7
votes
7answers
215 views

How to prove $(1-\frac1{36})^{25}\lt\frac12$?

How to prove the inequality? $(1-\frac1{36})^{25}\lt\frac12$ I'm in trouble. Thank you very much for your help
1
vote
1answer
45 views

To control first derivative with the function itself.

Let $f$ be a compactly supported positive $C^2$ function. I want to show that there exists $C$, such that for all $x\in \mathbb R$, we have $f'(x)^2< C f(x) $ by showing that for every point ...
1
vote
1answer
57 views

Does the following sequence converge?

Suppose $a_i>0$ for all $i$, $\frac{\sum_{i=1}^n a_i}{n}\to \infty$ and p>1. Let $$y_n = \frac{(\sum_{i=1}^n a_i)^p}{n^{p-1}\sum_{i=1}^n(a_i^p)}.$$ Is $y_n$ monotonic? How can you prove or disprove ...
2
votes
1answer
80 views

Homework on basic inequalities.

Let $a_j$ be a sequence of positive reals. Show that (a) $\left(\sum_{j=1}^\infty a_j\right)^\theta \le \sum_{j=1}^\infty a_j^\theta$ for any $0\le\theta\le1$. (b) $\sum_{j=1}^\infty a_j^\theta \le ...
3
votes
1answer
25 views

Triangle inequality and homomorphisms

Here is my situation: I have two homomorphisms $f$ and $g$ from a group $A$ into the complex numbers $\mathbb{C}$. I know that they are 'close' on a subset $B \subseteq A$. More formally there is an ...
2
votes
1answer
16 views

lower bound of modified Besselfunction

i'm looking for an lower bound for the modified Bessel function of the first kind $I_\nu(x)$ of a +ive real argument. There should be one of the form $$ce^{x^\alpha} \le I_\nu(x)$$.
5
votes
3answers
90 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?
3
votes
1answer
59 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 ...
4
votes
1answer
82 views

Prove the inequality for all $N$

Show that the following inequality holds for all integers $N\geq 1$ $$\left|\sum_{n=1}^N\frac{1}{\sqrt{n}}-2\sqrt{N}-c_1\right|\leq\frac{c_2}{\sqrt{N}}$$ where $c_1,c_2$ are some constants. I have ...
1
vote
1answer
26 views

differential inequality of continuous functions

Let $u:[0,+\infty)\to (0,+\infty)$ be a continuous function such that $\int_0^\infty u(x)dx<\infty$. Suppose there exist $a,b>0$ such that $\frac{du}{dx}\leq u(a+bu)$. Prove that ...
1
vote
1answer
33 views

inequalities concerning integration and measure

Let $f$ be a non-negative function on $\mathbb{R}^n$ such that $\int_{\mathbb{R}^n} f=1$. Let $p\in(0,1)$. Let $E$ be any measurable subset of $\mathbb{R}^n$. Prove that $$ \int _E f^p\leq ...
2
votes
2answers
43 views

All unit vector has bounded components?

If $\Vert v_i\Vert \leq 1$ for all $1\leq i\leq k$ with $\{ v_1,..,v_k\}$ is linearly independent THEN FOR ALL real numbers $\alpha_i$ with $$\Vert \sum_{i=1}^k\alpha_iv_i\Vert=1$$ we can find ...
1
vote
1answer
41 views

Inequality with a norm

I need help with the following: Let $A=\left(\begin{array}{cc}a & b \\c & d\end{array}\right)$, with $a\in\mathbb{R}$, $b\in(l^{1})^{*}$, $c\in l^{1}$, and $d\in L(l^{1},l^{1})$. Let $h\in ...
0
votes
1answer
30 views

Minimum value of the inequality

Give $a_{1}\geq a_{2}\geq ...\geq a_{n}> 0$ and a positive integer m . Find the minimum value of the following the inequality: $\left ( a_{1}+a_{2}+...+a_{n} \right )\left ( \frac{1}{a_{1}^{m}}+ ...
2
votes
3answers
61 views

Prove that $\exists k>0$ such that$\frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{n-1}}{a_{n}}<n-2014$

Consider a positive sequence $\{a_{n}\}$ such that $a_{n+1}>a_{n}$, and $\{a_n\}$ is unbounded. Show that there exits a positive integer $k$ such that, when $n>k$ ...
1
vote
1answer
89 views

Tricks to solve inequalities

I am wondering if there are some tricks to solve inequalities which are not manageable analytically. For example consider the inequality (say we restrict on positive $x$): $\displaystyle \frac {\text ...
4
votes
1answer
142 views

What $\alpha$ such that if $xy=\alpha$, then $e^{-x}+e^{-y}\geq 2e^{-\sqrt \alpha} $?

For every $ x,y \gt 0$, if $ xy=\alpha$, then we have $$e^{-x}+e^{-y}\geq 2e^{-\sqrt \alpha} $$ What are the possible values of $\alpha$? $2 < e^{1/(n+1)} + e^{-1/n}$ led to this problem. ...
1
vote
3answers
43 views

Question about sequences

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and suppose there exists $K \in \mathbb{R}$ with $0 \leq K <1$ such that for all $x,y \in \mathbb{R}$ with $x \neq y$. $$|f(x)-f(y)|<K|x-y|$$ ...
5
votes
1answer
63 views

An inequality of $L^p$ norms of linear combinations of characteristic functions of balls

Let $1<p<\infty$. Let $(a_n)_{n=1}^\infty$ be a sequence of nonnegative real numbers and $\{B_{r_i}(x_i)\}_{i=1}^\infty$ be a sequence of open balls in $\mathbb{R}^n$. Prove that there exists ...
6
votes
3answers
141 views

Something about $\frac{\log x}{x}$

Denote $\log x = \log_ex$. Let's consider the below function $$\frac{\log x}{x}$$. Apparently, It's maximum is $\frac{1}{e}$. and strictly increasing in $(0,e]$, strictly decreasing in $[e,+\infty)$. ...
1
vote
1answer
82 views

How prove this inequality $I=\int_{0}^{\sqrt{2\pi}}\sin{x^2}dx>0$

show that $$I=\int_{0}^{\sqrt{2\pi}}\sin{x^2}dx>0$$ follow is my methods: let $$x^2=t$$ then ...
3
votes
0answers
183 views

Prove that $\prod\limits_i(1+2\alpha_{i})\prod\limits_j(1-2\beta_{j})<\prod\limits_i(1+2x_{i})\prod\limits_j(1-2y_{j})$

Let $m,n\in N^{+}$ and $i=1,2,\ldots,n,\;j=1,2,\ldots,m\,$ and $\,x_{i},\alpha_{i},y_{j},\beta_{j}$ be real numbers such that $$0\le x_{i}<\alpha_{i}<\dfrac{1}{2},\qquad0\le ...
1
vote
1answer
46 views

Mathematical Proof (Apostol)

If $x > 0$, prove that there is a positive integer $n$ such that $\frac{1}{n} < x$ byy either contradiction or contrapositives. My attempts By contrapositives: Givens by contrapositive method ...
11
votes
4answers
182 views

How find this minimum of the value $f(1)+f(2)+\cdots+f(100)$

Give the positive integer set $A=\{1,2,3,\cdots,100\}$, and define function $f:A\to A$ and (1):such for any $1\le i\le 99$,have $$|f(i)-f(i+1)|\le 1$$ (2): for any $1\le i\le 100$,have ...
1
vote
1answer
76 views

The inequality about recurrence sequence

Sequence $(x_n)$ is difined $x_1=\frac {1}{100}, x_n=-{x_{n-1}}^2+2x_{n-1}, n\ge2$ Prove that $$\sum_{n=1}^\infty [(x_{n+1}-x_n)^2+(x_{n+1}-x_n)(x_{n+2}-x_{n+1})]\lt \frac {1}{3} $$ I found relation ...
1
vote
1answer
19 views

$|g(t_{1}) e^{-(t_{1}-x)^{2}}- g(t_{2})e^{-(t_{2}-x)^{2}}|\leq |f(t_{1}) e^{-(t_{1}-x)^{2}}- f(t_{2})e^{-(t_{2}-x)^{2}}| $?

Suppose $f, g: \mathbb R \to \mathbb C$ such that $|g(t_{1}) -g(t_{2})| \leq |f(t_{1})- f(t_{2})| $ for every $t_{1}, t_{2} \in \mathbb R.$ Take any $x\in \mathbb R$ and fix it. Edit: We also assume ...
1
vote
2answers
45 views

Check: Convergence of an infinite series

More a check than a question - I just need to ensure that my logic is correct (I always had trouble with this): Show whether the series $$\sum_{n=110}^{\infty}\frac{1}{3^{n}n^{3}}$$ Is divergent, ...
0
votes
1answer
24 views

Application of the Hoelder inequality

How to prove using Hoelder Inequality that, $$\sum\limits_{i=1}^n \mathbb{E} (O (|X_i| + |X_i|^3 )) \leq n \, \mathbb{E} (O ( |X|^3 ) ),$$ where $X = (X_1, X_2, \ldots X_n)$ are i.i.d. independent ...
0
votes
1answer
89 views

Changes in the hypotheses of a mean-value theorem

For $X \subset \mathbb{R}^d$ open, we define $$ C^1(X) := \left \{ f : X \to \mathbb{C} : f \text{ is a function s.t. } \frac{\partial f}{\partial x_j} \text{ exists and is continuous for } j = 1, 2, ...
0
votes
1answer
26 views

If F is real-entire, then how to write, $F(z)- F(w)$ in terms of $(z-w)$ and $(\bar{z}- \bar{w})$?

Define $F:\mathbb C \to \mathbb C$ such that $F(z)= \sum_{j,k=0}^{\infty}c_{j,k} z^{j} \bar{z}^{k}$ is an entire real analytic function on $\mathbb C$ with $F(0)=0.$ My question is :How to show: ...
3
votes
2answers
60 views

An inequality for some series

Consider real positive numbers $t_1,t_2,\cdots, t_n$ for some $n\in\Bbb N$, with $\sum_{i=1}^nt_i^2=n$, such that if $0<t_i<1$ then $$\frac{t_i}{\sin\left(\frac{t_i\pi}{1+t_i}\right)}<1$$ ...
0
votes
1answer
64 views

About the Gronwall inequality

If I have that $$||\eta_u(t)||\leq 1+C_1\int_0^t \frac{1}{||\eta(s)||}||\eta_u(s)||ds$$ and $$\sqrt{1-\frac{2\varepsilon}{C}}||u||\leq ||\eta(s)||\leq 2||u||$$ how to obtain using the Gronwall ...
0
votes
1answer
28 views

$C^{-1} (1+|x|^{2})^{\frac{s}{2}} \leq (1+|x|)^{\frac{s}{2}} \leq C (1+|x|^{2})^{\frac{s}{2}}$?

Let $s\in \mathbb R,$ and define $f: \mathbb R^{n}\to [0, \infty)$ such that $f(x)= (1+|x|^{2})^{\frac{s}{2}}, (x\in \mathbb R^{n})$ and $g:\mathbb R^{n}\to [0, \infty)$ such that $g(x)= ...
6
votes
4answers
94 views

A question about inequality ${(n+1)\over e^n}^n<n!$

How to prove the inequality $${(n+1)\over e^n}^n<n!$$ I have tried mathematical induction, but it doesn't work! Are there other methods to solve it?
0
votes
1answer
23 views

Nonexpansive Affine Operators in Hilbert spaces

Let $H$ be a Hilbert space with real inner product $\langle \cdot, \cdot \rangle : H \times H \rightarrow \mathbb{R}$. Let $T$ be an affine operator. Show that $T$ is nonexpansive, i.e., $\left\| ...
0
votes
0answers
38 views

about lp estimate of Schwartz function

A homework question that I really couldn't find how to start. Prove that for any $f$ in Schwartz class $ \lVert f \rVert_{q} \leq C_{p,q} \lVert \nabla f \rVert_{2}^{a} \lVert f \rVert_{2}^{1-a} $, ...
1
vote
0answers
49 views

Show that $\liminf\limits_{n\rightarrow\infty}a_{n}/a_{n-k}\leq \liminf\limits_{n\rightarrow\infty}a_{n}^{k/n}$

Here is an exercise: Let $\{a_{n}\}$ be a positive increasing sequence, can we prove that: $\liminf_{n\rightarrow\infty}\frac{a_{n}}{a_{n-k}}\leq \liminf_{n\rightarrow\infty}a_{n}^{k/n}$? Could ...
0
votes
4answers
135 views

Does $xy\geq x+y$?

I just see the GM-AM inequality. But I would like to compare $xy$ with $x+y$ for any $(x, y)\in\mathbb{R}^2$. It looks like $xy>x+y$ since the first one is multiplication and the second one is ...
1
vote
0answers
22 views

Proving an elementary inequality of real vectors related to the p-Laplacian [duplicate]

How would you prove the following inequality? $$ \left|\left|a\right|^{p-2}a-\left|b\right|^{p-2}b\right|\leq C_p\left|a-b\right|^{p-1} $$ where $a,b\in\mathbb{R}^{n}$ and $C_p>0$ is some constant ...
0
votes
0answers
62 views

Help with Gronwall's Inequality

I'm trying to solve an extra credit hw problem that has to do with Gronwall's Inequality. I understand (or think I do) how to find an upper bound when $p=0$, but I'm unsure of how to handle the extra ...
1
vote
2answers
39 views

How to show $(s+t)^p\le 2^{p-1}(s^p + t^p)$ for $p\gt1$

How to show $(s+t)^p\le 2^{p-1}(s^p + t^p)$ for $p\gt1$ I know how to prove for $s+t=1$ by $\min\lbrace t^p+(1-t)^p\rbrace=2^{1-p}$. But I do not know to how to generalize for $s+t\gt0$. Could you ...
0
votes
0answers
6 views

Local equivalency between the Euclidean norm and the CC norm

Let $X_1,...,X_m$ be a family of vector fields defined on an open and connected subset $\Omega$ of $\mathbb{R}^n$ with locally Lipschitz continuous coefficients. Assume that the vector fields are ...
0
votes
1answer
36 views

Integral inequality in $\Bbb R^n$

I came across this problem : Let $f\colon [a,b]\rightarrow \mathbb{R}^n$ a continuous vector valued function. Then it is true that: $$\left\Vert\int \limits_a ^b f(t) dt\right\Vert \leq \int ...
5
votes
3answers
209 views

Prove that the logarithmic mean is less than the power mean.

Prove that the logarithmic mean is less than the power mean. $$L(a,b)=\frac{a-b}{\ln(a)-\ln(b)} < M_p(a,b) = \left(\frac{a^p+b^p}{2}\right)^{\frac{1}{p}}$$ such that $$p\geq \frac{1}{3}$$ That is ...
0
votes
1answer
34 views

when one can expect $\left \| |x|^{2}x- |y|^{2}y \right \| \leq \frac{1}{2} \left \| (x-y) \right \|$?

Suppose $(X, \left\|\cdot \right \|)$ is Banach algebra with the property that $\left\| (|x|^{2}x) \right\| \leq \left\|x\right \|^{3};$ for every $x\in X.$ Let $y_{0}\neq 0 \in X$ and fix it; and ...
0
votes
1answer
15 views

Is this function biLipschitz?

Let $\Omega = (1,2) \times (1,2) \subseteq \mathbb{R}^2$ and define $f$ on $\Omega$ to $\mathbb{R}^2$ as follows: $$ f(x,y) = (x + y, x + y^2). $$ Does there exist a constant $C>0$ such that for ...
4
votes
1answer
246 views

Equality in Minkowski's inequality proof(no integrals)

So what I'm looking for is a proof for when does the equality hold in Minkowski's inequality? I'm talking about this form of inequality: $\left( \sum_{K=1}^n |x_k + y_k|^p \right)^{\frac{1}{p}} \leq ...
1
vote
1answer
75 views

the best constant in an inequality?

I learnt how to show the below inequality by C-S inequality: k is from $0$ to $\infty$ If $\sum a_{k}^{2}9^{k}\le 5$ then $\sum |a_{k}|2^{k}\le 3$. next,I tried to show that 3 is the best possible ...
1
vote
1answer
93 views

An application of Cauchy-Schwarz ineq. on infinite series

If $\sum a_{k}^{2}9^{k}\le 5$ then $\sum |a_{k}|2^{k}\le 3$. sums are from $0$ to $\infty$. could you please help with this question.
0
votes
1answer
36 views

Deriving Holder's Inequality

This is classwork: Let $m,n > 1$, where $1/m + 1/n = 1$. Fix $A > 0$ and have a function $f(x) = \frac{x^m}{m} + \frac{A^n}{nx^n}$ for $x > 0$. Show that $f$ has a global minimum at $x = ...