Questions on proving and manipulating inequalities.

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2
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1answer
31 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 ...
2
votes
3answers
52 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 ...
5
votes
0answers
54 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 $52 < a_{1371} < 65$ ?
0
votes
0answers
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: ...
2
votes
0answers
47 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 ...
0
votes
1answer
32 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 ...
0
votes
0answers
13 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% ...
3
votes
3answers
37 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 ...
0
votes
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 ...
0
votes
1answer
51 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 ...
2
votes
1answer
34 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 ...
1
vote
2answers
51 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 ...
2
votes
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 ...
8
votes
0answers
186 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 ...
0
votes
0answers
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 ...
0
votes
0answers
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.
0
votes
0answers
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 ...
0
votes
0answers
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 $P$ ...
0
votes
1answer
59 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 ...
1
vote
3answers
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{ ...
3
votes
4answers
232 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 ...
3
votes
1answer
52 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 ...
3
votes
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 ...
4
votes
2answers
69 views

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

Show: For any $k = 0,1,2,...$, $$ \sum_{i=0}^{i=k} \frac{(k+1)^2}{(i+1)^2 (k-i+1)^2} \leq \frac{4 \pi^2}{3}. $$
6
votes
2answers
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?
0
votes
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 ...
0
votes
3answers
31 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 ...
6
votes
3answers
128 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 ...
1
vote
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 ...
1
vote
1answer
36 views

Inequality for Three Variables in positive reals [on hold]

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}
1
vote
0answers
29 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 ...
0
votes
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: ...
0
votes
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
votes
3answers
85 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?
0
votes
2answers
67 views

Show that $\binom{n}{k}< \binom{n}{k+1}$ if and only if $k < (n-1)/2$ [closed]

Show that $\binom{n}{k} < \binom{n}{k+1}$ if and only if $k < \frac{n-1}{2}$ and then use this to deduce that the maximum of $\binom{n}{k}$ for $k=0,1,\dots,n$ is $\binom{n}{\lfloor ...
3
votes
1answer
22 views

Proving that $f(x) = \vert x \vert^{\alpha}$ is Holder continuous, inequality help

The definition of $\alpha$-Holder continuity for a function $f(x)$ at the point $x_0$ is tha there exist constant $L$ such that for all $x \in D$ \begin{equation} \vert f(x) - f(x_0) \vert \leq L ...
-2
votes
2answers
109 views

How to show $\log_2 x \cdot \log_{0.25} x \cdot \log_{0.125} x \cdot \log_{16} x > \frac {2}3$?

I was trying to solve $\log_2 x \cdot \log_{0.25} x \cdot \log_{0.125} x \cdot \log_{16} x > \frac {2}3$ and I keep getting a partial answer of $x>4$ though answer key suggests a more expanded ...
0
votes
0answers
15 views

Why $\dim(K\cap N(I)) +1 \geq \dim(K)$ if the index of the index form equals 1?

Let $c:[0,1]\rightarrow O$ a geodesic, $O$ Riemann manifold and let $\mathcal{W}$ the space of piecewise smoothl normal vector fields $W(t)$ along $c$ mit $W(0)=W(1)=0$. $N(I)$ is the nullspace of ...
1
vote
1answer
23 views

Inequality for positive functions

I'm saya. Please let me ask a question about an inequality. Let $f,g$ be a function such that $f,g$ : $\mathbb{R}\to [0,\infty)$ and there exists $C>0$ such that for all $x \in \mathbb{R}$ ...
3
votes
1answer
108 views

Find maximum of $P$

Let $$P = \frac{{{x^2}}}{{{x^2} + yz + x + 1}} + \frac{{y + z}}{{x + y + z + 1}} - \frac{{1 + yz}}{9}.$$ Find maximum of $P$ where $x, y,z$ are nonnegative real numbers such that ${x^2} + {y^2} + ...
3
votes
1answer
53 views

How to prove Godunova's inequality?

Let $\phi$ be a positive and convex function on $(0,\infty)$. Then $$\int_0^\infty \phi\left(\frac{1}{x}\int_0^x g(t)\,dt\right)\frac{dx}{x} \leq \int_0^\infty \phi(g(x))\frac{dx}{x}$$ The ...
0
votes
6answers
63 views

solving the inequalty

are there any ways to solve :$ x^4 -6x^3 +28x^2 -64x +96 >0$ ?
0
votes
2answers
37 views

proof of triangular inequality modified $|x+y|=|x|+|y|$ iff $|xy|>0$

$$|x+y|=|x|+|y| \iff |xy|>0$$ I tried to prove the above inequality but i cant find a way. I tried assuming the first condition is true and tried to derive the second part of it but it seems i ...
0
votes
0answers
26 views

An inequality with $\omega(n)$ [duplicate]

Prove: For any positive integer $k, N$, $$\left(\frac{1}{N}\sum\limits_{n=1}^{N}\left(\omega (n)\right)^k\right)^{\frac{1}{k}}\leq k+\sum\limits_{q\leq N}\frac{1}{q}$$ Where $\sum\limits_{q\leq ...
1
vote
3answers
65 views

Proof of $\sin2x+x\sin^2x \lt\dfrac{1}{4}x^2+2$

How can be proven the following inequality? $$\forall{x\in\mathbb{R}},\left[\sin(2x)+x\sin(x)^2\right]\lt\dfrac{1}{4}x^2+2$$ Thanks
1
vote
1answer
60 views

Ratio of 2 Gammas, approximation with power

Find all value of $\alpha$ such that $\lim\limits_{x\rightarrow +\infty}\left(\frac{\Gamma(x+\alpha)}{\Gamma(x)}-x^{\alpha}\right)=0$. (note: $\alpha$ is a constant with respect to $x$) By ...
0
votes
1answer
41 views

An inequality used to prove minkowski's inequality [closed]

How to prove the following : $$ |x+y| \leq 2(|x|^p + |y|^p)^{\frac{1}{p}}, x,y \in \Bbb R, p \geq 1$$
0
votes
1answer
37 views

Linear Programming?

An agriculture company has 80 tons of fertilizer Alpha and 120 tons of fertilizer Bravo. The company mixes these fertilizer into two products. Product Super requires 2 parts of fertilizer A and 1 part ...
2
votes
1answer
37 views

Solve: $\sum_{i=1}^n \max\left\{x-a_i,0 \right\}=1.$

Given $a_1,a_2,\ldots,a_n \in\mathbb{R}$. Solve the following equation on $\mathbb{R}$: $$\sum_{i=1}^n \max\left\{x-a_i,0 \right\}=1.$$ I am not sure that a closed-form solution exists, so iterative ...
0
votes
0answers
43 views

How can I cleverly use the reverse triangle inequality in this case?

Say, we have the following recurrence relation: $$x_{k+2}=4x_{k+1}+x_k(-3-2h\lambda)$$ with $x_0 $ given, $x_1=(1+h\lambda)x_0, h$ small (step size), and $\lambda<0$. Here is the context is case ...