Questions on proving, manipulating and applying inequalities.

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0
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6 views

Inequality for the square gradient of a power of absolute value

Let $U \subset \mathbb{R}^2$ be open, and let $f : U \to \mathbb{C}$ be a smooth complex-valued function which does not vanish anywhere on $U$. Let $r > 0$. Does the inequality $$\left| \nabla ...
1
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2answers
21 views

Trouble understanding inequality proved using AM-GM inequality

I am studying this proof from Secrets in Inequalities Vol 1 using the AM-GM inequality to prove this question from the 1998 IMO Shortlist. However, I'm lost on the very first line of the solution. ...
0
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0answers
17 views

An inequality $a_1\leq a_2\leq a_4 , a_1\leq a_3\leq a_4$

$a_1 , a_2, a_3 , a_4 , b_1 , b_2 , b_3 , b_4\in\Bbb R , p\in(0, 1)$. $a_1\leq a_2\leq a_4 , a_1\leq a_3\leq a_4 , b_1\leq b_2\leq b_4 , b_1\leq b_3\leq b_4 $. Show that $$ ...
4
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1answer
41 views

Inequality problem: Application of Cauchy-Schwarz inequality

Let $a,b,c \in (1, \infty)$ such that $ \frac{1}{a} + \frac{1}{b} + \frac{1}{c}=2$. Prove that: $$ \sqrt {a-1} + \sqrt {b-1} + \sqrt {c-1} \leq \sqrt {a+b+c}. $$ This is supposed to be solved using ...
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2answers
48 views

Inequality with Four Numbers

I am trying to prove the following inequality for real numbers $a,b,c,d$ all of which are greater than $1$ $8(abcd+1) > (a+1)(b+1)(c+1)(d+1)$ I tried the following approaches : Used the AM-GM ...
-2
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2answers
20 views

Inequality involving summation

Can someone help me with this inequality: $\sum_{i=1}^{n}{\dfrac{1}{\sqrt i}}\leq \dfrac{2n}{\sqrt{n}}$ Thank you.
2
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1answer
41 views

Problem understanding this specific AM-GM inequality proof

This is taken from Secrets in Inequalities by Pham Kim Hung So, this part of the proof involves proving that $f(n)$ implies $f(n-1)$. So we define a term as $a_n = \frac{s}{n-1}$. We define $s = a_1 ...
1
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1answer
40 views

Find the maximum and minimum of $\sum \limits_{i=1}^n x_i ^3$

Let $x_1,x_2, \dots ,x_n$ be a sequence of integers such that $i) -1\le x_i\le 2$ for $i=1,2,\dots,n$ $ii)x_1+x_2+\dots+x_n=19$ $iii){x_1}^2+{x_2}^2+\dots +{x_n}^2=99$ Determine the minimum and ...
0
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0answers
19 views

From inequality on derivatives to inequality on functions

What is the set of differentiable functions $f$ that satisfy the following inequalities for all $x\geq 0$: $0\leq f'(x)\leq e^{-x}$ Initially, I thought I should just integrate the inequality and ...
-1
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0answers
33 views

What is the maximum value of $M$ when $T$ is set of $\{2,4,8,16,… 2^n\}$ and $S$ is subset of $T$ by given conditions

Qns $T$ is set of $\{2,4,8,16,... 2^n\}$ and $S$ is a subset of $T$ if the sum of no two elements of $S$ is greater than $2^n-2$. let $m$ be $M$ number of elements in $S$. what is ...
2
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2answers
49 views

Find min of $M=\frac{1}{2+\cos2A}+\frac{1}{2+\cos2B}+\frac{1}{2-\cos2C}$

Find min of $$M=\frac{1}{2+\cos2A}+\frac{1}{2+\cos2B}+\frac{1}{2-\cos2C}$$, where $A, B, C$ are three angle of triangle $ABC$ Using Cauchy-Schwarz, we obtain: \begin{align*} M &= ...
3
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0answers
30 views

Inequality - Why do I not check for these other solutions?

Given, for example: $$ \Big(\frac{2x}{x-2}\Big)^{3x^2-x} \leq \Big(\frac{2x}{x-2}\Big)^{x^2+3x+6} $$ After checking when $x-2 \ne 0$ , the teacher taught us to check 3 cases: 1. $ ...
0
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0answers
16 views

Absolute Value Inequality of Differences

I'm hoping someone could give insight as to how I can improve my organization, and/or thought process. Show that $|a-b| \lt c$ if and only if $b -c \lt a \lt b + c$. By the statement $b - c \lt a ...
1
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6answers
119 views

Prove $((a+b)/2)^n\leq (a^n+b^n)/2$

Struggling with this proof. Prove that $$\left(\frac{a+b}{2}\right)^n≤\frac{a^n+b^n}{2},$$ where $a$ and $b$ are real numbers such that $a+b≥0$ and $n$ is a positive integer. What technique would ...
2
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3answers
67 views

Prove the equation has unique class of solutions

Find the solutions of equation: $$ x^y + y^x = 1 + xy \quad x,y \in \mathbb{R} \quad x,y >0 $$ My quest First, $x=1$ or $y=1$ gives us obvious solutions, so let's suppose $x \not =1$ and $y \not= ...
0
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0answers
7 views

How can a nonconvex polytope be defined (not by an LMI)?

A convex polytope can be defined by an LMI (linear matrix inequality) or a list of points. How can a nonconvex polytope be defined?
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0answers
30 views

Absolute Value Inequality Proof

I realize this is almost identical to another question I posted, but I wanted to ask what the distinction between the two is -- comprehension-wise (other than the $\lt$ vs. $\le$). Show that $|b| ...
11
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0answers
171 views

Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator

Prove that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator. I did in the following way. Are there other ways? Proof : Let $f(x)=e\pi\frac{\ln x}{x}$. Then, ...
1
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4answers
37 views

Prove that triangle inequality $|a + b| \le |a| + |b|$ holds when $(a + b) \ge 0, a \ge 0, b < 0$

This is what I did: $a + b \ge 0 \rightarrow |a + b| = a + b$ $a \ge 0 \rightarrow |a| = a$ $a + b \le a + |b|$ $b \le |b|$ Which is true $\forall b$. Is this a formal enough way of proving ...
0
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1answer
22 views

Upper bound of the function

here you can read my first question on this topic, namely: $$\text{if } f\left(\frac{x}{3}\right)-f\left(\frac{x}{4}\right)\le Ax+B\ln x-C, $$ where $f(x)$ is my function and $A$,$B$,$C$ are ...
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4answers
35 views

Epsilon-Delta Limit Proof: Arccos(x) Inequalitiy

I'm studying a Calculus proof using notes (proving that $\lim_{x \to 1} \cos(x) = \cos(1)$ from the definition of limit). The text says that we get from: $\cos(1) −\epsilon < \cos(x) < ...
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2answers
25 views

Can we relax the triangle inequality for $\| v \|$ = $\|v - v_0 + v_0\|$?

Given some vector $v$ on vector space $X$ with a norm $\| \cdot \|$ Then $\| v \|$ = $\|v - v_0 + v_0\|$ where $v_0$ is some other vector is it legal to then write $\| v - v_0 + v_0 \| = \|v -v_0\| ...
0
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1answer
39 views

Jensen's inequality problem [on hold]

I want to know an example of a infinite measure space $(\Omega, \mathcal{F},\mu)$, real valued function $g$ and convex function $\phi$ defined on the real line s.t. $$\phi\left(\int g d\mu\right) > ...
3
votes
1answer
51 views

Russian MO 2004 Question involving the AM-GM inequality

I'm reading Secrets in Inequalities by Pham Kim Hung, and I'm having trouble understanding this proof from a problem from the 2004 Russian MO. Let a,b,c be positive real numbers and $a + b +c = 3$. ...
2
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2answers
36 views

Rearrangement and Cauchy

Let $a_1, \ldots, a_n$ be distinct positive integers. I want to prove that $$\frac{a_1}{1^2} + \frac{a_2}{2^2} + \cdots + \frac{a_n}{n^2} \geq \frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{n}.$$ ...
0
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1answer
57 views

Find the minimum value of $P=\frac{1}{2-x}+\frac{1}{2-y}+\frac{1}{2-z}$

Let $x,y,z$ be positive real numbers such that $x^3+y^3+z^3=3$. Find the minimum value of $$P=\frac{1}{2-x}+\frac{1}{2-y}+\frac{1}{2-z}.$$ I think that we need to show that $\dfrac{1}{2-x} \ge ...
0
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1answer
39 views

How to prove triangle inequality in How to Prove It Sec. 3.5 Question 12c?

(a) Prove that for all real numbers $a$ and $b$, $$|a| \le b \text{ iff } -b \le a \le b.$$ (b) Prove that for any real number $x$, $$-|x| \le x \le |x|.$$ (Hint: Use part (a).) (c) Prove that ...
2
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2answers
26 views

Simplifying a complication max operation

I have dervied an inequality and have arrived to the following $$\max\{1, \frac{b}{2}+1\} \leq \max\{a, \frac{b}{2}+ \frac{a}{2}\}$$ I am trying to simplify further and arrived to the following ...
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3answers
84 views

$\frac {1}{2n+1}<\frac {1}{n^2+1}+\frac {1}{n^2+2}+\cdots+\frac {1}{n^2+n}<\frac {1}{2n}$

Given that $n \in \mathbf Z$ and also $n\ge 2$ show that - $$\frac {1}{2n+1}<\frac {1}{n^2+1}+\frac {1}{n^2+2}+\cdots+\frac {1}{n^2+n}<\frac {1}{2n}$$ No idea about this. Please help.
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0answers
12 views

Showing a bound exists

I was able to derive the following differential equations I have to work with for a function $V$: $$ \begin{align*} dV(x_1,x_2,x_3,x_4) &= ...
3
votes
3answers
35 views

An upper bound for a function

I am trying to find an upper bound $b\ge f(x)~\forall x\ge0$ for the following function: $$f(x)=\frac{x}{(w+ux^2)^2},$$ where $w,u>0$ are parameter values. I am interested in the positive domain ...
0
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4answers
45 views

Rearrange the inequalities

I have the two inequalities: $$ f\left(\frac{x}{3}\right)-f\left(\frac{x}{4}\right)\le Ax+B\ln x-C $$ $$ f\left(\frac{x}{3}\right)-f\left(\frac{x}{4}\right)+f\left(\frac{x}{6}\right)\ge Dx-B\ln x+E ...
3
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1answer
52 views

Prove that $\max a_i \le 4 \min a_i$

Let $a_1,...,a_n$ be given positive reals, such that: $$\sum a_i \times \sum \frac1{a_i} \le (n + \frac12)^2$$ Prove that $\max \{a_i\} \le 4 \min \{a_i\}$ I don't know exactly how to approach ...
0
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1answer
24 views

simplifying a max operation

In some derivation I arrived to I had reached the following assuming $\alpha,\beta >0$ $$\max\{1,1+\beta \}\leq \max\{\alpha,\alpha+\beta\}$$ Obviously the following condition can be deduced ...
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1answer
63 views

For nonnegative numbers $x + y + z = \pi$, prove that $1 \le \cos x+\cos y+\cos z \le \dfrac 3 2 $

Let $x,y,z$ be nonnegative real numbers and $x+y+z=\pi$.Prove the inequality $$1 \le \cos x+\cos y+\cos z \le \dfrac 3 2 $$ I tried to put $z=\pi-x-y$ and then calculate the extremas of two ...
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2answers
24 views

How do you find redundant constraints for a feasible region?

I've found a few papers that deal with removing redundant inequality constraints for linear programs, but I'm only trying to find the non-redundant constraints that define a feasible region (i.e. I ...
1
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2answers
50 views

Proof that Right hand and Left hand derivatives always exist for convex functions.

Definition A function $f$ is convex on an interval if for $a,x, \text{and} \;b$ in the interval with $a\lt x\lt b$, we have $$\frac{f(x)-f(a)}{x-a}\lt \frac{f(b)-f(a)}{b-a}.$$ While reading the ...
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1answer
25 views

Finding a function such that $(n-2)/2 + f(n-1) \leq f(n)$

By bounding a certain quantity defined on real numbers by $f(n)$ I derived the following inequality arising from an inductive argument. $ (n-2)/2 + f(n-1) \leq f(n).$ A solution to the above ...
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0answers
11 views
+300

Proof of multivariate regression plane maximizes correlation in normals

I am doing a homework sheet as practice for an upcoming course in multivariate statistics and been stuck on the following problem: Let ...
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1answer
38 views

$t\in (0,1)$ and $f(tx+(1-t)y)\le tf(x)+(1-t)f(y)$. Show that if strict inequality holds for even one $t$, then it holds for all $t$.

This is a part of a solution to a problem in showing that if $f$ is continuous and satisfies the condition $f([x+y]/2)\lt [f(x)+f(y)]/2$, then $f$ is convex. Let $t\in (0,1)$. We have the weak ...
4
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2answers
99 views

Given $a, b,c,d$ non-negative and $a+b+c+d=4$. Prove that $a^3b+b^3c+c^3d+d^3a+23abcd \le 27.$

This is a problem on Mathlinks.ro and it have had no solution. So I hope it would have a nice answer and as simply as possible. Given $a, b,c,d$ are non-negative numbers and $a+b+c+d=4$. Prove ...
0
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4answers
87 views

About the limit $\lim_{n\to +\infty}\frac{n^k}{n!}$ for a fixed $k\in\mathbb{N}$.

Given a natural number $k$ and some real number $\epsilon>0$, I have to prove that there exists a natural number $n$ such that $\frac{n^k}{n!}<\varepsilon$. I tried to develop for ...
1
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4answers
67 views

Prove that $\left|(|x|-|y|)\right|\leq|x-y|$

Prove that $\left|(|x|-|y|)\right|\leq|x-y|$ Proof: $$\begin{align} \left|(|x|-|y|)\right| &\leq|x-y| \\ {\left|\sqrt{x^2}-\sqrt{y^2}\right|}&\leq \sqrt{(x-y)^2} ...
0
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3answers
47 views

Why is this implication true?

$\dfrac{4x^2}{(1+x^2)^4} < 1 \to \dfrac{2\mid x \mid}{(1+x^2)^2} < 1$ . This is from a textbook i'm using (Advanced Engineering Mathematics, 10th ed., Kreyszig). It looks like the left hand side ...
1
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1answer
34 views

Simple Inequality for Proving Equivalent Besov Seminorms

For $f\in L^{p}(\mathbb{R}^{n})$, $1\leq p<\infty$, and $h\in\mathbb{R}^{n}$, define the quantity $$I_{p}(h):=\left(\int_{\mathbb{R}^{n}}\left|f(x+h)-f(x)\right|^{p}dx\right)^{1/p}$$ and define ...
18
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11answers
619 views

Which is greater, $98^{99} $ or $ 99^{98}$?

Which is greater, $98^{99} $ or $ 99^{98}$? What is the easiest method to do this which can be explained to someone in junior school i.e. without using log tables. I don't think there is an ...
1
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2answers
73 views

How to find the minimum of the $a^2+b^2+c^2+a+b+c+ab+bc$

(1): Let $a,b\in R$, Find this minimum of the value $a^2+b^2+ab+a+b$ I have prove $a^2+b^2+ab+a+b\ge-\dfrac{1}{3}$ (2): Let $a,b,c\in R$, find the minimum of the value $$a^2+b^2+c^2+a+b+c+ab+bc$$ I ...
1
vote
0answers
30 views

Proof of Gruss inequality

I've been reading articles that use the Gruss inequality for some time now, but I can't seem to find a proof of it anywhere. The only source I could find that actually has the proof is the original ...
1
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1answer
43 views

Prove that if $|a_1\sin x+a_2\sin2x+a_3\sin3x+\cdots+a_n\sin nx|\leq|\sin x|$ for $x\in R,$then $|a_1+2a_2+3a_3+…+na_n|\leq1$

Prove that if $|a_1\sin x+a_2\sin2x+a_3\sin3x+\cdots+a_n\sin nx|\leq|\sin x|$ for $x\in R,$then $|a_1+2a_2+3a_3+\cdots+na_n|\leq1$ When we try to differentiate it on both sides wrt $x$,then modulus ...
1
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1answer
47 views

Proving inequality involving convex combination

What are the steps in proving this following inequality: for $p,q \in [0,1]$: $$ p \log \frac{p}{q}+(1-p) \log \frac{1-p}{1-q} \ge 2(p-q)^2 $$ I'm just looking for an outline of the approach to tackle ...