Questions on proving and manipulating inequalities.

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15 views

How can I find condidions when one sum is greater than the other?

I have the following two sums: $$ \sum_{j\neq i} Q_{ij}u_i\int g(z, k) f(z; i, j) \textrm{d}z $$ and $$ \sum_{j\neq i} Q_{ij}u_j\int g(z, k) f(z; i, j) \textrm{d}z, $$ where $k$ is a parameter and the ...
6
votes
2answers
70 views

Looking for a “job description” for Hölder's inequality

Here's an example of what I mean by "job description" in the post's title: triangle inequality: to be used, whenever the (unsigned) distances between adjacent points in a sequence $x_0, x_1, x_2, ...
3
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1answer
54 views

A possible inequality related to binomial theorem (or, convex/concave functions)

Let $x, \ y, \ p$ be any real numbers with $x>0$, $y>0$, and $p>1$. The question is about (most probably) an elementary inequality: Is it always true that $x^p+y^p\leq (x+y)^p$ ? Note ...
0
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2answers
49 views

Find all $\alpha$ such that for any $x>-1$ we have $\ln(1+x)\leq x-\frac{x^2}{2}+\alpha x^3$

Here's a small problem I'm trying to solve: Find all $\alpha$ such that for any $x>-1$ we have $\ln(1+x)\leq x-\frac{x^2}{2}+\alpha x^3$ After moving some things to the left side we have: ...
2
votes
1answer
31 views

Finding the maximum of sum of coefficients of a polynomial

Suppose $p(x)=ax^2+bx+c$ is a quadratic polynomial with real coefficients and $|p(x)| \leq 1$ for all values of $x$ in the range $[0,1]$. Prove that maximum possible value of $|a|+|b|+|c|$ is $17$. ...
-3
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1answer
63 views

Proof of $\dfrac{a}{b+2c}+\dfrac{b}{c+2a}+\dfrac{c}{a+2b}\ge 1$ if $a, b, c\in \mathbb{R_+}$ [closed]

$a$, $b$, $c$ are all positive real numbers. Show that: $\dfrac{a}{b+2c}+\dfrac{b}{c+2a}+\dfrac{c}{a+2b}\ge 1$. Thank you.
3
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1answer
52 views

Absolute Value of Complex Integral

Let $[a,b]$ be a closed real interval. Let $f:[a,b] \to \mathbb{C}$ be a continuous complex-valued function. Then $$\bigg|\int_{a}^{b} f(t)dt \ \bigg| \leq \int_{a}^{b} \bigg|f(t)\bigg| dt,$$ where ...
1
vote
2answers
73 views

$x^9 - 2x^7 + 1 > 0$

$x^9 - 2x^7 + 1 > 0$ Solve in real numbers. How would I do this without a graphing calculator or any graphing application? I only see a $(x-1)$ root and nothing else, can't really factor an ...
1
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0answers
39 views

How to prove that the absolute value of the second derivative of J_0(x) is bounded?

$\lvert J_0^{''}(x)\rvert < \dfrac {1} {2}$ How do I prove the above inequation? From the above graphic it is clear that $\lvert J_0^{''}(0)\rvert=0.5$ is the maximum that the function can ...
0
votes
3answers
49 views

Upper and lower bounds for $\log(\frac{n}{n-1})$

If we have a value $x>1$ and a constant $c$ then by making $k=O(\log x)$ we have $$x\cdot (1-c)^k \leq 1$$ Now, if $c$ is not a constant, but a function such as 1/n, what is the smallest k for ...
2
votes
1answer
72 views

How could I continue to show the inequality?

Let $g: [0, \pi]\rightarrow \mathbb{R}$ a $C^{\infty}$ function for which the following stands: $$g(0)=0 \ \ , \ \ g(\pi)=0$$ I have to show that $$\int_0^{\pi}g^2(x)dx \leq ...
0
votes
1answer
39 views

Determine where the hessian matrix is indefinite for $f(x,y) = x^y$

I need to find the area in which the hessian matrix for $f(x,y) = x^y, x > 0$ is indefinite. I did so: 1) $\begin{pmatrix} y\cdot(y-1)\cdot x^{y-2} & x^{y-1} \cdot (1+y\cdot ln(x)) \\ x^{y-1} ...
1
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0answers
44 views

How prove $\frac{a^2}{(a+b)^2}+\frac{b^2}{(b+c)^2}+\frac{c^2}{(c+a)^2} \ge \frac{3}{4}+\frac{(a-b)(b-c)(a-c)}{(a+b+c)^3-3abc} $?

Let $a \ge b \ge c >0$ . How prove $\frac{a^2}{(a+b)^2}+\frac{b^2}{(b+c)^2}+\frac{c^2}{(c+a)^2} \ge \frac{3}{4}+\frac{(a-b)(b-c)(a-c)}{(a+b+c)^3-3abc} $? Maby simple way?
0
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0answers
17 views

I always have some doubts regarding the inequalities in cases where the function become Complex in the field for the real numbers

Consider this inequality $x + \log\left(x \right)> \log\left (x\right) - 2$ Does this inequality has $-1$ as its solution ? It will be very helpful for me.
1
vote
1answer
132 views

How to solve the exponential inequality $x+3^x<4$

How to solve the inequality $$x+3^x<4$$ This problem is found in Spivak's calculus, ch 1 - the highly praised work - which is supposed to be a gentle introduction for beginners in mathematics. ...
2
votes
2answers
49 views

How to solve inequality of the form $(x-a)(x-b)(x-c)\ge 0$?

How to solve an inequality of the form $$f(x)=(x-a)(x-b)(x-c)\ge 0$$ for $$ a,b,c,x,f(x) \in \mathbb R $$ WITHOUT testing if an $f(x)$ within an interval between the roots is actually bigger or lesser ...
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2answers
68 views

Prove that if $\sum \limits_{k=1}^{n} x_n=1$ then $\sum \limits_{k=1}^{n} x_n^2 \geq {1 \over n}$

Prove that if $\sum \limits_{k=1}^{n} x_n=1$ then $\sum \limits_{k=1}^{n} x_n^2 \geq {1 \over n}$ where $\{x_k\}_1^n$ are real numbers which are not all the same. I tried to prove it by induction. ...
0
votes
2answers
50 views

prove inequality $\sum_{i=0}^{\infty} \frac{1}{(i+n)^2} < \int_0^{\infty} \frac{1}{(x+n)^2}dx$

How can I prove that $$\sum_{i=0}^{\infty} \frac{1}{(i+n)^2} < \int_0^{\infty} \frac{1}{(x+n)^2}dx$$ where n is a natural number? I mean, intuitive seems obvious, since the terms inside the ...
4
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1answer
29 views

A Question about Nested Maximizations

I am working on labor demand models where firms have to choose the optimal level of employment by maximizing profits. In particular, I have faced the following problem: Maximize with respect to $l$ ...
0
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1answer
56 views

Proving uniqueness of limits

IF: given $r$ there exist $k,m$ such that: for all $a>0$ there exist $b>0,c>0$ such that: $$\begin{align*}0<|x-r|<b&\implies|A(x)-k\,\,|<a\\ ...
2
votes
2answers
39 views

Inequality with Algebra $\sup_{x,y,z\in A}\left(\left(13x+\frac{5}{x}\right)+\left(13y+\frac{5}{y}\right)+\left(13z+\frac{5}{z}\right)\right)= 63$

let $A=\{(x,y,z)|x,y,z\in [\frac{1}{2},2],xyz=1\}$ Maybe have $$f=\sup_{x,y,z\in A}\left(\left(13x+\dfrac{5}{x}\right)+\left(13y+\dfrac{5}{y}\right)+\left(13z+\dfrac{5}{z}\right)\right)= 63?$$ Here ...
5
votes
1answer
61 views

if $a+b+c\le 1$ prove that $3(a+b+c)-(a^2+b^2+c^2-ab-bc-ac)\ge (\sqrt{a}+\sqrt{b}+\sqrt{c})^2$

let $a,b,c\ge 0$ such $a+b+c\le 1$,prove $$3(a+b+c)-(a^2+b^2+c^2-ab-bc-ac)\ge (\sqrt{a}+\sqrt{b}+\sqrt{c})^2\tag{1}$$ I conjecture: Let $a_{i}\ge 0,i=1,2,\cdots,a_{1}+a_{2}+\cdots+a_{n}\le ...
0
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0answers
24 views

Inequality, derivates

$f,g:[0,\infty)$ functions $f(0) \lt g(0)$ and $f'(x) \lt g'(x)$ for any $x\in [0,\infty)$. Does this means that $f(x) \lt g(x)$ for any $x\in [0,\infty)$? And can I use this without proof? I need it ...
2
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1answer
32 views

inequality problem in Spivak Calculus

Can somebody please explain why it immediately follows that for $$ (x-1)^2+1>0 $$ all x are admissible? (This problems is found in Spivak's Calculus, 3rd edition, Chapter 4, problem (v) )
0
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1answer
21 views

Power series expansion requirements

Hello stackexchange folks :) I have a question regarding the assumptions made right before you choose to expand or approximate a function by a power series. Specifically I have the function: ...
0
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1answer
17 views

Small inequality

Proposition: Let $x_1,x_2,x_3$ and $a_1,a_2,a_3$ be real numbers and $x_1+x_2+x_3=0$. Prove that $a_1x_1+a_2x_2+a_3x_3=0$. My solution: Case 1: If $x_1,x_2,x_3\geq0$, then ...
5
votes
1answer
85 views

Counterexamples to the Matrix norm AM-GM inequality?

I am new here and this my first question, I hope I am being as clear as possible and apologize in advance for any misunderstandings. I am researching the Arithmetic-Geometric Mean (AM-GM) inequality ...
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3answers
49 views

$a^2+b^2+c^2=(abc)^2-2\leq 6$. Proof or counter-example needed for $a,b,c\gt 0$

I was working on an inequality proof in which I need to use the following inequality to conclude. $$\forall~a,b,c\gt 0~,~a^2+b^2+c^2=(abc)^2-2\implies a^2+b^2+c^2\leq 6$$ I can't think of any ...
5
votes
1answer
91 views

Prove that: $2(ab+bc+ca)-a^2-b^2-c^2\le6$.

Let $a,b,c>0$ such that: $\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}+\dfrac{1}{c^2+1}=1$. Prove that: $2(ab+bc+ca)-a^2-b^2-c^2\le6$. I have no idea for solve this problem.
2
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3answers
52 views

Prove $a^\alpha b^\beta + c^\alpha d^\beta \leq (a+c)^\alpha (b+d)^\beta$

I'm trying to prove that the sum of two log-convex functions is log-convex. I've figured out that this can be done by proving: $a^\alpha b^\beta + c^\alpha d^\beta \leq (a+c)^\alpha (b+d)^\beta$ for ...
0
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2answers
58 views

How to determine that if $\frac{n}{\sqrt{n^3+2}}>\frac{n+1}{\sqrt{(n+1)^3+2}}$ for any n is natural number

How to prove that $\frac{n}{\sqrt{n^3+2}}>\frac{n+1}{\sqrt{(n+1)^3+2}}$ for any n is natural number I have that problem when tried to determine an alternating series conditional converge or not. ...
0
votes
2answers
28 views

Prove that $-x^2 \leq x^n \leq x^2$ for $-1<x<1, n\in \mathbb N, n \geq3$

Prove that $-x^2 \leq x^n \leq x^2$ for $-1<x<1, n\in \mathbb N, n \geq3$ I have no idea how to do this, I don't even know how to begin. Please help!
12
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3answers
64 views

Prove that $4x^2-8xy+5y^2\geq0$ - is this a valid proof?

I need to prove that $4x^2-8xy+5y^2\geq0$ holds for every real numbers $x, y$. First I start with another inequality, i.e. $4x^2-8xy+4y^2\geq0$, which clearly holds as it can be factorized into ...
0
votes
1answer
23 views

Invalid range from inequality

We were given this function and asked to give Range. $$f(x)~=~\dfrac{x^2}{x^2+1}$$ Now I took 3 cases and deduced that $\text{Range} = \left[~0,\infty ~\right)$ Now it is obvious that if we divide ...
0
votes
1answer
55 views

Prove the inequality, $\root3\of4\sin^2(x/2)<3(\sin x+1-x)^{2/3}$

Prove that $$\left(\sin^2{\frac{x}{2}}\right) \cdot \frac{\sqrt[3]{4}}{3} \cdot \frac{1}{{(\sin x + 1 - x})^{\frac{2}{3}}} <1$$
2
votes
3answers
50 views

Power function of fixed numbers.

Prove that $3^x-4^x+2x4^{x-1}\le0$, where $x\in[-0.5,0]$.Here is it's plot. I tried to do it by first and second derivative test but it involves $log$ which make the expression more complicated.
3
votes
1answer
61 views

Estimate the integral of $(1+x^2)^{-\alpha}$, where $\alpha>1/2$

I'm reading a proof of a theorem, and there's one step I couldn't understand why. It said that for all $a>0$ and $\alpha>1/2$, $$ \int_{a}^{\infty}(1+x^2)^{-\alpha} \ \mathrm dx ...
1
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1answer
23 views

Is this inequality of real numbers true?

Let $\alpha\in (0,1/2)$ be a parameter. Is it true thet for every $x>y>0$ real numbers we have $$y^{-\alpha} - x^{-\alpha} \leq C y^{-\alpha -\frac{1}{2}} (x-y)^{1/2}$$ for some constant ...
2
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0answers
22 views

Obtain an inequality of real numbers

Let $x,y> 0$ be real numbers such that $x>y$. Let $\alpha \in (0,1/2)$ be a parameter then I obtained the following inequality: $$y^{-\alpha} - x^{-\alpha} \leq C ...
1
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2answers
67 views

About primes and Euler's totient function.

Is the number of primes $< n$ itself less than the number of positive integers that are less than $n$ and relatively prime to $n$?
1
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2answers
61 views

About an inequality using $\ln \ln n$ related to Robin's inequality.

Is $\ln\ln n < \sigma(n)/n$, where $\sigma(n)$ is the sum of the divisors of $n$? Or is this grossly innaccurate or too vague? (The comments about blocking my questions is totally unfair. I ...
0
votes
0answers
26 views

A Integral inequality.

For any positive integer $n \in {\mathbb{N}^ + }$, prove inequality $$\int_{ - \pi }^\pi {\left| {\cos \left( {\frac{{2n + 1}}{2}t} \right)} \right| \cdot \left| {\frac{1}{{\sin \left( {\frac{t}{2} + ...
4
votes
5answers
139 views

Inequality in Algebra: $1 \leq x_1 x_2 \cdots x_n$ implies that $2^{n} \leq (1 + x_1)(1+x_2) \cdots (1 + x_n).$

How do I prove that if $x_1, \ldots, x_n$ are positive real numbers, then $$1 \leq x_1 x_2 \cdots x_n \text{ implies that } 2^{n} \leq (1 + x_1)(1+x_2) \cdots (1 + x_n).$$ I attempted a proof by ...
2
votes
4answers
44 views

Domain of the function $f(x) = \sqrt{\frac{3^x-4^x}{x^2-4x-4}}$ will be?

I tried solving this question by $1.$ $-1$ and $4$ will not be in domain because denominator can not be zero . $2.$ Either both denominator and numerator will be positive or negative so that whole ...
2
votes
0answers
19 views

Determining asymptotics of a function given a series of difference-like inequalities

I have a function $f: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ and I know it satisfies the following properties. $f(x) \leq \frac{\log{\sqrt{2}}}{2x}$ and for all $A \geq 1$ and $B \geq ...
1
vote
0answers
41 views

Measure of set satisfying acute version of AM/GM

Let $N>2$ be integer. Take a $N$ real numbers $x[k] \in [0,1]$. By the AMGM inequality we have that their geometric mean is less or equal to their sum divided by $N$. My question is: what is the ...
-1
votes
1answer
40 views

Is $\pi(y)\pi(x+k)\ge\pi(x)\pi(y+k)$? [closed]

Is it true that for some fixed $k\ge2$ and for all sufficiently large $x$ and $y$ with $y\ge x$ we have, $$\pi(y)\pi(x+k)\ge\pi(x)\pi(y+k)$$ where $\pi(x)$ is the prime counting function. I am ...
3
votes
6answers
64 views

Prove that $2^n(n!)^2 \leq (2n)!$

Prove that $2^n(n!)^2 \leq (2n)!$ One can also use the following result to prove the above: $2 · 6 · 10 · 14 · · · · · (4n − 2) = \frac{(2n)!}{ n!}$. The above relation gives, $(2n)!=2^n n! ...
12
votes
2answers
242 views

How to prove that $\frac{a}{\sqrt{a+b}}+\frac{b}{\sqrt{b+c}}+\frac{c}{\sqrt{c+a}}\leq\frac{3\sqrt{3}}{4}$?

Let $a,b,c>0: (a+b)(b+c)(c+a)=ab+bc+ca$. How to prove that $$\frac{a}{\sqrt{a+b}}+\frac{b}{\sqrt{b+c}}+\frac{c}{\sqrt{c+a}}\leq\frac{3\sqrt{3}}{4}$$
1
vote
1answer
47 views

Complex Number inqualities

Although the inequalities are not defined on complex numbers. But does the inequality $x < 4 + 5i$ be said to possess any solutions ? Where $ i = \sqrt{-1}$.