Questions on proving, manipulating and applying inequalities.

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1
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3answers
77 views

About a complex number $z\in S^1$ fulfilling an inequality

Let $a,b,c$ be complex numbers, such that $\|a\|=\|b\|=\|c\|=1$. I have to show that there exists a complex number $z$, with $\|z\|=1$, such that: $$\dfrac{1}{\|z-a\|^2}+\dfrac{1}{\|z-b\|^2}+\dfrac{1}{...
0
votes
2answers
91 views

How to prove $(\cos (x)+1)^{\sin (x)+1}>(\sin (x)+1)^{\cos (x)+1},(0<x<\frac{\pi}{4})$

Q: How to prove $$(\cos (x)+1)^{\sin (x)+1}>(\sin (x)+1)^{\cos (x)+1},(0<x<\frac{\pi}{4})$$ What should I do here? I don't even know where to start from. Please help me by giving me a hint.
2
votes
2answers
41 views

Interior of a preimage of a continuous function

Let $ f:\mathbb{R}^n\rightarrow \mathbb{R} $ be convex. Let there exist a point $ x_0 $ with $ f(x_0)<0 $. Prove that $$ \operatorname{int}\left\lbrace f(x)\ge 0 \right\rbrace = \left\...
-3
votes
2answers
74 views

proving $\frac{1}{n+3}+\frac{1}{n+4}+…+\frac{1}{2n+4}>\frac{1}{2}$

how can one prove that: $\frac{1}{n+3}+\frac{1}{n+4}+...+\frac{1}{2n+4}>\frac{1}{2}$ For all natural $n$, without using induction? thank you.
0
votes
2answers
79 views

$\left| x \right| \le 3\left[ {\sqrt x } \right]$ [closed]

Let $\left| x \right| \le 3\lfloor {\sqrt x } \rfloor$. What is the answer to this inequality?
-2
votes
2answers
45 views

Real Analysis (Proof)

I'm thinking that maybe this is an application of the Mean Value Theorem. But I'm not sure how to do it. Please help. >.< i) Let $a>0$ and $n>2$. If $$\frac{a}{1+2a}< \frac{1}{n}$$ , ...
0
votes
1answer
79 views

Three inequalities with a common constraint

A question struck me today and to be able to answer it I wrote it down as a mathematical expression and have been able to simplify the question to the following: Given $$x > 1,y > 1,z > 1$$ $...
0
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0answers
12 views

Do points on the graph itself count as part of the solution set to a system of inequalities?

How many points $(x,y)$ with integer values for $x$ and $y$ lie in the solution set of the system: $y > -2x+1$, $y>3x-1$, $y=3$ $2$ points that lie within the intersection of the graphs are ...
1
vote
2answers
113 views

Proof of inequality between sums

Let $\{y_i\}_{i=1}^N, \{z_i\}_{i=1}^N$ be two sets of real numbers s.t. $y_i, z_i \ge 0$, $\sum_{i=1}^N y_i = 1$, $\sum_{i=1}^N z_i \le 1$. I have been asked to show that $$ \sum_{i=1}^N y_i \log \...
0
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1answer
45 views

Minimizing a strictly convex function with inequality constraint

So we've been learning about the Kuhn Tucker conditions in my non-linear optimization course and I've been having trouble with this problem: QUestion: description here Question: a strictly convex ...
4
votes
0answers
91 views

Prove $\left(\frac{a+1}{a+b}\right)^a+\left(\frac{b+1}{b+c}\right)^b+\left(\frac{c+1}{c+a}\right)^c \geqslant 3$

$a,b,c \geqslant 0,$$ a+b+c=3$, and $(a+b)(b+c)(c+a) \neq 0$ , prove $$\left(\frac{a+1}{a+b}\right)^a+\left(\frac{b+1}{b+c}\right)^b+\left(\frac{c+1}{c+a}\right)^c \geqslant 3$$ I try Bernouli's ...
0
votes
1answer
58 views

Proof of an inequality involving general triangles

In any triangle the following inequality holds: $$\dfrac{9abc}{a+b+c}\ge4S\sqrt{3}$$ where $a,b,c$ are the sides of the triangle and $S$ the area. How the previous inequality can be proven?
0
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0answers
23 views

Solutions to Diophantine Moving Window Inequations

I am interested in finding the number of non-negative integer solutions, $N(m,h,u)$, to this system of inequations $$ \left\{ \matrix{ 0 \le x_{\,0} + x_{\,1} + \cdots + x_{\,m} \le u \hfill \...
-1
votes
0answers
25 views

Check $r^t\geq \dfrac{\cos\left( \frac{u-1}{2}t\right)}{\cos\left( \frac{u+1}{2}t\right)}$, for $\forall r>1, \ u\in[0,1]$, $t\in [0,\pi/2]$.

Check if there exists an interval $I\subseteq [0,\pi/2]$ for which the following inequality $$r^t\geq \dfrac{\cos\left( \frac{u-1}{2}t\right)}{\cos\left( \frac{u+1}{2}t\right)}, \ \ \ \ \ \forall r>...
2
votes
1answer
91 views

Bounds for $\sum_{n\geq 2}\frac{n+2}{(n^2-1)(n+3)}$

Use the integral test to prove this inequaliy I calculated the integral $\int_{2}^{\infty}\frac{2+x}{(x^2-1)(x+3)}dx$ How can I use the integral test to show that $0.45 < \sum_{n=2}^{\infty}\frac{...
0
votes
1answer
65 views

Find the Smallest Value

Find the smallest value of $$a + \frac {1}{(a-b)b} $$ where a>b>0 I found this question in AM-GM inequality problems but I am stuck at this
0
votes
0answers
15 views

An example of a bounded domain $\Omega\subset \left\{ 0<\Re s< 1\right\} $ for which $\Re \zeta(s)$ is non-negative

Denoting the complex variable $s=\sigma+it$ (and we know that $\mathbb{C}$ and $\mathbb{R}^2$ are isomorphic, thus $s\equiv(\sigma,t)\in\mathbb{R}^2$) one has for $0<\Re s=\sigma<1$ that $$\zeta(...
2
votes
0answers
40 views

Prove this by inequality with four variables inequality

Let $a,b,c,d>0$ show that $$\color{blue}{\left(\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{d}+\dfrac{d^2}{a}\right)^2\ge 4(a^2+b^2+c^2+d^2)+\dfrac{8}{3}[(a-b)^2+(a-c)^2+(a-d)^2+(b-c)^2+(b-d)^2+(c-d)^...
1
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3answers
58 views

For $\triangle ABC$, prove $( \sin A + \sin B )( \sin B +\sin C )( \sin C + \sin A) > \sin A \sin B \sin C$

In $\triangle$ ABC, prove that $$( \sin A + \sin B )( \sin B + \sin C )( \sin C + \sin A) > \sin A \sin B \sin C$$ I have tried the formula A.M.- G.M. relation with $\sin A$, $\sin B$, and $\...
2
votes
2answers
36 views

Bound on integrable nonnegative function $F$ given inequality with compactly supported continuous functions.

Full Question: Suppose that $F$ is a nonnegative function that is integrable on $\mathbb R$ and there is a constant $C$ such that $\int_\mathbb R Ff \leq C\int_\mathbb R f$ whenever $f$ is a ...
4
votes
1answer
70 views

Ways to squeeze $e$ by hand

Let $a$ and $b$ be the lower and upper bound of $e$, respectively. Both $a$ and $b$ are rational numbers. Without using a calculator and without knowing the value of $e$, find $a$ and $b$ where $b-a&...
0
votes
1answer
36 views

Approximation of $|x|^p$

Let $a,b$ be reals and $1\leq p$. How do I prove that there exists $O(\epsilon)$ such that $|a+b|^p - |a|^p \leq \epsilon |a|^p + O(\epsilon) |b|^p$ for a given $\epsilon >0$? Direct taylor ...
7
votes
3answers
140 views

Proving $\pi \gt e+\frac{1}{e} \gt \pi-\frac{1}{\pi} \gt e$

I created this problem for myself as a fun exercise. I want to prove the following statement: $$\pi \gt e+\dfrac{1}{e} \gt \pi-\dfrac{1}{\pi} \gt e$$ I found that the following upper/lower bounds ...
1
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4answers
119 views

How to resolve $|x|-|x+1|\leq 7$?

How can I resolve this problem : $|x|-|x+1|\leq 7$ ? The answer is true for all $\mathbb{R}$ but I am having some trouble proving it.
1
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0answers
27 views

How to prove an inequality that sum of variables is larger or equal than sum of squares? [duplicate]

How one can prove that if $x,y,z>0$ and $x^2+y^2+z^2=3$ then $x+y+z\geq (xy)^2+(yz)^2+(zx)^2$? I tried AM-GM but without success.
0
votes
1answer
22 views

Contradictory system of linear inequalities

Assume that $(F_i)_i$ is a system of linear inequalities in $n$ variables, of the form $F_i(x_1,\ldots, x_n) > 0$, where $F_i(x_1, \ldots, x_n) = a_{i,1}x_1 + a_{i,2} x_2 +\ldots + a_{i,n} x_n + ...
0
votes
0answers
45 views

How to solve a modular inequality with optimization?

I have this: $x\le y$ $ y\lt m$ $x^2\mod m < y$ $y$ and $m$ are given. I am trying to maximize the value of $x$. Any advice on how to approach this?
1
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3answers
97 views

Prove $\forall n \in \mathbb{N}: \int_{0}^{\frac{\pi}{2}} |\frac{\sin(nx)}{x}|dx \geq \frac{2}{\pi}\sum_{k=1}^{n}\frac{1}{k}$

As my further preparation to Putnam competition, I came across such inequality to prove: $$\forall n \in \mathbb{N}: \int_{0}^{\pi} \left|\frac{\sin(nx)}{x}\right|dx \geq \frac{2}{\pi}\sum_{k=1}^{n}\...
0
votes
1answer
41 views

Can this be proved using inequalities?

Consider real numbers X1,X2 ......Xn, and Q1,Q2 ......Qn. ALL X's are not zero. Similarly, ALL Q's are not zero. If (i) X1+X2+........+Xn=0 and if (ii) ...
3
votes
3answers
60 views

prove $\lim_{n\to\infty}{\frac{a^n}{n!}}=0$

I have the proof but i don't understand one part. The proof (for $a>1$) goes as follows: there exists $k \in N$ such that$a<k, \frac{a}{k}<1$ and since $\frac{a}{k+i}<\frac{a}{k}, i \in N$...
0
votes
0answers
45 views

Existence of solution for $Mx \ge b$

Is there an analytical way to determine if there is a solution for $x \in R^{4}$ in the following matrix inequality? $Mx \ge b$ where $M \in R^{N\times 4}$ and $b \in R^{N}$ I know that I can solve ...
1
vote
0answers
41 views

Prove Inequality

I'm trying to prove the inequality $$\left(\frac{\sum_j (x_j+y_j)^p}{\sum_j (x_j+y_j)^r}\right)^{\frac{1}{p-r}} \le \left(\frac{\sum_j x_j^p}{\sum_j x_j^r}\right)^{\frac{1}{p-r}} + \left(\frac{\...
6
votes
2answers
111 views

Show that $x^2 + y^2 + z^2 \ge 35$ if $x+3y+5z \ge 35.$

Show that $x^2 + y^2 + z^2 \ge 35$ if $x+3y+5z \ge 35.$ I have tried everything (proof by contradiction, etc.) but I can't seem to get it. The book didn't give any constraints whatsoever. Any hints ...
0
votes
1answer
61 views

Solving an exponential inequality problem [duplicate]

How do I prove the following inequality : $$\Bigg(\frac{2}{\alpha^2} \, \big( e^{\alpha x} - e^{\alpha y} \big) \, + \, e^{\alpha y} (y^2 - x^2) \; \Bigg) \geq 0 $$ given, $x, y \geq 0$ ? Can ...
2
votes
4answers
71 views

Convex integral inequality

I cannot prove that if $f(x)$ is convex on $[a,b]$ then $f\Big(\frac{a+b}2\Big) \le \frac1{b-a}\int_a^b f(x)\,dx \le \frac{f(a)+f(b)}2 .$
1
vote
1answer
24 views

If $x \in (a, b)$ and $xy + z(1-y) \in (a,b)$, what can I say about $y$ and $z$?

Suppose that: $$x \in (a, b)$$ and $$xy + z(1-y) \in (a,b),$$ where $(a,b) \subseteq (0,1)$, while $y$ and $z$ are real number. What can I say about $y$ and $z$? Of course, when $y = 1$, then the ...
0
votes
1answer
49 views

Four workers in a construction company

There are four workers $A,B,C,D$ in a construction company with different skills. It takes $D_{AB}$ days when $A$ & $B$ works together to finish a construction, $D_{BC}$ days when $B$ & $C$ ...
1
vote
3answers
102 views

Minimizing $\cot^2 A +\cot^2 B + \cot^2 C$ for $A+B+C=\pi$

If $A + B + C = \pi$, then find the minimum value of $\cot^2 A +\cot^2 B + \cot^2 C$. I don't know how to solve it. And can you please mention the used formulas first. What I can see is that if one ...
2
votes
2answers
38 views

How do I find the domain/range of functions algebraically?

I've been having trouble when trying to find the domain/range of functions algebraically. Here is an example: $P(x)=\frac{1}{3+\sqrt{x+1}}$ Finding the domain: $x+1\ge0$ $x\ge-1$ Therefore, $x \...
7
votes
1answer
130 views

Inequality problem, for positive $a,b,c$, if $abc=1$, then $\frac{1}{1+a+b^2}+\frac{1}{1+b+c^2}+\frac{1}{1+c+a^2}\leq1$

I need help or guidance in solving this inequality that I am battling for 3 days now. I have tried everything that comes to mind, but I am stuck. The inequality is as $$\sum_\textrm{cyc}\frac{1}{1+a+...
2
votes
4answers
55 views

Inequality: powers of small numbers

If $\epsilon \approx 0 $ then which one is greater $1-\epsilon^{k}$ or $(1-\epsilon)^{k}$ where $k \in \mathbb{Z}_{> 0}$ is a positive integer.
0
votes
3answers
87 views

Find the minimum of the value $n$ such that $(1-0.03)^n<0.03$

How can I find the smallest positive integer $n$ such that $$(1-0.03)^n<0.03$$ without the help of a computer?
3
votes
1answer
29 views

For positive self adjoint $T$, show $|\langle Tx,y\rangle|^2 \le \langle Tx,x\rangle \langle Ty,y\rangle$

As in title, $T$ is a positive self adjoint, bounded linear operator on a Hilbert Space $X$ and I'd like to show $$|\langle Tx,y\rangle|^2 \le \langle Tx,x\rangle \langle Ty,y\rangle$$ Self adjoint ...
-1
votes
1answer
44 views

Reverse Holder continuity

Consider a function $f(x)$ with a point-wise Holder exponent $\beta \leq 1$. Definition of point-wise Holder exponent: $$ \beta_x: = \sup \left\lbrace \beta: \limsup_{h \rightarrow 0^+} \left|\frac{...
0
votes
1answer
65 views

Prove inequality: $\frac13\left(x^3+y^3+z^3\right)\ge xyz+\frac34|(x-y)(y-z)(z-x)|$

For any nonnegative numbers $x,y,z$ prove inequality: $$\frac13\left(x^3+y^3+z^3\right)\ge xyz+\frac34|(x-y)(y-z)(z-x)|$$ My work so far: I used formulas $x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-...
3
votes
3answers
84 views

Why does $n \geq 2$ imply that $\frac n 2 < n$?

It has been a while since I did math proof in school, and I just can't figure out why $$n \geq 2 \text{ implies that } \frac n 2 < n$$ Anything would help! Thanks.
3
votes
4answers
88 views

Prove $a^2+b^2+c^2 \ge a+b+c$ if $abc=1$, and $a$, $b$, $c$ are positive real numbers

Prove $a^2+b^2+c^2 \ge a+b+c$ if $abc=1$, and $a$, $b$, $c$ are positive real numbers It is in the exercises of the AM-GM inequality chapter of a book, and that is why I believe it will be solved by ...
0
votes
3answers
30 views

Need help showing $(a^p + b^p) \le (a^2 + b^2)^{p/2}$, where $p \ge 2$, and $a,b \ge 0$.

I've been so far trying to show: $(\frac{a^2}{a^2 + b^2})^{p/2} + (\frac{b^2}{a^2 + b^2})^{p/2} \le 1.$ Also, it holds true that $\frac{a^2}{a^2 + b^2} \le 1$ and $\frac{b^2}{a^2+b^2}\le 1.$ I'm ...