Questions on proving, manipulating and applying inequalities.

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14
votes
3answers
173 views

Show that $\frac{x}{3!}-\frac{x^3}{5!}+\frac{x^5}{7!}-\cdots\leq \frac{1}{\pi}$.

My problem is to show that $$\frac{x}{3!}-\frac{x^3}{5!}+\frac{x^5}{7!}-\cdots\leq \frac{1}{\pi}$$ for all $x\in\Bbb R$. I was thinking of first finding the max and then show that its less ...
0
votes
0answers
41 views

Inequality Steps

I am stuck in the following inequality. $$2\sum_{q=1}^N(q-1)\frac{1}{q^2}<2 \sum_{q=1}^\infty\frac{1}{q^2}$$ Can anyone help to get the inequality? N is a fixed natural number.
1
vote
2answers
37 views

Concentration property of entropy

Let $X$ be a random variable taking its values in $A = \{a_1,\ldots,a_n\}$ such that $Pr[X = a_i] = p_i$ for all $1 \leq i \leq n.$ The entropy of $X$ is defined as $$H(X) = -\sum_{i=1}^n p_i ...
0
votes
5answers
109 views

How to resolve $n>(1+\frac{1}{n})^n$?

I'm trying to prove that $\forall n\geq 3, n^{n+1}>(n+1)^n$. I came that this is true for $n>(1+\frac{1}{n})^n$. WolphramAlpha gives $n>2.293166...$ but I failed to compute it analytically.
9
votes
1answer
60 views

inequality $\max\{a_1,a_2,\cdots,a_n \}\leq {n^2}^{n-1}.$with Egyptian fraction

Let $a_1,a_2,\cdots,a_n $ be positive integer such that$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=1.$ Prove that$$\max\{a_1,a_2,\cdots,a_n \}\leq {n^2}^{n-1}.$$ This Problem from:1
2
votes
1answer
69 views

The proof of theorem 3.19 from baby Rudin

If $s_n\leqslant t_n$ for $n\geqslant N$, where $N$ is fixed, then $$\liminf_{n\to \infty} s_n\leqslant \liminf_{n\to \infty} t_n$$ $$\limsup_{n\to \infty} s_n\leqslant \limsup_{n\to \infty} t_n$$ ...
0
votes
1answer
30 views

Smoothing Inequalities

Can anyone explain to me how the "smoothing argument for inequalities" works? I know that basically it can be used to prove an inequality $f(a_1,a_2,\cdots,a_n)\geq C$ subject to the constraint ...
1
vote
3answers
49 views

How can I prove that if $\lim_{n \to \infty}s_n=s$ then $|s_n-s|< \epsilon$ is equivalent to $s-\epsilon <s_n <s+ \epsilon$

My professor casually mentioned this in class and told us to prove it if we weren't convinced, however, I cannot find how to prove it.
6
votes
2answers
310 views

Is this proof of Cauchy Schwarz inequality circular or valid?

I'm a college freshman learning linear algebra on my own, and I'm in the section on inner products. I noticed a proof of the Cauchy Schwarz inequality for vectors in my book, and it seems to contain ...
3
votes
5answers
219 views

How to prove $3^\pi>\pi^3$ using algebra or geometry?

It's a question of a some time ago test, I've found a way to solve the problem using calculus, but always I've thought that exist a solution with algebra and geometry. Thank you for your time.
0
votes
1answer
30 views

Super algebraic decaying series

For $a>0$, $b > 1$ and $c \geq 1$ it holds $$F(a,b,c):=\sum_{j=a+1}^\infty \exp(-b \log(j)^c) \leq \int_{a}^\infty \exp(-b \log(x)^c) \, d x < \infty.$$ I am looking for upper bounds on $F$. ...
-4
votes
1answer
33 views

How to solve for this inequalities? Which is greater? $p$ or$k$? [closed]

$p + |k| > |p| + k$ Is $p > k$ or $k > p$ ? Please Explain.
4
votes
1answer
65 views

An inequality for condition $x_{1}+x_{2}+\cdots+x_{n}=1$ [closed]

Let $x_{i}>0,i=1,2,\cdots,n$, and such $x_{1}+x_{2}+\cdots+x_{n}=1$, show that $$\left(\sum_{i=1}^{n}\dfrac{1}{1-x_{i}}\right)\left(1-\sum_{i=1}^{n}x^2_{i}\right)\le n$$ since ...
2
votes
7answers
106 views

How is $\left|\frac{xy}{\sqrt{x^2+y^2}}\right| \leq \frac{\sqrt{|xy|}}{\sqrt{2}}$

$$\left|\frac{xy}{\sqrt{x^2+y^2}}\right| \leq \frac{\sqrt{|xy|}}{\sqrt{2}}$$ Does this apply in general, because here in this example i have $x \to 0, y \to 0$. Some inequality is used I believe to ...
1
vote
1answer
22 views

Upper bound for area on sphere

Consider the sphere $\mathbb{S}^{n-1}:= \{x \in \mathbb{R}^n : \|x\|_2=1\}$, and let $A^\epsilon_x:= \{z \in \mathbb{S}^{n-1}:\langle z,x \rangle \ge \epsilon\}$ where $x \in \mathbb{S}^{n-1}$. Note ...
2
votes
1answer
104 views

Inequality deduced from martinagles

Let $c$ be a positive real constant, and let $x_i,i \in \{1,2,...,n\}$ be real numbers such that $$ |x_i|\le c,\forall i \in \{1,2,...,n\}.$$ Let $p_i,i \in \{1,2,...,n\}$ be positive reals ...
1
vote
1answer
24 views

Prove the following inequality envolving $L^{1}$ and $L^{2}$ norms

I want to prove the following: The uniform, $L^1$ and $L^2$ norms on $C([0,1])$ satisfy $$||f||_1 \le ||f||_2 \le ||f||_u$$ The thing is How Can I prove that $$||f||_1 \le ||f||_2 ?$$ Note: $ ...
1
vote
1answer
82 views

How to prove this $a(a-1)^2+b(b-1)^2\ge (a-1)^2(b-1)^2(a+b-1),a,b\ge 0$ [closed]

Assmue that $a,b\ge 0$,show that $$a(a-1)^2+b(b-1)^2\ge (a-1)(b-1)(a+b-1)$$
1
vote
1answer
52 views

On an estimation of binomial coefficient

On page 14 of the book 'Proofs from THE BOOK', there is an estimation presented as: $$\binom{2n}{n}\le \prod_{p\le \sqrt{2n}}\ 2n. \prod_{\sqrt{2n}<p\le \frac{2}{3}n}\ p. \prod_{n<p\le 2n}\ p, ...
8
votes
3answers
219 views

Need help with an inequality

$1<\frac{1}{1001}+\frac{1}{1002}+...+\frac{1}{3001}<1\frac{1}{3}$ The first part is trivial with $AM-HM $ inequality. Having problem with the second part.
2
votes
2answers
71 views

two variable nonhomogeneous inequality

Let $$x\ge 0,y\ge 0,x\neq 1,y \neq 1$$Prove the inequality $$\dfrac {x}{(y-1)^2} +\dfrac {y}{(x-1)^2} \ge \dfrac {x+y-1}{(x-1)(y-1)} $$
1
vote
1answer
57 views

Do $\ell_p$ distances contract when scaling down onto unit $\ell_q$ ball?

More specifically, is it true that $\Vert\frac{x}{\max\{1, \|x\|_q\}}-\frac{y}{\max\{1, \|y\|_q\}}\Vert_p\leq\Vert x-y\Vert_p$ for all $x, y$ when $p\geq 2$? ($q$ is the obvious norm parameter ...
4
votes
2answers
83 views

Ambiguity with Logs and Inequalities

Solve the following inequality: $$0.8^x > 0.4$$ Method 1 (Using the Common Logarithm) $$\log_{10}0.8^x > \log_{10}0.4$$ $$x\log_{10}0.8 > \log_{10}0.4$$ Because $$\log_{10}0.8 < 0$$ ...
0
votes
0answers
42 views

By $\sum_{j=1}^N a_{ij}\leq N$ is it possible to infer $\max_i|\textbf{p}_i(t)|\leq C \max_i|\textbf{p}_i(0)|$?

Consider $$\frac{d}{dt}\textbf{p}_i(t)=\bar{\textbf{p}}_i(t)-\textbf{p}_i(t), \ \ \ \bar{\textbf{p}}_i(t):=\sum_{j=1}^N a_{ij} \textbf{p}_j(t), \ \ \textbf{p}_i(t)\in\mathbb R^d.$$ and $A=\{a_{ij}\}$, ...
3
votes
2answers
68 views

Prove this inequality $(a_{1}a_{2}\cdots a_{n})\sqrt{1-a_{n+1}}+\sqrt{n-1}\cdot a_{n+1}<\sqrt{n}$

Assmue that $a_{i}\in (0,1),i=1,2,3,\cdots,n$,show that $$(a_{1}a_{2}\cdots a_{n})\sqrt{1-a_{n+1}}+\sqrt{n-1}\cdot a_{n+1}<\sqrt{n},$$ I've tried many things but all have failed.
3
votes
2answers
65 views

If $x,y,z>0$ and $xyz=32,$ Then the minimum of $x^2+4xy+4y^2+4z^2$ is

If $x,y,z$ are positive real no. and $xyz= 32\;,$ Then Minimum value of $$x^2+4xy+4y^2+4z^2$$ is $\bf{My\; Try::}$ Here I have Used $\bf{A.M\geq G.M}$ Inequality So $$\displaystyle ...
2
votes
3answers
100 views

nice classical nonhomogeneous inequality

Let $a,b,c$ be positive reals and $abc=1$. Prove that $$10(a^4+b^4+c^4)+21\ge 17(a^3+b^3+c^3).$$ I have found a solution using MV and I'm wondering if there is a nice solution.
3
votes
1answer
59 views

Sharper than Mean Value Inequality

Prove that $$|x\ln x-y\ln y| \le |x-y|^{1-1/e}$$ for $0<y<x\le 1$ Using the Mean Value theorem, all what I found that there exist $c\in (y,x)$ such that $$|x\ln x-y\ln y| \le ...
0
votes
1answer
47 views

prove the following inequalities

Let $a_1,a_2,\dots,a_{100}$ be real numbers, each less than one, satisfy $$a_1 +a_2+ \dots+ a_{100}\gt 1$$ $\mathbf (i)$ Let $n_0$ be the smallest integer $n$ such that $$a_1+a_2+\dots+a_n\gt ...
1
vote
1answer
60 views

Is there a “greater than about” symbol?

To indicate approximate equality, one can use ≃, ≅, ~, ♎, or ≒. I need to indicate an approximate inequality. Specifically, I know A is greater than a quantity of approximately B. Is there a way to ...
2
votes
1answer
71 views

How to show that $1/\cosh(x) < \sinh(x)/x < x/\sinh(x)$?

I was going through an old exam paper and I saw this question. How to show that for $0 < x < \pi/2$, $$\frac{1}{\cosh(x)} < \frac{\sin(x)}{x} < \frac{x}{\sinh(x)}\;?$$ I can see ...
-2
votes
1answer
21 views

problem from inequalities chapter [closed]

prove that if $(1+a_1)(1+a_2)...(1+a_n)=2^n$, then $a_1a_2a_3...a_n\le1$
1
vote
2answers
55 views

Inequality about logarithm

I have tried to prove the following inequality: $$ \left(1+\frac{\log n}{n}\right)^n \gt\frac{n+1}{2}, \mbox{for}\;n\in\{2,3,\ldots\} $$ which seems to be correct (confirmed by numerical result). ...
1
vote
1answer
21 views

How prove this $\max_i|\textbf{p}_i(t)|\leq \max_i|\textbf{p}_i(0)|$

Let $$\sum_{j=1}^N a_{ij}=1, \ \ \ \ i=1,\dots,N$$ and consider $$\frac{d}{dt}\textbf{p}_i=\bar{\textbf{p}}_i-\textbf{p}_i, \ \ \ \bar{\textbf{p}}_i:=\sum_{j=1}^N a_{ij} \textbf{p}_j.$$ Prove that ...
1
vote
3answers
38 views

Show that $1/\sqrt{1} + 1/\sqrt{2} + … + 1/\sqrt{n} \leq 2\sqrt{n}-1$ [duplicate]

Show that $1/\sqrt{1} + 1/\sqrt{2} + ... + 1/\sqrt{n} \leq 2\sqrt{n}-1$ for $n\geq 1$ I attempted the problem but I get stuck trying to show that if the statment is true for some $k\geq1$ then $k+1$ ...
1
vote
1answer
25 views

Why are these points included in the following example if we are getting < or > sign?

When we are asked to find the values of x for which $ f(x) $ is increasing or decreasing we put $ f'(x)>0 $ or $ f'(x)<0 $. so we get strictly greater than or less than sign. Hence our answer ...
3
votes
2answers
53 views

Exponential of Average Less than Average of Exponential

I am aiming to show the following inequality: $$\exp\left[\frac{1}{b-a}\int_{a}^bf(x)dx\right] \leq \frac{1}{b-a}\int_a^b \exp[f(x)]dx$$ where $f(x) \in C([a,b])$. Intuitively, this makes sense ...
0
votes
1answer
27 views

Minimise the result of an expression

What is the minimum value the expression ${a} + 3{b} + 3{c} + {d}$ can have if $$a, b, c, d \in \mathbb{N}$$ $${a} \neq {b} \neq {c} \neq {d}$$ and the sum of any two variables is not equal to ...
2
votes
2answers
65 views

Prove that if $|x-2|<0.001$, then $|\frac{1}{x}-\frac{1}{2}|<3\times 10^{-3}$

I still have difficulties with absolute value, and even if I manage to solve questions and problems, I do that awkwardly. So, please show me if this is the way to answer this question. Thank you in ...
1
vote
1answer
50 views

Triangle nequality with $\prod_{cyc}\left(2\sin^2{A}+\frac{1}{\sin^2{A}}\right)$

In $\Delta ABC$,show that $$\left(2\sin^2{A}+\dfrac{1}{\sin^2{A}}\right)\left(2\sin^2{B}+\dfrac{1}{\sin^2{B}}\right)\left(2\sin^2{C}+\dfrac{1}{\sin^2{C}}\right)\ge \left(\dfrac{17}{6}\right)^3$$ This ...
0
votes
1answer
41 views

An upper-bound problem of sum of positive numbers

I came across the following problem of inequality. If $ \ \ \sum_{i=1}^{n}x_i^3\leq S$ then find the value of $K$ such that $\sum_{i=1}^{n}x_i\leq K$. It is given that $x_i>0,\forall i\in ...
3
votes
3answers
146 views

How to prove this inequality $ \left|\frac{{\sin x}}{x} - \frac{{\sin y}}{y}\right| \le \sqrt {2\left|\frac{1}{x} - \frac{1}{y}\right|}$

$$ \left|\frac{{\sin x}}{x} - \frac{{\sin y}}{y}\right| \leqslant \sqrt {2\left|\frac{1}{x} - \frac{1}{y}\right|} $$ I tried to prove this by using mean value theorem, but I failed. Could you help ...
1
vote
1answer
70 views

Will $\kappa_1, \kappa_2, m$ cardinals. Given $\kappa_1 \leq \kappa_2$. prove: $\kappa_1 \cdot m \leq \kappa_2 \cdot m$

Will $\kappa_1, \kappa_2, m$ cardinals. Given $\kappa_1 \leq \kappa_2$. prove: $\kappa_1 \cdot m \leq \kappa_2 \cdot m$. Hi, I would be happy if someone could help me with this. What I did until ...
3
votes
2answers
109 views

Prove that $\left (1+\frac ba \right )\left (1+\frac ac \right )\left ( 1+ \frac cb\right )\ge 4+3\sqrt2$

Given $a,b,c>0$ and $a^2\ge b^2+c^2$. Prove that $$\left (1+\frac ba \right )\left (1+\frac ac \right )\left ( 1+ \frac cb\right )\ge 4+3\sqrt2$$ This is my try: I expanded the LHS, and I have ...
1
vote
1answer
41 views

what is the upper bound of $\max \mathbf{w}^T\mathbf{x}_i$

I need to find an equation for the upper bound of $\max \mathbf{w}^T\mathbf{x}_i, \; i=1, \dots N$. where $\mathbf{w}$ and $\mathbf{x}_i$ are two vectors. I need to find a function $f$ which holds ...
6
votes
3answers
162 views

Proving that $\sum_{i=1}^n\frac{a_i}{1-a_i}\leq\sum_{i=1}^n\frac{b_i}{1-b_i}$

Suppose $a_1,a_2,...,a_n>0$ and $\sum_{i=1}^na_i=1$. Define $b_1,b_2,...,b_n$ by $b_i=\frac{a_i^2}{\sum_{j=1}^n(a_j^2)}$. Show that ...
0
votes
2answers
42 views

an unclear step in a textbook solution of quadratic inequality

We have a quadratic inequality $$Ax^2+Bx+C>0$$ After solving it for cases where $B^2-4AC > 0$, my textbook turns to cases where $B^2-4AC < 0$: Using the perfect square method, let's ...
2
votes
1answer
34 views

An inequality for Lucas numbers

What are some methods to demonstrate the inequality $$\sum_{k=1}^{n} \sqrt{ \binom{n-1}{k-1} \, \frac{L_{k}}{k} } \leq \sqrt{L_{2n}}$$ where $L_{m}$ are the $m^{th}$ Lucas number ?
2
votes
2answers
29 views

Positive entropy of system of mixed substance (mathematical viewopoint)

If two substances respectively having mass $m_1$ and $m_1$, constant pressure specific heat capacities $c_{p1}$ and $c_{p2}$, and temperatures $T_1$ and $T_1$ are mixed at constant pressure, reaching ...
0
votes
1answer
43 views

Proving a simple Algebraic Inequality

I came up with a simple algebraic problem, which I have been spending sometime to prove it. Let $a,b,c,d$ be positive real numbers and $w_1 + w_2 = 1$ where both $w_1$ and $w_2$ are strictly ...