Questions on proving and manipulating inequalities.

learn more… | top users | synonyms (1)

2
votes
2answers
80 views

How to prove this inequality? $(a+b+c=1)$

Show that if $a,b,c$ are positive reals and $a+b+c=1$, then the following must hold: $$\frac{2(a^3+b^3+c^3)}{abc}+3 \geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$ What I have tried is using $abc \leq ...
0
votes
1answer
332 views

Square root's inequality [closed]

Could anyone help me solve this inequality? I would be really grateful if you showed how to calculate it with steps. Thank you in advance. $\sqrt{x+2} + \sqrt{x-5} \ge 5 - x$
0
votes
1answer
28 views

Simple inequality with exponential

I have bounded $A$ by $$ e^{-\epsilon c}(\cosh c)^n $$ for any $c>0$, and if I'm correct the minimum occurs when $\tanh c=\epsilon/n$. By the right choice of $c$, I want to show that $$ A\le ...
6
votes
1answer
41 views

How to prove this inequality $(\sum_i x_i y_i)^2 - \sum_i x_i^2y_i^2 \leq 1-1/n$?

How to prove this inequality: $$(x_1y_1+x_2y_2+ \cdots + x_ny_n)^2 - (x_1^2y_1^2+x_2^2y_2^2+\cdots+x_n^2y_n^2)\leq 1-\frac{1}{n},$$ where $x_i,y_i \geq 0,i=1,2,\ldots,n$, and ...
0
votes
1answer
26 views

Convex optimization problem: linear equality and inequality constraints

When linear equality constraints can be converted in an inequality constraints for a strongly convex optimization problem? I mean, I got the same solution for both the following problem: 1) $\min_x ...
0
votes
1answer
28 views
1
vote
1answer
36 views

Condition on Inequalities

Here $$X=\frac{p}{b+q}+\frac{b^2 p (1+q)+2 b p \left(a p+q^2\right)+q \left(a (-1+p) p+q+p q^2\right)}{\left(b+a p+b q+q^2\right)^2}$$ $$Y=\frac{p^2}{b+a p+b q+q^2}$$ Where $a$, $b$, $p$ and $q$ are ...
0
votes
1answer
56 views

Prove that $1+\sum_{m=1}^n \frac{1}{m!}\geqslant 2\ $ for $n=2,3,4,5,\ldots$

I was given Prove that $1+\sum_{m=1}^n \frac{1}{m!}\geqslant 2\ $ for $n=2,3,4,5,\ldots$ I understand that $1+\sum_{m=1}^n \frac{1}{m!}>2$, but I don't see how $1+\sum_{m=1}^n ...
1
vote
1answer
15 views

$g(x)=\frac{x}{m}-1+\log m$, $g(x) \ge log x$

Let $m \in \mathbb N$ and $g(x)=\frac{x}{m}-1+\log m$, if $m-\frac{1}{2} \le x<m+\frac{1}{2}$. How could I show that $g(x) \ge \log x$?
2
votes
1answer
55 views

$x\over(1-x)$ $y\over(1-y)$ $z\over(1-z)$ >= 8 when $x ,y ,z $ are positive proper fractions and $x+y+z = 2$

Q. Prove that $x\over(1-x)$ $y\over(1-y)$ $z\over(1-z)$ $\geq$ 8 when $x ,y ,z $ are positive proper fractions and $x+y+z = 2$ What I did: From A.M. G.M. inequality, $(x+y+z)\over3$ $\geq$ ...
3
votes
1answer
27 views

Inequality involving sums and products (related to multi-stage rockets)

Let $a_i$ and $b_i$ be strictly positive real numbers for $i=1,\ldots, n$. I wonder whether the following inequality holds in general or does not:$$\frac{\sum_{i=1}^n a_i+\sum_{i=1}^n ...
1
vote
0answers
50 views

Is this integral function bounded?

Let $n \in\mathbb{N},n\geq2$, $\gamma\in\mathbb{R},\gamma<n-1$. Let $\Omega$ be a open and bounded subset of $\mathbb{R}^n$ regular enough ($C^2$ I think). It is known that ...
2
votes
2answers
37 views

Solving an inequality $B<n!$ without a calculator or gamma function?

Is there a way to solve $B<n!$ where $B$ is some very large real number (suppose for example $B=10^{17}$) without a calculator or gamma function? At the very least, to find the nearest integer for ...
-1
votes
1answer
28 views

Approximation using $1-x \le e^{-x}$

Suppose I want to approximate a number $p_k = (1-\frac{1}{365}) \cdot ... \cdot (1-\frac{k-1}{365})$. $k$ is a natural number. The book I'm reading says it can be done using the fact that $1-x \le ...
2
votes
1answer
63 views

Proving the inequality $\frac{1}{k!}+\frac{1}{(k + 1)!}+\frac{ 1}{ (k + 2)! }+…\leq {(\frac{e}{k})}^k$

In the first part of the question we showed that $P(X \geq k)\leq E(e^{tX}e^{-kt})$ for all $t \geq 0$ and real $k$ by the use of Markov's inequality. This wasn't too bad. Now, in the second part, ...
1
vote
0answers
24 views

Can we choose $g$ so that $\|(g\widehat{(f^{3})})^{\vee}\|_{L^{p}} \leq C \|g_{1}f\|_{L^{2}}^{r} \|(g_{2}\hat{f})^{\vee}\|_{L^{s}}$?

Let $f, f^{2}, f^{3}\in L^{q}(\mathbb R)\cap C_{0}(\mathbb R)$ where $ q\geq p, \ \text{and}$ and $C_{0}(\mathbb R)$ is the class of continuous functions vanishing at infinity. My Questions: ...
1
vote
3answers
78 views

Solve $\frac{(x-1)^{204}(x+3)^5(x-4)^{2015}}{(x+5)^{102}}\ge 0$

Solve $\frac{(x-1)^{204}(x+3)^5(x-4)^{2015}}{(x+5)^{102}}\ge 0$ Just wanted to share a nice and quick technique i learnt for such problems.
0
votes
2answers
47 views

Solving inequality equation involving sum of binomial coefficients

I have a function $f(k,\,i)$ involving binomial coefficients: $$f(k,\,i)\,=\left(\begin{matrix}k+i \\ k\end{matrix}\right)=\frac{(k+i)!}{k!\,i!}$$ And the following sum over this function (expansion ...
0
votes
0answers
30 views

Polynomial optimization and AM-GM inequality

I want to maximize the function $f(\mathbf{x},\mathbf{y}) = \sum \limits_{k=1}^{K}p_k(\mathbf{x})q_k(\mathbf{y})$, where $0 < p_k(\mathbf{x}) \leq \delta_k$ and $0 < q_k(\mathbf{y}) \leq ...
1
vote
3answers
80 views

Easy inequality going wrong

Question to solve: $$\frac{3}{x+1} + \frac{7}{x+2} \leq \frac{6}{x-1}$$ My method: $$\implies \frac{10x + 13}{(x+1)(x+2)} - \frac{6}{x-1} \leq 0$$ $$\implies \frac{4x^2 -15x-25}{(x-1)(x+1)(x+2)} ...
5
votes
3answers
84 views

Prove that $\det(AA^T+I)\ge 1$

If $A$ is a matrix with real entries, prove that $$\det(AA^T+I)\ge 1.$$ I tried using the eigenvalues. One thing came into my mind: maybe $AA^T$ is positive definite (I don't know whether this is ...
0
votes
1answer
31 views

Proving that $\phi_a(z) = (z-a)/(1-\overline{a}z)$ maps $B(0,1)$ onto itself.

I want to prove that if $\phi_a: B(0,1) \to \Bbb C$ is given by $\phi_a(z) = (z-a)/(1-\overline{a}z)$ with $|a| < 1$, then $|\phi_a(z)| < 1$. Resist the itch on your finger urging you to close ...
2
votes
1answer
23 views

Prove the set is closed with respect to its norm…

Let $V$ be a normed vector space over R. Let $W$ be a proper closed subspace of $V$. We say $w^*$ is a best approximation in $W$ to $v^* \in V$ if $\|v^*-w^*\| \leq \|v^*-w\|$ for all $w \in W$. ...
4
votes
0answers
93 views

By which way I can prove $\arctan\left(2^{\sqrt{3}-\sqrt{7}}\right)<\frac{61}{125}$ [closed]

By which way I can prove $$\arctan\left(2^{\sqrt{3}-\sqrt{7}}\right)<\frac{61}{125}$$
2
votes
0answers
81 views

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

Let $x$ be a real number. Find the minimum value of $$P=\frac{\sqrt{3(2x^2+2x+1)}}{3}+\frac{1}{\sqrt{2x^2+(3-\sqrt{3})x+3}}+\frac{1}{\sqrt{2x^2+(3+\sqrt{3})x+3}}$$ This is a problem from 2015 ...
1
vote
1answer
34 views

How can I prove Holder Ineqaulity for $0<p<1$

$0<p<1$ $\dfrac{1}{p}+\dfrac{1}{p'}=1$ if , $f \in L^{p}$ and $0<\int_{\Omega}\vert g(x) \vert^{p'}dx < \infty$ then $$\int_{\Omega}\vert f(x)g(x) \vert dx \geq (\int_{\Omega}\vert ...
3
votes
0answers
63 views

How prove $\sigma(4^p-1)<(2^{p+1}-1)^2$

If $p$ is an odd prime numbers, show that $$\sigma(4^p-1)<(2^{p+1}-1)^2$$ where $\sigma(n)$ stands for the sum of divisors. I thought of using the formula for $\sigma(n)$: If ...
0
votes
2answers
50 views

$\|(g\widehat{(f|f|^{2})})^{\vee}\|_{L^{2}} \leq C \|f\|_{L^{2}}^{r} \|(g\hat{f})^{\vee}\|_{L^{2}}$ for some $r\geq 1$?

Let $g\in C_{c}^{\infty} (\mathbb R)$, and $f, |f|^{2}f\in L^{2}(\mathbb R)\cap C_{0}(\mathbb R)$ (where $C_{c}(\mathbb R)$ is the class of smooth functions with compact support and $C_{0}(\mathbb R)$ ...
2
votes
1answer
29 views

Maximization Lemma in Matrix Algebra

I am having a problem with understanding an English sentence underlined in red below. Can somebody let me understand what it is saying? and what is maximized?
1
vote
1answer
14 views

How does this inequality of a complex function hold

I cannot figure out how $\Re[g(\lambda)]\leq |\lambda|$ implies $|g(\lambda)|\leq|2 r-g(\lambda)|$ where $\lambda$ is an arbitrary complex number s.t. $|\lambda|\leq r$, and $g$ is an entire function. ...
3
votes
2answers
23 views

If $n\ge2$, prove that $\frac {n!}{n^n} \le ({\frac 1 2})^k$, where $k$ is the greatest integer $\le \frac n 2$.

Using only precalculus knowledge, if $n\ge2$, prove that $\frac {n!}{n^n} \le ({\frac 1 2})^k$, where $k$ is the greatest integer $\le \frac n 2$. (taken from Apostol's Calculus I, page 46) I don't ...
1
vote
3answers
11 views

Show that $f$ is $3$-Lipschitz w.r.t the second component

Let $$f(x,y) = \frac{xy}{1 + x^2 + y^2}$$ and $$D = \{(x,y) \in \mathbb R^2 \ / \ x^2 + y^2 \le 1\}$$ Show that $\forall$ $(x,y_1), (x,y_2)$ $\in D$, $|f(x,y_1) - f(x,y_2)| \le 3|y_1 - y_2|$ ...
2
votes
5answers
84 views

How to properly solve this inequality $2^x < \frac{3}{4}$?

How to properly solve this inequality? $$2^x < \frac{3}{4}$$ I know that it will be something like that: $$ x \stackrel{?}{\ldots} \log_2\frac{3}{4} $$ But I don't know how to decide if it should ...
1
vote
3answers
227 views

Mean Value Theorem Inequality Contradiction(?)

I am trying to show: $e^x > 1+x+\frac{x^2}{2}$ for $x>0$ using Mean Value Theorem (MVT). My method is as follow: consider the function $f(x)=e^x - (1+x)$. We know $f'(x) = e^x -1 > 0 ...
0
votes
0answers
22 views

Inequalities in binomial and normal distrubutions

Example Q Foo is normall distrubuted like $$X\sim N(100,15^2)$$ foo of 110 is required. Does that mean that I should find: $$P(X\gt 109) $$ or $$P(X\gt 110) $$ or $$P(X\ge 110) $$ I feel ...
0
votes
3answers
79 views

Proving the inequality $(x+y)^\alpha \leq x^\alpha + y^\alpha$ [duplicate]

Why does this inequality hold? $$(x+y)^\alpha \leq x^\alpha + y^\alpha$$ where $x$, $y\geq 0$ and $\alpha \in (0,1)$. Thanks in advance!
1
vote
3answers
58 views

Prove that if $a,b \in \mathbb{R}$ and $|a-b|\lt 5$, then $|b|\lt|a|+5.$

Im trying to prove that if $a,b \in \mathbb{R}$ and $|a-b|\lt 5$, then $|b|\lt|a|+5.$ Ive first written down $-5\lt a-b \lt5$ and have tried to add different things from all sides of the inequality. ...
2
votes
2answers
96 views

Entropy upper bound inequality for Sub-Gaussian Random Variable

We say that the random variable $Z$ is $\sigma^2$-subGaussian if $\mathbb{E} \exp(tZ) \leq \exp(t^2\sigma^2)/2$. Define the $(x\log x)$-entropy (or simply the entropy) of a nonnegative random ...
0
votes
1answer
52 views

Is this inequality true, and does it have a name?

$$ \sqrt{\left(\sum_{i=1}^n a_i^2\right)} \leq \sum_{i=1}^n \sqrt{a_i^2}$$ If my proof is not incorrect, then it is true for $n=2$, but before proving the general case I'd like to know whether it is ...
2
votes
1answer
24 views

Norm of Operator Proof

I'm stuck on this problem that I can't seem to figure out. Here's the problem. To note, equation 2.42 says that $$||T|| = \sup \{ ||Tu||: u \in C([a,b]), ||u|| = 1 \}$$ where $T$ is defined, ...
0
votes
1answer
11 views

Relation between a function and its norm

While reading up on Sturm-Liouville system theory, I came across something I didn't fully understand. At one point, in the midst of proving the existence of solutions to the Sturm-Liouvill problem, ...
0
votes
1answer
42 views

The meaning of 'worst case'

When giving bound on convergence rate, complexity and so on, people sometimes will specify it by 'worst case'. What is the meaning of 'worst case'?
4
votes
4answers
71 views

Prove that if $0\leq a,b$ and $a+b=1$ then $x^ay^b\leq ax+by$ for $x, y >0$

Would like help getting started, unfamiliar with proving inequalities in general.
1
vote
4answers
165 views

Compute $\sqrt[7]{0.999}$ to three decimal places.(From Gelfand's Algebra text.)

After a brief introduction to roots(imaginary numbers were not introduced yet), this question is asked. I am apparently expected to find the answer using elementary algebraic manipulation. I have ...
2
votes
7answers
188 views

Which number is bigger: $\sqrt[10]{2}$ or $1.2$?

What is the general method for finding such inequalities? I have some more problems of this kind in the text I am using.
1
vote
4answers
64 views

Show that $\ln (x) \leq x-1 $

Show that $\ln (x) \leq x-1 $ I'm not really sure how to show this, it's obvious if we draw a graph of it but that won't suffice here. Could we somehow use the fact that $e^x$ is the inverse? I mean, ...
0
votes
1answer
36 views

Why is $\left|e^{iat} e^{-st}\right| \le \left| e^{iat}\right| \left| e^{-st}\right| \le e^{-st}$ true?

My text states: $\left|e^{iat} e^{-st}\right| \le \left| e^{iat}\right| \left| e^{-st}\right| \le e^{-st}$ as $\left| e^{iat} \right|=1$ where $a,s,t \in \mathbb{R}$ I thought the last ...
1
vote
2answers
29 views

complex numbers- proving the equality part in the Cauchy–Schwarz inequality using Lagrange identity

I need to discuss the equality case of: $$ \left | \sum_{k=1}^{n} z_{k}w_{k} \right |^{2} \leq \left ( \sum_{k=1}^{n}\left | z_{k} \right |^{2}\right )\left ( \sum_{k=1}^{n}\left | w_{k} \right ...
3
votes
2answers
326 views

Inequality involving Square Root

$\sqrt{3x-x^2}<4-x$ I know I can't simply square both sides of an inequality. I have narrowed down the possible values of x => x belongs to [0,3] because the expression inside the square root ...
2
votes
0answers
64 views

Can Waring problem be solved with triangle inequality?

When we calculate the difference $\frac{3^k-1}{2^k-1}-\left(\frac{3}{2}\right)^k,$ we get $\frac{3^k-2^k}{4^k-2^k}.$ Then solving: $1-\frac{3^k-2^k}{4^k-2^k}>\frac{3^k-2^k}{4^k-2^k},$ we get ...