-1
votes
0answers
51 views

How prove that $\ln2>{\left(\frac{2}{5}\right) }^{\frac{2}{5}}$? [duplicate]

How prove that inequality $\ln2>{\left(\frac{2}{5}\right) }^{\frac{2}{5}}$?
0
votes
0answers
56 views

Bound for this integral

Using the fact that $$\sqrt{(1+y^2)} - \sqrt{(1+x^2)} \geq \frac{x}{\sqrt{1+x^2}}(y-x)$$ for each $x,y\in \mathbb{R}$. We need to show that $$L(k)- L(h) \geq \int_a^b \frac{h'}{\sqrt{1+{h'}^2}} ...
2
votes
4answers
78 views

How to find a function $\phi(x)$ such that $\sqrt{1+y^2} - \sqrt{1+x^2} \geq \phi(x) (y-x)$ for each $x,y\in \mathbb{R}$

How to find a function $\phi(x)$ such that $\sqrt{1+y^2} - \sqrt{1+x^2} \geq \phi(x) (y-x)$ for each $x,y\in \mathbb{R}$. Here are some of my ideas: Also by applying Mean Value theorem, we know that ...
1
vote
2answers
72 views

$-\varepsilon\log(x)\overset{?}{\geq} -\log(\varepsilon x)$

I'm refering to this proof: http://en.wikipedia.org/wiki/Quantum_relative_entropy#The_result In there it's stated that "Since the matrix $(P_{ij})_{ij}$ is a doubly stochastic matrix and $-\log$ is a ...
3
votes
0answers
62 views

Upper bound for the sums of powers of factors

Fix $\alpha \in \,]0,1]$. Is it true that for each sufficiently large positive integer $n$, if $n = x_1 \cdots x_j$, for some integers $x_1, \ldots, x_j \geq 2$, with $j \geq 2$, then $$x_1^\alpha + ...
2
votes
1answer
79 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 ...
4
votes
2answers
74 views

Binomial expansion inequality

In a paper I am reading, there is a step that seems to come from the following inequality: $$(1+x)^\alpha \le 1+2^\alpha x,$$ where $0<x<1$. (Also, $3\le \alpha \le 9/2$ in the context of the ...
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
1answer
91 views

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 ...
0
votes
1answer
26 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 ...
2
votes
1answer
45 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 ...
0
votes
1answer
54 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
38 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
54 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 ...
0
votes
0answers
26 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
1answer
81 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 ...
3
votes
1answer
34 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 ...
6
votes
2answers
57 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?
6
votes
3answers
137 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 ...
5
votes
3answers
87 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
57 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
2answers
38 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 ...
2
votes
2answers
111 views

Proving $\displaystyle \frac{\sin^3x}{x}\lt 0.69$ for any $x\gt 0$

Question : How can we prove strictly that the following inequality holds for any $x\gt0$?$$\frac{\sin^3x}{x}\lt 0.69$$ This seems difficult though it doesn't look so. Can anyone help?
11
votes
0answers
112 views

Stronger version of AMM problem 11145 (April 2005)?

How to show that for $a_1,a_2,\cdots,a_n >0$ real numbers and for $n \ge 3$: ...
0
votes
1answer
60 views

$|x|^p+|y|^p\geq |x+y|^p$ for $0<p\leq 1$ [closed]

How to prove such inequality: $|x|^p+|y|^p\geq |x+y|^p$ for $0<p\leq 1$ and $x,y \in \mathbb{R}$?
0
votes
0answers
25 views

calculate the sup of the max of 3 functions

Let a function be the variable, how the calculate the following expression? $$\inf_{c(t) \in C[-1,0]} \max \{ \max_{-1 \leq t \leq 0} |c(t)| , \max_{0 \leq t \leq 1} | \int_{0}^{t} c(v-1) +1 dv +c ...
6
votes
4answers
204 views

Inequality for harmonic means

Prove that for real numbers $a_1 ,a_2 ,...,a_n >0$ the following inequality holds $$\frac{1}{a_1 } +\frac{2}{a_1 +a_2 } +...+\frac{n}{a_1 +a_2 +...+a_n }\leq 4\cdot \left(\frac{1}{a_1} ...
2
votes
1answer
39 views

How to prove this integral inequality?

Here is a problem: Let $B_r=\{ (x_1,x_2,\cdots,x_n)\in \mathbb{R}^n: x_1^2+x_2^2+\cdots+x_n^2<r^2\}.$ Let $f$ be a $C^1$ real function on $B_2$. Prove that $$\inf_{a\in R}\int_{B_2} ...
12
votes
2answers
155 views

$|f(x)|\leq \sqrt{\frac{\pi}{3}\int_0^\pi f'^2}$

Let $f\in C^1([0,\pi],\mathbb R)$ such that $\displaystyle\int_0^\pi f(t) dt=0$ Prove that $\forall x\in [0,\pi],\displaystyle|f(x)|\leq \sqrt{\frac{\pi}{3}\int_0^\pi f'^2(t)dt}$ Failed ...
6
votes
6answers
138 views

Show $(x+y)^a > x^a + y^a$ for $x,y>0$ and $a>1$

This is a pretty straightforward question. I want to show $(x+y)^a > x^a + y^a$ for $x,y>0$ and $a>1$. One way would be this. WLOG, suppose $x \leq y$. Then: $(1+\frac{x}{y})^a ...
6
votes
4answers
152 views

How can we prove $\int_1^\pi x \cos(\frac1{x}) dx<4$ by hand?

Is there any way we can prove this definite integral inequality by hand: $$ \int_{1}^{\pi}x\cos\left(1 \over x\right)\,{\rm d}x < 4 $$ I don't where to start even, please help. That ...
0
votes
2answers
60 views

Prove Minkowski's inequality directly in finite dimensions

The typical way I've seen to prove Minkowski's inequality is to take $$ |f(x) + g(x)|^p \leq |f(x)||f(x)+g(x)|^{p-1} + |g(x)||f(x)+g(x)|^{p-1}, $$ integrate over $x$, and apply Holder's inequality to ...
1
vote
1answer
48 views

A particular quadratic minimization problem

Given $n^2$ constants $a_{11},a_{12},\ldots,a_{1n},a_{21},\ldots,a_{nn}$ and $n^2$ non-negative variables $x_{11},x_{12},\ldots,x_{1n},x_{21},\ldots,x_{nn}$. Find the minimum value of $$\sum_{i=1}^n ...
3
votes
2answers
102 views

what type of functions has $f(x+y) \geq f(x) + f(y)$

I was working on some $L^p$ inequalities and stumbled up on this. I know that $$f(x+y) \geq f(x) + f(y)$$ if $f$ is convex and monotone increasing. Does this hold for "only if"? And is there a name ...
1
vote
1answer
25 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 ...
5
votes
1answer
91 views

Show that $\sum_{k=1}^{n}\frac{x_k^2}{x_k^2-2x_k\cos(\frac{2\pi}{n})+1}\geq 1$

Let $n>2$ an integer and $x_k>0$ with $x_1\cdot x_2\cdots x_n=1$ Show that $$\sum_{k=1}^{n}\frac{x_k^2}{x_k^2-2x_k\cos(\frac{2\pi}{n})+1}\geq 1$$ I tried an induction without succeed, ...
4
votes
1answer
124 views

Lower bound for $(x^c-1)^{1/c}$

I have been trying to find a lower bound for $x>1$, $c>0$: $$ \Large(x^c-1)^{1/c} $$ My strategy is to find a lower bound for $(x^c-1)^{1/c}$ which can hopefully get rid of some of the $c$ ...
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 ...
6
votes
2answers
131 views

How to arrange $e^3,3^e,e^{\pi},\pi^e,3^{\pi},\pi^3$ in the increasing order?

For these six numbers, $e^3,3^e,e^{\pi},\pi^e,3^{\pi},\pi^3$, how to arrange them in the increasing order? This problem is taken from the today test: National Higher Education Entrance Examination. ...
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
87 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 ...
1
vote
1answer
53 views

$||f + g||_{L^p} = ||f ||_{L^p} +||g||_{L^p} $ for $p\in\mathbb{R}^+\setminus\{ 1\}$ and $f,g\geq 0$

$||f + g||_{L^p} = ||f ||_{L^p} +||g||_{L^p} $ for $p\in\mathbb{R}^+\setminus\{ 1\}$ and $f,g\geq 0$, then $f = Cg$ for some non-negative constant $C$. First assume $||f ||_{L^p} +||g||_{L^p} = 1$, ...
0
votes
0answers
20 views

$\sum (a_i + b_i)^p $ and $\left(\sum a_i\right)^p +\left(\sum b_i\right)^p $ for $a_i, b_i \geq 0,p\geq 1$

Is there a relation between $\sum (a_i + b_i)^p $ and $\left(\sum a_i\right)^p +\left(\sum b_i\right)^p $ for $a_i, b_i \geq 0,p\geq 1$? At first sight, when $a_i = b_i$, we have $$2^p\sum a_i^p ...
1
vote
2answers
86 views

$L^p$ norm and triangle inequality

I thought about this while studying about $L^p$ spaces, as the standard triangle inequality does not hold for $0<p\leq 1$. But we have the variant $$||f+g||_{L^p}^p \leq ||f||_{L^p}^p+||g||_{L^p}^p ...
2
votes
1answer
76 views

Tighter logarithmic inequality

There is a well-known lower bound for $$ x\log{1+x\over x}\geq {x\over1+x} $$ for $x\geq0$. I know a tighter lower bound on the same domain $$ x\log{1+x\over x}\geq{2x\over1+2x}\geq {x\over1+x}. $$ It ...
4
votes
1answer
139 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
1answer
24 views

Inequalities involving x and y.

I am asked to prove: $(x-y)^3 \ge x^3-3x^2y$ where $x,y$ are real and $0 < y < x$ I am told Bernoulli's inequality may help. I have however reduced this to $3xy^2 - y^3 \ge 0$. I have ...
5
votes
1answer
62 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 ...
1
vote
1answer
64 views

Prove $x_n \leq x_{n+1}$ for all $n$ by induction

Prove $x_n \leq x_{n+1}$ for all $n$ by induction. I am reading this example from "Understanding Analysis" by Abbott (page 10). He says the multiple across the inequality by $1/2$ and then add 1 to ...