1
vote
0answers
24 views

Integral Hölder bound

I was wondering if it is possible to find the following bound or if not, find a counterexample of it. Let $f\in C_0^1$ (compactly supported continously differentiable, in particular $\alpha$-Hölder ...
1
vote
0answers
26 views

Strange inequality

I found the inequality $\beta e - \frac{3}{2} n \ log(e+Bn)+ \frac{5}{2} \ n \ log(n) + const \cdot n \geq \frac{\beta e}{2}+ \beta n $ in a textbook,provided that either $e$ or $n$ is large. We ...
1
vote
4answers
56 views

Real Between Rationals

Let $x$ be a real number. Show that, for any $\varepsilon>0$, there exist two rationals $q$ and $q'$ such that $q<x<q'$ and $|q-q'|<\varepsilon$ How should I approach this prove?
3
votes
1answer
58 views

How can I prove that $a^{2} < b^{2} $implies that $a < b$ in the Real Numbers? [on hold]

The answer to my question doesn't seem to exist elsewhere on the internet. I have the sets $ A=\{ a : a\in R: a > 0,\ a^2 < 3\} $ and $ B=\{ b: b\in R: b>0,\ b^2 > 3\} $, and I'm just ...
1
vote
0answers
43 views

Limit superior does not increase when a sequence is replaced by its sequence of averages [duplicate]

Let $\{x_n\}$ be a bounded sequence of real numbers, and define a new sequence $\{\sigma_n\}$ by $$\sigma_n=\frac 1n\sum_{i=1}^nx_i.$$ Prove that $\limsup \sigma_n\le \limsup x_n$. I am confused on ...
1
vote
3answers
28 views

Show that for all real numbers a and b, ab <= (1/2)(a^2+b^2)

so as in the title, I have the following theorem to prove. Theorem Show that for all $a$, $b$ $\epsilon$ $R$, that the following inequality holds, $\begin{equation} ab \leq \frac{1}{2}(a^2 + b^2) ...
1
vote
1answer
49 views

Show that $ \limsup x_n ≤ \limsup y_n$ and $\liminf x_n ≤ \liminf y_n$

Let $(x_n)$ and $(y_n)$ be bounded sequences such that $x_n ≤ y_n$ for all $n \in \mathbb{N}$. Show that $\limsup x_n ≤ \limsup y_n$ and $\liminf x_n ≤ \liminf y_n$.
1
vote
1answer
111 views

How prove there exsit $C_{0}$ such $e^{-\pi^2 k^2\beta}-(\cos{(k\alpha\pi)}+\pi k\sin{(k\alpha\pi)})>C_{0}$

Question: let $$\delta_{\alpha\beta}(k)=e^{-\pi^2 k^2\beta}-(\cos{(k\alpha\pi)}+\pi k\sin{(k\alpha\pi)})$$ if $\alpha\in Q$ and $\alpha>0,\beta>0,k\in N^{+}$, show that there ...
1
vote
1answer
21 views

generalization of midpoint-convex

Let f : (a,b) → R is a midpoint-convex function (I didn't say continuity). Here I'd like to verify following inequality ""directly"". f( (x1+x2+x3)/3 ) ≤ (f(x1)+f(x2)+f(x3))/3 .. I can easily ...
0
votes
0answers
31 views

Inequality in $\mathbb{Z}^2$

Let $k=(k_1,k_2)\in\mathbb{Z}^2$. Denote $|k|\leqslant n$ when $|k_1|,|k_2|\leqslant n$. I need help to show $$|\sum_{k+l+m=0}_{|k|,|l|,|m|\leqslant ...
1
vote
1answer
21 views

When can I expect the derivative of an inequality to always hold true?

Let $f,g$ be real-valued functions. Suppose I have an inequality $$f(x)>g(x)$$ for $x\in D$, where $D\subseteq\mathbb{R}$ is some domain. After some "tinkering" I see that we can not always expect ...
2
votes
3answers
47 views

Showing that $x+ cos x - 1 > 0$ for all $x > 0$

I got this problem: Show that for all $0<x$, $0<x+cos x - 1$ I tried to show it several times but none worked. I showed that $lim_{x\to\infty} (x+cos x - 1) = \infty$ by using the Squeeze ...
0
votes
1answer
42 views

Verification of proof that $x^p+y^p<(x+y)^p$

I came across the inequality $x^p+y^p<(x+y)^p$, $x,y \in \mathbb{R}^+$, $p>1$ during working on an assignment. While I think I have a proof down, I'm not 100% confident about the steps taken. ...
2
votes
1answer
34 views

Supremum of sum of two sequences: $\sup (x_n+y_n) \le \sup x_n + \sup y_n$

Prove that $\sup\{x_n+y_n\}\leq \sup\{x_n\}+\sup\{y_n\}$, if both sups are finite. Furthermore, prove that $\limsup\{x_n+y_n\}\leq \limsup\{x_n\}+\limsup\{y_n\}$ if both limsups are finite.
1
vote
1answer
36 views

Inequalities with function $e^{x^2+e^{x^2}}$

Let $f(x)=e^{x^2+e^{x^2}}$ for $x\in\mathbb{R}$. How to prove that for any $a,b>0$, $a\neq b$ the following inequalites hold $$(b-a)f(\frac{a+b}{2}) < \int_a^b f(x)\ dx < ...
0
votes
5answers
82 views

Prove or give a counterexample: For all $x > 0$, $x^2 + 1 < (x+1)^2 \le 2(x^2 + 1)$

I am working on the following problem from Lay's Analysis with an Introduction to Proof: Prove or give a counter example: For all $x > 0$ we have $x^2 + 1 < (x+1)^2 \le 2(x^2 + 1)$ Now, ...
0
votes
2answers
36 views

Inequalities with logarithms and limits

For my analysis homework, I am to show that $\lim_{n \to \infty} \frac{3^n}{n!} = 0$ using the epsilon definition. My approach is to invoke the squeeze theorem and show that the above sequence is less ...
1
vote
3answers
65 views

$(x+y)^c\le x^c+y^c$ for $0<c\le1$ [duplicate]

The statement I'm trying to prove is: $(x+y)^c\le x^c+y^c$ whenever $0\le x,y$ and $0\le c\le1$. This comes up in the proof that $|x|_*^c$ is an absolute value whenever $0<c\le1$ and $|x|_*$ ...
1
vote
1answer
64 views

$L_2$ error between a non-negative monotone function and its mean?

I have been recently trying to prove a lemma which seems true in every single example I have tried, yet that I didn't manage to prove so far unless making extra (not desirable) assumptions. A ...
3
votes
1answer
54 views

Are some of the Real number axioms redundant?

We are taking a course in Real Analysis with the text: "Elementary Analysis, the theory of Calculus", by Kenneth Ross. I have also been reading a little bit of Beckenbach and Bellman's book, ...
1
vote
1answer
62 views

The expression $1 + x^2 +(-T_px+y)^2 +z^2$ is bounded below by a constant multiple of $(1+x^2+y^2+z^2)$

Suppose $T_p > 0$. Is there a simply way to show that $1 + x^2 +(-T_px+y)^2 +z^2 \geq C (1+x^2+y^2+z^2)$, for all $(x,y,z) \in \mathbb R^3$, where $C>0$.
1
vote
1answer
46 views

Proving inequality $\frac{1}{2}e^x\left(2+e^x\right) > \left(1+e^x\right)\ln(1+e^x)$

Do you have any ideas on how to show the following inequality? $$\frac{1}{2}e^x\left(2+e^x\right) > \left(1+e^x\right)\ln(1+e^x)$$ It's not about the convexity of any of those functions. ...
3
votes
1answer
90 views

Measure of the set where a trigonometric polynomial with zero mean is non-negative

Suppose $f$ is a real trigonometric polynomial of degree $N$ with constant term $0$. What lower bounds can we place on the measure $\mu$ of the set $\{ t \in S^1 : f(t) \geq 0 \}$, independent of the ...
2
votes
1answer
99 views

Equality about limsup.

Suppose $\sum_{n=1}^\infty \mathbb P(A_n)=\infty$,then: $$\limsup_{n\to\infty}\frac{(\sum_{k=1}^n \mathbb P(A_k))^2}{\sum_{i,k=1}^n\mathbb P(A_i\cap A_k)}=\limsup_{n\to\infty}\frac{\sum_{1\le ...
11
votes
5answers
243 views

Proof: $\cos^p (\theta) \le \cos(p\theta)$

I came across this problem when I was at a book store inside of a book made to prepare Berkeley graduates to pass a mandatory exam. I wanted to buy the book, but, alas, I didn't have the money (forty ...
1
vote
1answer
30 views

A fundamental inequality.

I have read in a textbook($p\ge1 ,X_k $ are r.v.): $$\|\sum_{n=0}^k|X_n|\|_p\le(k+1)\|X_k\|_p\quad\text{where}\mathbb E|X_n|^p\le\mathbb E|X_k|^p \quad\text{for} \quad 0\le n\le k $$ I tried to ...
1
vote
1answer
43 views

AN application of Schwarz inequality.

In the proof of Chung-Erd$\ddot{o}$s inequality: Let $X_k=1_{A_k}$,then: $$(\mathbb E(\sum_{i=1}^nX_k))^2\le\mathbb P(\sum_{i=1}^nX_k\gt0)\mathbb E[(\sum_{i=1}^nX_k)^2]$$ The textbook said this ...
0
votes
2answers
44 views

Using the mean value theorem to prove inequalities

Using the Mean Value Theorem, show that for any $t>0$, $$\left|e^{-x^2/2t}-e^{-y ^2/2t}\right|\leqslant \frac{|x-y|}{t}$$ for all $x,y$ with $|x|,|y|\leqslant 1$. My attempt. Without loss of ...
1
vote
2answers
90 views

prove that $ a^2 + b^2 + c^2 \ge 2\left( {a^3 b^3 + a^3 c^3 + c^3 b^3 + 4a^2 b^2 c^2 } \right)$

I have: let $$a;b;c$$ be non-negative real numbres with sum 2.prove that $$a^2 + b^2 + c^2 \ge 2\left( {a^3 b^3 + a^3 c^3 + c^3 b^3 + 4a^2 b^2 c^2 } \right)$$ I should determine whether this ...
3
votes
1answer
59 views

How prove $\sqrt{x}+\sqrt{y}+\sqrt{\frac{x+y+2}{xy-1}}\ge 2\left( \frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}+\sqrt{\frac{xy-1}{x+y+2}}\right)$

How prove $\sqrt{x}+\sqrt{y}+\sqrt{\frac{x+y+2}{xy-1}}\ge 2\left( \frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}+\sqrt{\frac{xy-1}{x+y+2}}\right)$ for $8x\ge13, 8y \ge 13$?
0
votes
0answers
37 views

Properties of Sup, Inf on sets.

I understand this proof is a bit long-winded, but I am only concerned with it is correct or not. It seems sound to me. Claim: If $A,B \subset \mathbb{R}$ and are non-empty, $a \leq b, \forall a \in ...
4
votes
2answers
74 views

Lower bound for $2\sin^2(y\pi)$

I was trying to understand the proof of a theorem, and the author uses the fact that if $y \in \mathbb{Q} \cap(0, \frac{1}{2}]$, then $$2\sin^2(y\pi) \geq \frac{8}{n^2},$$ where $y=\frac{p}{q}$, ...
3
votes
1answer
58 views

Matrix inequalities question

Let $A, B \in \mathbb{R}^{n \times n}$. Assume that: $$ 0 \preccurlyeq 2 A^\top A \preccurlyeq A^\top + A $$ $$ B^\top + B \preccurlyeq 0 $$ Is the following inequality true? $$ A B + B^\top ...
0
votes
0answers
36 views

Why $\alpha\le L_1$ for any $L_1>L$ implies $\alpha\le L$? [closed]

First part Second part Let $\alpha=\limsup|s_n|^{1/n}$ and $L=\limsup\left|\frac{s_{n+1}}{s_n}\right|$. We need to prove $\alpha\le L$. This is obvious if $L=+\infty$, so we assume $L<+\infty$. ...
0
votes
0answers
59 views

Equality case in Hölder's inequality

How can I show that $$\left(\int{p(x)^{1-\sigma}\mathrm dx}\right)^{\frac{1}{1-\sigma}}\cdot \left(\int y(x)^\frac{\sigma-1}{\sigma}\mathrm dx\right)^{\frac{\sigma}{\sigma-1}}=\int p(x) ...
1
vote
1answer
48 views

Two sequences defined by recurrence relations satisfy $x_n/y_n<\sqrt{7}$ for all $n$

Let $(x_n)_{n\geq 1}$ and $(y_n)_{n\geq 1}$ be two sequences such that: $$x_{n+1}=x_n^2+1 \quad \text{ and } \quad y_{n+1}=x_n y_n$$ with $x_1=2$ and $y_1=1$ Prove that for all $n$ ...
0
votes
0answers
28 views

Minkowski inequality for $0<p<1$

I'm trying to prove this, $$\left ( \sum_{i=1}^{n}(x_i+y_i)^p \right) \geq \left ( \sum_{i=1}^{n}(x_i)^p \right)^\frac{1}{p} + \left ( \sum_{i=1}^{n}(y_i)^p \right)^\frac{1}{p} $$ for $0<p<1$. ...
6
votes
1answer
68 views

How to show $ \left(\frac{1-x}{2}\right)^p+\left(\frac{1+x}{2}\right)^p \leq \frac{1+x^p}{2}$ [duplicate]

When $p\geq 2$ and $0\leq x\leq1$, how does one show the inequalities $$ \left(\frac{1-x}{2}\right)^p+\left(\frac{1+x}{2}\right)^p \leq \frac{1+x^p}{2}$$ and $$ 2(1+x^p)\leq (1+x)^p + (1-x)^p \ ?$$ ...
2
votes
1answer
69 views

How prove $(\ln{\frac{1-\sin{xy}}{1+\sin{xy}}})^2 \geq \ln{\frac{1-\sin{x^2}}{1+\sin{x^2}}}\ln{\frac{1-\sin{y^2}}{1+\sin{y^2}}}$

How prove that if $x, y \in (0,\sqrt{\frac{\pi}{2}})$ and $x \neq y$, then $(\ln{\frac{1-\sin{xy}}{1+\sin{xy}}})^2 \geq \ln{\frac{1-\sin{x^2}}{1+\sin{x^2}}}\ln{\frac{1-\sin{y^2}}{1+\sin{y^2}}}$?
2
votes
1answer
58 views

How prove $(x+\sqrt{x^{2}-1})^{n}+(x-\sqrt{x^{2}-1})^{n}\leq 2(1+n(x-1))^{n}$ for $n\in\mathbb{N}$?

Let $x\ge 1$. How prove that $(x+\sqrt{x^{2}-1})^{n}+(x-\sqrt{x^{2}-1})^{n}\leq 2(1+n(x-1))^{n}$ for $n\in\mathbb{N}$?
2
votes
3answers
32 views

Minimum of $\max(1-2x+y,1+2x+y,x^2-y^2)$ for $x,y(y\ge 0)$?

We define $$S(x,y)=\max(1-2x+y, \, 1+2x+y, \, x^2-y^2)$$ on $\mathbb{R}\times\mathbb{R}_{\geq 0}$. How do we find the minimum value of $S(x,y)$?
2
votes
1answer
115 views

How prove that $\max(|f(1)|,|f(2)|,|f(3)|,|f(4)|)\geq \frac{1}{2}$ if $f(x) = \cos(Ax)+\cos(Bx)$?

Let $ A, B$ be real numbers and $ f(x) =\cos(Ax) + \cos(Bx)$. How prove that $ \max(|f(1)|,|f(2)|,|f(3)|,|f(4)|)\geq \frac{1}{2}$?
0
votes
0answers
51 views

Sigma Notation Inequality

Given two sets of nonnegative real numbers: $$\{a_1, a_2, ..., a_N\}, \{b_1, b_2, ..., b_N\}$$ Are there any conditions for which the following inequality is true? $${1\over N} \sum_{i=1}^N ...
2
votes
1answer
59 views

How prove $-\sqrt{2}\log(\cos x)\leq\sqrt{x\tan x-\sin^{2}x}$?

How prove that $-\sqrt{2}\log(\cos x)\leq\sqrt{x\tan x-\sin^{2}x}$ for all $x\in\left [ 0,\frac{\pi}{2}\right)$?
1
vote
1answer
38 views

Don't understand inequality in order to prove Algebraic Limit Theorem

I'm self-studying from the book Understanding Analysis by Stephen Abbott and I'm stuck on Theorem 2.3.3 on page 45, i.e., the Algebraic Limit Theorem. In particular, letting $\lim a_n = a$ and $\lim ...
0
votes
1answer
36 views

Putting a bound on some probability inequality

Assume that we have the following polynomial: $$ax^2 + bx =c$$ and a, b, c are i.i.d uniform random variables in [0, 1]. I'm trying to calculate the probability that the root is real, and that ...
0
votes
2answers
66 views

Using mean value theorem to show that $\cos (x)>1-x^2/2$

I have a question, by applying the mean value theorem to $f(x)=\frac{x^2}{2}+\cos (x)$, on the interval [0,x], show that $\cos (x)>1-\frac{x^2}{2}$. We know that ...
0
votes
0answers
32 views

Help with a simple function inequality

Let $R\geq 1$, $f(x)$ a function and $1_A$ the indicator function of the set $A$; is this inequality true? $$\frac{\vert f(x)\vert}{R}1_{R\leq\vert x\vert\leq 2R}\leq\frac{\vert f(x)\vert}{1+\vert ...
1
vote
1answer
33 views

Clarification: how to get the following asymptotics

I'm having some trouble justifying some steps in a paper. Let $a_n$ be an increasing sequence of integers satisfying $n! \le a_n \le 2(n!)$, and let $f:\mathbb{N} \to \mathbb{N}$ be a function ...
0
votes
1answer
51 views

How prove $\frac{x^{3}+y}{y^{3}+x}-1\geq \ln \frac{(x^{2}+1)^{2}}{x}-\ln \frac{(y^{2}+1)^{2}}{y}$?

How prove $\frac{x^{3}+y}{y^{3}+x}-1\geq \ln \frac{(x^{2}+1)^{2}}{x}-\ln \frac{(y^{2}+1)^{2}}{y}$ where $x, y\geq 1$?