2
votes
1answer
147 views

How to prove that there exist a concave function and $\gamma\in[0,1]$ and some other numbers which satisfy an inequality

I'm working on an economics paper, and in the model I've made I've basically gotten myself a little bit stuck. I need to show that there exists a nondecreasing concave function $u$ and numbers $P$ and ...
-2
votes
2answers
125 views

Prove $\frac{a^2}{b^2+c^2} + \frac{b^2}{c^2+a^2} + \frac{c^2}{a^2 + b^2} \geq \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}$ [closed]

Let $a,b,c \in \mathbb{R_+}$. Prove that $$\frac{a^2}{b^2+c^2} + \frac{b^2}{c^2+a^2} + \frac{c^2}{a^2 + b^2} \geq \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}. $$
5
votes
3answers
152 views

A proof problem from a first time real analysis course

Given a continuous function $g:[a,b]\to\Bbb R$, if there exists a number $K>0$ s.t. for all $x\in[a,b]$, $|g(x)| \le K \int_a^x |g|$, prove $g(x)=0$ for all $x\in[a,b]$. And I tried to derive some ...
4
votes
1answer
221 views

Prove $\frac{1}{a(1+b)} + \frac{1}{b(1+c)} + \frac{1}{c(1+a)} \geq \frac{3}{ (abc)^{\frac{1}{3}}\big( 1+ (abc)^{\frac{1}{3}}\big) }$ using AM-GM

I need to proof this inequality by AM-GM method. Any ideas how to do it? $$\frac{1}{a(1+b)} + \frac{1}{b(1+c)} + \frac{1}{c(1+a)} \geq \frac{3}{ (abc)^{\frac{1}{3}}\big( 1+ (abc)^{\frac{1}{3}}\big) ...
5
votes
1answer
113 views

Inequality :$7a+5b+12ab\le9$

If we assume that $a,b$ are real numbers such that $9a^2+8ab+7b^2\le 6$ ,how to prove that : $$7a+5b+12ab\le9$$
2
votes
5answers
146 views

What is this inequality called?

Apparently it's a famous inequality taught in 1st year calculus but I have never even seen it before nor know it has a name. $x + y \geq 2\sqrt{xy}$ It looks like it is just saying $(x + y)^2 \geq ...
0
votes
1answer
295 views

Inequality. $a\sqrt{2b+c^2}+b\sqrt{2c+a^2}+c\sqrt{2a+b^2} \leq 3\sqrt{3}$

Could you help me please with the following inequality $a,b,c$ non-negative numbers such that: $a+b+c=3.$ Prove that: $$a\sqrt{2b+c^2}+b\sqrt{2c+a^2}+c\sqrt{2a+b^2} \leq 3\sqrt{3}.$$
2
votes
1answer
160 views

Double Inequality. $ab+bc+ca \leq \frac{1}{8}\sum{\sqrt{(1-ab)(1-bc)}} \leq a^2+b^2+c^2$

Let $a,b,c$ be non-negative numbers such that $a+b+c=1.$ Prove that : $$ab+bc+ca \leq \frac{1}{8}\sum_{cyc}{\sqrt{(1-ab)(1-bc)}} \leq a^2+b^2+c^2.$$ Thanks:)
3
votes
2answers
305 views

Inequality. $a^7b^2+b^7c^2+c^7a^2 \leq 3 $

Let $a,b,c$ be positive real numbers such that $a^6+b^6+c^6=3$. Prove that $$a^7b^2+b^7c^2+c^7a^2 \leq 3 .$$
1
vote
2answers
85 views

Prove $(1+z)^a \geq 1+az$, for $z>-1, a>1$, using mean value theorem

Prove $(1+z)^a \geq 1+az$, for $z>-1, a>1$, using the mean value theorem Hint says try using: $f(z)=(1+z)^a$ I've tried messing around with this, but I can't seem to get 1 + az on the RHS. ...
1
vote
1answer
86 views

Inequality. $\frac{a}{(b+c)^4}+\frac{b}{(c+a)^4}+\frac{c}{(a+b)^4} \geq \frac{3}{2(a+b)(b+c)(c+a)}$

For $a,b,c >0$ prove that : $$\frac{a}{(b+c)^4}+\frac{b}{(c+a)^4}+\frac{c}{(a+b)^4} \geq \frac{3}{2(a+b)(b+c)(c+a)}.$$ I don't know how should I start. It is difficult for me because the ...
1
vote
1answer
65 views

Inequalities of summations

I am thinking if the following condition is in general true: $\frac{n}{m} \leq \frac{\sum_{i = 1} ^ {n} a_i}{\sum_{i = 1} ^ {m} b_i}$, when $n \leq m$ and $a_i \geq 0$, $b_i \geq 0$ but i cannot find ...
0
votes
2answers
275 views

Inequality. $\sum{\sqrt{x^2+xy+y^2}}\geq \sum{\sqrt{2x^2+xy}}.$

Let $x,y,z >0$. Prove that: $$\sum_{\text{cyc}}{\sqrt{x^2+xy+y^2}}\geq \sum_{\text{cyc}}{\sqrt{2x^2+xy}} .$$ Thanks for your help :)
3
votes
1answer
2k views

General Proof for the triangle inequality

I am trying to prove: $P(n): |x_1| + \cdots + |x_n| \leq |x_1 + \cdots +x_n|$ for all natural numbers $n$. The $x_i$ are real numbers. Base: Let $n =1$: we have $|x_1| \leq |x_1|$ which is clearly ...
2
votes
0answers
44 views

Randomized Solution to a System of Inequalities

Given a set of $\mathbf v_i \in \{0,1\}^k$ for $i=1,\dots,n$ and a vector $\mathbf x \in [0,1]^k$, we want to decide if the following inequality holds or not: $$ \mathbf x \le \sum_{i=1}^n \alpha_i ...
1
vote
1answer
204 views

Trigonometric Inequality. $\tan{A}+\tan{B}+\tan{C} \geq \frac{s}{r}$

For any acute-angled triangle $ABC$ show that $$\tan{A}+\tan{B}+\tan{C} \geq \frac{s}{r},$$ where where $s$ and $r$ denote the semi-perimeter and the inradius, respectively. Merci :)
0
votes
2answers
2k views

Solving Inequalities with variables in the denominator

Ok so I've got an exam in 4 hours and I can't ever figure out these problems. Here's the problem I'm struggling with now: $$\frac{x}{3x - 5} \leq \frac{2}{x - 1}.$$ I've learned this so many times ...
3
votes
2answers
235 views

Inequality. $\sum{(a+b)(b+c)\sqrt{a-b+c}} \geq 4(a+b+c)\sqrt{(-a+b+c)(a-b+c)(a+b-c)}.$

Let $a,b,c$ be the side-lengths of a triangle. Prove that: I. $$\sum_{cyc}{(a+b)(b+c)\sqrt{a-b+c}} \geq 4(a+b+c)\sqrt{(-a+b+c)(a-b+c)(a+b-c)}.$$ What I have tried: \begin{eqnarray} ...
1
vote
1answer
242 views

Chernoff inequalities for the sum of Exponential RVs

These two well-known Chernoff bounds for the sum of RVs $X=\sum_{k=1}^{n}X_k$ in mulitplicative form, $\mathbf{P}(X \leq (1- \delta)\mathbf{E}X) \leq e^{-\frac{\delta^2 \mathbf{E}X}{2}}\\ ...
0
votes
2answers
34 views

An inequality involving a product

Let $x_1\in(0,1)$ and $a_1,\ldots,a_n\ge-1$ reals. We know that \begin{equation} \prod_{i=1}^n (1+x_1a_i) < 1 \end{equation} Does it then also hold true that \begin{equation} \prod_{i=1}^n ...
2
votes
4answers
925 views

How to prove Cauchy-Schwarz Inequality in $R^3$?

I am having trouble proving this inequality in $R^3$. It makes sense in $R^2$ for the most part. Can anyone at least give me a starting point to try. I am lost on this thanks in advance.
1
vote
1answer
130 views

Inequality. $3\left(IA^2+IB^2+IC^2\right) \geq AB^2+BC^2+CA^2$

(Korea 1998) Let $I$ be the incenter of a triangle $ABC$.Prove that: $$3\left(IA^2+IB^2+IC^2\right) \geq AB^2+BC^2+CA^2.$$ Please help me to improve this kind of inequalities. Thanks :)
-1
votes
3answers
143 views

absolute value inequality

I would like to know how to solve the inequality $$|x^2-y^2|\leq 2x+2y-4xy.$$ I have tried to solve it by myself and searched in the internet, but didn't come up with an answer. Thanks in advance. ...