Questions on proving and manipulating inequalities.

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Determining the best possible constant $k$, for an Integral Inequality

If $f : [0,\infty) \to [0,\infty)$ is an integrable function, then what is the best possible constant $k$, for which the following ineqality holds: $$\int_0^{\infty}f(x)dx \leq ...
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1answer
47 views

Inequality using AM-GM

$$ \frac ab + \frac bc + \frac ca \geq a + b + c $$ where $abc = 1$ Using AM-GM I can get $$ \frac ab + \frac bc +\frac ca + ab + ac + bc \geq 2(a + b + c), $$ but I can't show $ab + bc + ac \leq a ...
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79 views

How prove this inequality $f(x)\ge f(0)$

Question: let $a>b>c>0,n\in N^{+},n\ge 2$ is give numbers,show that: ...
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1answer
49 views

Typo in Spivak's explanation of limits in Calculus?

Here's what he says (including the preceding paragraph): "To show in general that f [(where f(x)=1/x)] approaches 1/a near a for any a we proceed in basically the same way, except that, again, we ...
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1answer
45 views

Do there exist two vectors in a Hilbert space such that $(x,y)\geqslant k\|x-y\|^{-2}$?

Let $H$ be a Hilbert space, $(x,y)$ denote the inner product of the elements $x,y\in H$, $\|x\|$ denote the norm of $x\in H$, and $k>0$. Do there exist such $x,y\in H$ that $$ ...
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1answer
17 views

Relation between the mean value inequality over an area and over a surface

Suppose that $f$ is a locally integrable function on $\mathbb{R}^{N}$ $(N\geq2)$ such that for all $x$ in $\mathbb{R}^{N}$ and all positive real number $r$ we have \begin{equation} f(x)\leq ...
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2answers
44 views

If $x,y \in (0,\frac{\pi}{2})$ then expression $\sin x +\cos y +\tan^2y+\cot^2x+5>\ldots?$

Problem : If $x,y \in (0,\frac{\pi}{2})$ then expression $\sin x +\cos y +\tan^2y+\cot^2x+5$ is always greater than : (a) $\ 7 $ (b) $\ 8 $ (c) $\ 9 $ (d) $\ $none of these Solution : We ...
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1answer
31 views

An angular inequality

In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on ...
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2answers
79 views

Minimum of $|\det(X+iC)|$

Let $C$ be a fixed real $n\times n$ matrix, $X$ be an arbitrary real $n\times n$ matrix. Find the minimum value of: $$|\det(X+iC)|=\sqrt{\det(X+iC)\det(X-iC)}$$ When $n=1$ it's clear that the ...
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2answers
34 views

Is $3(2k+1)(2^{2k+1}-1)>(2^{k+3}-1)(2^{k+1}-1)$?

Let $k$ be an integer. I need to prove that: $$3(2k+1)(2^{2k+1}-1)>(2^{k+3}-1)(2^{k+2}-1)$$ where $k>a$ for a suitable $a$. thanks in advance.
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How find this minimum of the possible value $\max_{1\le j\le n}{\left(\displaystyle\sum_{i=1}^{35}x_{i,j}\right)}$

give the positive integer $n\in N^{+}$, and $x_{i,j}(1\le i\le 35,1\le j\le n)$, such $$\begin{cases} x_{i,j}=0 (or )1&1\le i\le 35,1\le j\le n\\ \displaystyle\sum_{j=1}^{n}x_{i,j}=27& 1\le ...
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3answers
108 views

Is $-|x|\le\sin x\le|x|$ for all $x$ true?

I have seen in Thomas' Calculus that says to prove $\lim_{x\rightarrow0}\sin x=0$, use the Sandwich Theorem and the inequality $-|x|\le\sin x\le|x|$ for all $x$. My question is how could the ...
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1answer
34 views

Proving an inequality about a $\sin x$ and $\exp x$ [on hold]

Show that $$\frac{\sqrt 2}{2}\ \sin x\geq e^x-1$$ for any $x\leq 0$.
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83 views

Polynomial P(x) such that [on hold]

Let $P(x)$ be a real polynomial with degree $n$ such that $|P(x)| \lt 1$ for all $|x| \le 1$. Prove that $P(2) \lt 4^n$.
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1answer
29 views

Proof About the Product of Two Integers

I reading about of proof of the claim "If $a \ge 0$ and $b > 0$, then $a \le ab$. The proof the author is employing is inductive. I understand the basis case; however, I do not understand the proof ...
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1answer
40 views

Generalization of Bernoulli's Inequality

Is it possible to generalize Bernoulli's Inequality to $(x+y)^n \geq x + ny$ provided $x+y \geq 0 $ and $x \geq 1$ and $n$ is a positive natural number? I was thinking that the proof follows by ...
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3answers
56 views

Proving an inequality about a sequnce with Cauchy-Schwarz

show that $$\sum\limits_{i=1}^n \frac{x_i}{i^2} \geq \frac{1}{1} + \frac{1}{2} + \dots +\frac{1}{n}$$ where $x_1,x_2,\dots,x_n$ are natural numbers and all of them are different numbers(no such a ...
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1answer
68 views

An inequality about a sequence

Let $(a_n)$ be a sequence such that $a_0=1 , a_1=2 , a_{n+1}=a_n+\dfrac {a_{n-1}}{1+ a_{n-1}^2} , \forall n \ge1 $ , then is it true that $52 < a_{1371} < 65$ ?
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1answer
22 views

Limits for expected value in a proof

I have a small step in a proof, that I'm not sure if I got it right. We have given the function $f(s):=\mathbb{E}[e^{\lambda S (s-1)}]$ where $S$ is a random variable such that: ...
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18 views

Minimum in complex inner product vector space

I'm stuck at this problem, can someone give me a hint? Let $x_i$ and $y_i$ ($i=\overline{1,n}$) be vectors in an infinite dimensional vector space $V$ with inner product $(,)$ satisfy: ...
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1answer
62 views

Prove $\sum\limits_{\mathrm{cyc}}\sqrt{a^2+bc}\leq{3\over2}(a+b+c)$ with $a,b,c$ are nonnegative

Hope someone can help on this inequality using nonanalytical method (i.e. simple elementary method leveraging basic inequalities are prefered). Prove ...
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1answer
39 views

cauchy schwarz inequality extreme

cauchy schwarz inequality states that: (case of real numbers) $$ \left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right) $$ and we ...
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0answers
14 views

Max Earning Probability of the Portfolios [on hold]

I have a problem about deriving Chebyshev Inequality From A.D. Roy Safety first Portfolios model He would like to Min Prob(Rp≤RL)≤α It's meant u want min probability of Rp less equal than -5% ...
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3answers
39 views

Prove that $2n+1 \leq 2^n$ for $n \geq 3$ using mathematical induction.

Question: $2n+1 \leq 2^n$, for all $n \geq 3$ I've tried: Basis: $P(3) = 7 \leq 8 $, so basis step is valid Pick an arbitrary value from the universe, $k \geq 3$ Inductive Step: $2k + 1 \leq ...
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1answer
23 views

An application of Holder's inequality to show one norm is smaller than another

Let $p(s) = r(s) + m-1$ where $r:[0,T) \to [q,\infty)$ where $q \geq 2$ and $m > 1$ is fixed. Let $\text{Vol}(\Omega) = 1$. Then can we show that $$\lVert u \rVert_{L^{r(s)}(\Omega)} \leq ...
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1answer
52 views

determing constant in inequality with nonnegative numbers

Let $ r \geq 1$ be an integer. Prove that there exists a constant $ C_r = C(r)>0$ such that for any non-negative real numbers $ a_1, a_2, \cdots, a_n \in [0, \infty)$ the following inequality ...
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1answer
35 views

Given $a_{m*n} \leq a_m + a_n$, show that there exists $C$ such that $a_n \leq C log(n)$

Given $\{a_n\}$ is non decreasing, non negative and $$a_{m\cdot n} \leq a_m + a_n,$$ show that there exists $C$ such that $a_n \leq C \log(n)$ for $n\geq 2$. First taking $n=2^k$, we see that ...
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2answers
52 views

How can I show that $n^{n+2}<(2n)!$ for any integer $n$.

When I was try to show that the series $\sum_n \frac{n^n}{(2n)!}$ is convergent using comparison test, I stuck at the point $n^{n+2}<(2n)!$ I think it can be show using mathematical induction. If ...
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1answer
31 views

is my induction proof sufficient?

question; prove that $\forall\ n\ge4, n\in \mathbb{Z}, \ n!\gt n^2$. my work; let $n=4$ then $4!=24 \gt 4^2=16.$ true. now assume $n! \gt n^2$ is true for all $n\le k$ so now assume $k! \gt ...
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202 views

How prove this matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrix $A,B,C\in M_{n}(C)$ is Hermitian matrix and is Positive definite matrices ,such $$A+B+C=I_{n}$$show that $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge ...
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15 views

exponential inequality for sum of dependent random variables

I have proved an inequality for the expectation in the context of dependent random variables. Can you please confirm it and give me some feedbacks? If $X_1,X_2,X_3,\ldots,X_m$ are $m$ dependent mean ...
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23 views

Prove natural log between two finite harmonic sums [duplicate]

Prove for n in the naturals we have: $$\sum_{k=2}^n 1/k \le \ln(n) \le \sum_{k=1}^{n-1} 1/k$$ Intuitively this makes sense to me but I can't for the life of me figure out how to start this proof.
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29 views

Can we find some expressions for $p$ and $q$?

Let $f\colon\mathbb R\to\mathbb R$ be a real analytic function. Assume also that $f$ has a zero at $s=1$ of order $m$. Assume that there exists an integer $r$ such that ...
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43 views

An extremal problem using AM-GM inequality

Let $x, y, z$ be nonnegative real numbers and such that $$ x^2+y^2+z^2=2. $$ Find the maximum value of $$ P=\frac{x^2}{x^2+yz+x+1}+\frac{y+z}{x+y+z+1}-\frac{1+yz}{9}. $$ My attempt. I guess that $P$ ...
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1answer
65 views

An inequality between $\int_{a}^{b}f(x)g(x)dx$ and $\int_{a}^{b}|f(x)-g(x)|dx$

Does there exist an inequality between $\int_{a}^{b}f(x)g(x)dx$ and $\int_{a}^{b}|f(x)-g(x)|dx$ or an inequality between $\int_{a}^{b}f(x)g(x)dx$ and $\int_{a}^{b}(f(x)-g(x))^2dx$ ? Thank you very ...
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44 views

Positive integral everywhere implies positive function a.e

I would like to get feedback on my demonstration of this simple statement : Let $f$ be an integrable function on the measure space $(X,S,\mu)$. \begin{align} \text{If }\int_E f \, d\mu \geq 0\text{ ...
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4answers
236 views

inequality method of solution

Im looking for an efficent method of solving the following inequality: $$\left(\frac{x-3}{x+1}\right)^2-7 \left|\frac{x-3}{x+1}\right|+ 10 <0$$ I've tried first determining when the absolute value ...
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1answer
54 views

How prove this inequality $\sum_{k=0}^{n}\frac{\sum_{i=k}^{n}\binom{i}{k}}{k+1}\cdot\left(-\frac{1}{e}\right)^{k+1}<1$

show that $$\sum_{k=0}^{n}\left(\dfrac{\displaystyle\sum_{i=k}^{n}\binom{i}{k}}{k+1}\cdot\left(-\dfrac{1}{e}\right)^{k+1}\right)<1$$ maybe this inequality is from Mathematical olympiad, I think ...
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1answer
32 views

need to prove an inequality with absolute value to the power of positive number

I need help to prove the inequalities in the following cases $ ||x|^p-|y|^p|\leq \begin{cases} |x-y|^p & \mathrm{if} \, 0<p<1\\ p|x-y|(x^{p-1}+y^{p-1}) & \mathrm{if} \, 1\leq p<\infty ...
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2answers
71 views

Show that the following series is less than $4 \pi^2 / 3$.

Show: For any $k = 0,1,2,...$, $$ \sum_{i=0}^{i=k} \frac{(k+1)^2}{(i+1)^2 (k-i+1)^2} \leq \frac{4 \pi^2}{3}. $$
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55 views

How to prove that for $n \in \mathbb{N}$ we have $\sum_{k=2}^n \frac{1}{k}\leq \ln(n) \leq \sum_{k=1}^{n-1} \frac{1}{k}$

Prove that for $n \in \mathbb{N}$ we have $\sum_{k=2}^n \frac{1}{k}\leq \ln(n) \leq \sum_{k=1}^{n-1} \frac{1}{k}$ by using Riemann integral?
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1answer
24 views

Zero Product Theorem - an inequality question regarding the denominator

Hey guys so I have this question: $(12x + 4)^{-1} < 0$ At first look I thought it would just always get back to 0 < 0 with loss of my x, which can't be. I Had a look around and the best I can ...
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3answers
33 views

Simplifying and understanding inequalities in two variables

given an equation: $$\frac{x-y}{x+y}\ge0$$ What are the steps to simplify this into an understandable group of inequalities which will yield a solution set as a group of areas? I already know that ...
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3answers
129 views

Asymptotic behaviour of the integral of the quadratic mean of the coordinates on the hypercube

I have to compute the limit $\lim_{n\to +\infty}I_n$, where: $$\qquad I_n=\int_{[0,1]^n}\sqrt{\frac{1}{n}\sum_{i=1}^n x_i^2}\,d\mu.$$ I believe that its value is just $\frac{1}{\sqrt{3}}$, since the ...
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2answers
30 views

What will be the range of $f(x)= \frac{12}{\sqrt{(15-2x-x^2)}}$

Here's my try: Since the denominator involves a square root so I solved the following inequality: $15-2x-x^2>0$ which gives a solution set of $x=(-5,3)$. This is the domain of $f(x)$. However since ...
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1answer
36 views

Inequality for Three Variables in positive reals [closed]

Let $a,b,c$ be positive real numbers. Prove that \begin{equation} \frac{ab+c^2}{a+b}+\frac{bc+a^2}{b+c}+\frac{ac+b^2}{a+c}\geq a+b+c \end{equation}
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31 views

Decomposition of polynomials and inequality

This was asked in comment here by @23rd : If $f$ is a polynomial with $\deg f=n\ge2$, then there exist polynomials $g$ and $h$, such that $$f(x)=2xg(x)−h(x)$$ $$\deg g\le n−1, \quad \deg h \le ...
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1answer
27 views

Inequality involving fractions and several variables

Based on my previous two questions (that failed), I am trying once more. What are some simplified conditions for which: ...
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0answers
29 views

AM-GM-HM on expression with parameters?

This question, which I believe is easier to answer, is related to my previous question: Finding a value that makes an expression negative I am persistent - and need some ideas to help me prove ...
5
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3answers
86 views

prove that $a^b\ge{b}^a$ where $a\le{b}$.

prove that $a^b\ge{b}^a$ for all $a,b\ge3$. given that $a\le{b}$. I was trying to solve the question by graph. Can anyone help me please?