Questions on proving, manipulating and applying inequalities.

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How do I show that $a^p + b^p > (a + b)^p$?

A question on my math homework asks us to show that if $0 < p < 1$ and $a, b > 0$, then $a^p + b^p > (a + b)^p$. I have no idea how to do this, any pointers?
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3answers
59 views

Prove $(x^2-2)\sin(x)+2x\cos(x)\geq 0$ for $x$ on $[0,\pi/2)$

Originally comes from the question How to prove SinA/A+sinB/B+SinC/C<(9*(3)^.5)/2pi. The goal is proving $\frac{\sin(x)}{x}$ concave down on $[0,\pi/2)$, which I find it non-trivial.
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1answer
54 views

Establishing a inequality

I need to establish an inequality containing $19$ terms in $7$ variables. My problem arises in the context of proving that the HIV-only Quasi Disease-Free Equilibrium of a HIV-TB co-epidemic model ...
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1answer
239 views

upper bound on a matrix norm

what is the smallest upper bound for the following norm $\|\left(\lambda\ I +A\ A^T\right)^{-1}\|<?$. where, A is a rectangular matrix, $\lambda>0$ is a scalar. (any possible norm)
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1answer
37 views

Proving that $1+x^py^{1-p} \le (1+x)^p(1+y)^{1-p}$

I'd like to show that $1+x^py^{1-p} \le (1+x)^p(1+y)^{1-p}$ for all $x,y \ge 0$ and $p \in [0,1]$. I started out by defining a function $f(x,y)=(1+x)^p(1+y)^{1-p}$ and checking for critical points, ...
1
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1answer
143 views

How to prove SinA/A+sinB/B+SinC/C<(9*(3)^.5)/2pi

Only for an acute angle triangle. $A$,$B$,$C$ are angles of a triangle. This isnt sine rule form. Ive tried Cauchy Schwarz theorem , A.M, G.M form but am unable to get the above result. Could someone ...
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2answers
56 views

Question about Mobius function.

Let $N \in \mathbb{N}.$ I would know if is it true that $$-\underset{k\mid N}{\sum}\mu\left(k\right)\log\left(k\right)>0.$$I know that $$-\mu\left(k\right)\log\left(k\right)=\underset{r\mid k}{\sum}...
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7answers
249 views

How can I prove $(a+b+c)!>a!b!c!$

In fact, I couldn't prove the inequality because I don't know which method is used for this. The condition for this inequality is $$(a+b+c)>1$$
2
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1answer
34 views

How to justify that $d_{\mid\cdot\mid} ^{\alpha} (x,y) = |x-y|^{\alpha}$ (with $0<\alpha <1$) is a distance on $\mathbb{R}$

I know directly that : $\forall x,y \in \mathbb{R}$, $d_{\mid\cdot\mid}^\alpha (x,y)=0$ $\Leftrightarrow$ $x=y$ $\forall x,y \in \mathbb{R}$, $d_{\mid\cdot\mid}^\alpha (y,x) = d_{\mid\cdot\mid}^\...
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2answers
179 views

Inequality $(1+x_1)(1+x_2)\ldots(1+x_n)\left(\dfrac{1}{x_1}+\dfrac{1}{x_2}+\cdots+\dfrac{1}{x_n}\right)\geq 2n^2.$

Let $n\geq 2$, and $x_1,x_2,\ldots,x_n>0$. Show that $$(1+x_1)(1+x_2)\ldots(1+x_n)\left(\dfrac{1}{x_1}+\dfrac{1}{x_2}+\cdots+\dfrac{1}{x_n}\right)\geq 2n^2.$$ For $n=2$, this reduces to $(1+x_1)(1+...
0
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1answer
86 views

proving limits with Epsilon-Delta definition

Say I've got $\lim_{x\to a} f(x)= A$ and $\lim_{x\to a} g(x)= B$ How do I do the following: prove that if A < B, then there is the existence of a $\delta$ such that when $0 < |x-a| < \...
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3answers
63 views

$\varphi(n) \leq 5$, where $\varphi$ is the Eulerian function

If $\varphi(n) \leq 5$, then can we find a bound for $n$ itself, where $\varphi$ is the Eulerian function?
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2answers
95 views

Maximum of $\sin A\sin B\cos C+\sin B\sin C\cos A+\sin C\sin A\cos B$ in triangle

What is the maximum value of $$\sin A\sin B\cos C+\sin B\sin C\cos A+\sin C\sin A\cos B,$$ where $A,B,C$ are angles in a triangle? We can rewrite as $$-\sin A\sin B\sin(A+B)+\sin B\sin(A+B)\cos A+\...
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1answer
169 views

How could I proof that there can not be equality in Chebyshev's inequality?

For $k>0$. I have gotten the expresion $F(\mu+\sigma k)-F(\mu-\sigma k) = 1-1/k^2$ for all $k>0$. I can not see why this equality is not possible for any continuous RV, what does this mean for ...
2
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3answers
154 views

floor ceiling proof

Hi I would like to proof without using induction that: $$ \left\lceil\frac{n}{m}\right\rceil \leq \frac{n+m-1}{m} $$ I tried: $$ \left\lceil\frac{n}{m}\right\rceil \leq \frac{n}{m}+\frac{m}{m}-\...
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1answer
56 views

How to deal with a sum in the denominator

In a book, I saw the following inequality: $$e^{-\frac{\lambda^2}{2 (\lambda + \Delta)}}\leq e^{- \min\{\lambda, \frac{\lambda^2}{\Delta}\}/4},$$ for some (specific) $\lambda, \Delta \geq 0$ Does ...
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3answers
80 views

Conjectured Inequality

I noticed that for positive integers $(a,b,c)$, it happens that $ab+bc+ca$ is always greater than $abc+1$. Is this true, and if it is, how would I prove it?
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1answer
33 views

Riemann Integrables

Let $f: [a,b] \rightarrow \mathbb{R}$ and $g: [a,b] \rightarrow \mathbb{R}$ Prove the following. $f \le M \implies \int^b_a f(x)dx \le M(b-a)$ $f+g \in \mathbb{R[a,b]}$ and $\int^b_a [f(x)+g(x)]dx =...
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1answer
47 views

Integral of the log is less than the integral of the log of the average value

This is an interesting property that I came across while reading an old proof on this website. The poster didn't really explain it, so I thought I might ask. We suppose $u$ is a positive measure on ...
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2answers
265 views

How are some equations and inequalities called identities (how do they have infinite solutions?)?

I've seen identities, which means they have infinite solutions.$\longleftarrow$This is incorrect; see the comments below. Examples are$$4x+6=2(2x+3)$$$$9q-6\lt9q+3$$$$12u\le3(4u)$$$${x\over 9}={x\...
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2answers
3k views

Why do some equations or inequalities have no solution?

I've seen some equations and inequalities that have no solution. Examples of these are$$3m+4=3m-9$$$$128y-10\lt128y-25$$$$10t+45\ge2(5t+23)$$The third example evaluates to$$10t+45\ge10t+46$$using the ...
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2answers
460 views

A conjecture concerning primes and algebra

A monoid morphism $\psi:\mathbb Z_+\!\!\rightarrow\mathbb Z_+$ is defined by an arbitrary function $f:\mathbb Z_+\!\!\rightarrow\mathbb Z_+$ and defines a group homomorphism $\varphi:\mathbb Q_+\!\!\...
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1answer
35 views

Easy (?) estimation about prime powers

Let $N_k$ be some integers with $\sum_{k\mid n}kN_k=p^n$. How can I prove $$\frac{p^n}{n}-\frac{2p^{n/2}}{n}\leq N_n?$$
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1answer
38 views

Reason behind solution in this inequality with absolute values

Solve the inequality $|3x-2|-|x+2|>x$ When $|x+2|<0$: $-(3x-2)+(x+2)>x\iff x <\frac{4}{3}$ When $|x+2|>0\land |3x-2|<0$: $-(3x-2)-(x+2)>x\iff x < 0$ When $|3x-2|>0$: $...
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1answer
45 views

prove $\limsup_{n\to\infty}\frac{\sqrt{2({{x_1}^2+{x_2}^2+…+{x_n}^2})}}{n}\leq\limsup_{n\to\infty}\frac{x_n}{\sqrt{n}}$

Let $x_1,x_2,x_3$,... be a sequence of nonnegative real numbers. Prove that $\limsup_{n\to\infty}\frac{\sqrt{2({{x_1}^2+{x_2}^2+...+{x_n}^2})}}{n}\leq\limsup_{n\to\infty}\frac{x_n}{\sqrt{n}}$ I try:...
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3answers
187 views

How to prove the inequality $2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$?

Prove that for any positive integer $n$, $$2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$$ Progress I think Riemann sum should be used for ...
1
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1answer
341 views

Real Analysis Riemann Integration - Strict Monotonicity for Integrals

If $f,g$ are Riemann integrable on $[a,b]$, and $f(x) < g(x)$ for all $x \in [a,b]$, prove that $$ \int_a^b f(x) \,dx < \int_a^b g(x) \,dx$$ This is a strict inequality. I know how to prove the ...
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3answers
48 views

Show that $a \lt \frac{a + b}{2} \lt b$ for $a\lt b $ and $a, b \in \mathbb{R}$

How can I prove this statement true? I have tried saying starting like this: $a = 0; \qquad b>0.$ But I don't know where to proceed from here
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1answer
86 views

Tricky Substitution to get AM-GM inequality

So, I'm reading the literature to find different proofs of the AM-GM inequality, the following proof quite hit me, and I don't seem to understand at all. The proof is as follows: For any positive ...
2
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1answer
68 views

What's bigger, the sum of powers or the power of the sum?

Do we know if $(\sum\limits_{i=1}^n a_i)^k \geq \sum\limits_{i=1}^n a_i^k$ for any $k\geq1$?
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2answers
79 views

The minimum value of $\frac{(x+\frac{1}{x})^6-(x^6+\frac{1}{x^6})-2}{(x+\frac{1}{x})^3+x^3+\frac{1}{x^3}}$

Problem : The minimum value of $$\frac{(x+\frac{1}{x})^6-(x^6+\frac{1}{x^6})-2}{(x+\frac{1}{x})^3+x^3+\frac{1}{x^3}}$$ Can I use this in numerator and denominator : The minimum value of $a +\frac{...
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3answers
64 views

Finding the range of equation. Any tricks?

I m working on the following problem For real numbers $a,b$, if $a+ab+b=3$, then find the range of $m=a-ab+b$. Is there any inequalities here to use?
2
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2answers
88 views

Applications of the limit $n$th root of $n$.

I have the following question: Given that I've already proven that $$ \lim_{n \rightarrow \infty} \sqrt[n]{n}= 1,$$ let $a_n = \sqrt[n]{n}$. I want to prove that $ a_n > a_{n+1}$. My textbook ...
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2answers
86 views

Is it true that $|x^a - y^a| \leq |(x-y)^a|$ on $[0,1]$, where $a\le 1$?

It looks to me like for a function $f(x) = x^a$ on the domain $[0,1]$ where $a \leq 1$, and $x,y$ are points in the domain, $|x^a - y^a| \leq |(x-y)^a|$ I would like to use this in a proof and so if ...
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1answer
81 views

$ (1+\sin{x})^{\cos{x}} + (1+\cos{x})^{\sin{x}} > 3x $

How do I show that, for $ 0 < x < \dfrac{\pi}{4} $ (first quadrant), the inequality $ (1+\sin{x})^{\cos{x}} + (1+\cos{x})^{\sin{x}} > 3x $ is valid? I've tried Bernoulli's, but it took me to ...
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4answers
46 views

Prove $(n!)/n^n \leq 1/2^{k}$, where $k$ is the floor of $n/2$.

I suppose the natural way to prove this is by induction. When I follow the rather natural steps $$\frac{(n+1)!}{(n+1)^{n+1}} = \frac{n!}{(n+1)^{n}} \leq \frac{n!}{(n)^{n}}$$ in order to apply the ...
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1answer
56 views

Proving a bound on $f:(0,\infty)\rightarrow \mathbb{R}$, supposing another bound holds

Let $f:(0,\infty)\rightarrow \mathbb{R}$ be such that $$f(x)\leq e^{C \cdot t^2\cdot s^2-t\cdot x\cdot s},\quad \quad(1)$$for all $t\in[0,1]$ and for some positive $s,C$. I have to prove the bound $$...
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2answers
25 views

If the graph of the function f(x) = $2x^3+ax^2+bx , a,b \in N$ cut the x -axis at three distinct points,

If the graph of the function f(x) = $2x^3+ax^2+bx , a,b \in N$ cut the x -axis at three distinct points, then find the maximum values of a+b. Please suggest as I am not getting any clue on this how ...
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3answers
938 views

If a,b,c $>0$ and a+b+c=1, then find the maximum / minimum value of the following

If a,b,c $>0$ and a+b+c=1, then find the maximum / minimum value of the following : (a) abc (b) $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ (c) $(1+\frac{1}{a})(1+\frac{1}{b})(1+\frac{1}{c})$ ...
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1answer
60 views

Is it correct to approach this with Holder Inequality? What am I doing wrong?

I know that $ \forall n\in N, a_0 + a_1 + ... + a_n = 1$, with $a_0, a_1, ... a_n > 0$ and $f(t) = a_0t^n+a_1t^{n-1}+...+a_n, \forall t\in R$ I have to prove that for every $x > 0$ $$ f^2\left(...
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1answer
31 views

If $x\in \mathbb{R}^n$ and is a unit vector, why is $\sum\limits_{j,k=1}^n |x_j||x_k| < n^2$?

This is an excerpt of a larger proof: Other pertinent information: $A$ is a positive definite $n \times n$ matrix The set $C$ is the unit sphere I don't get the last inequality: $\gamma \sum \...
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1answer
56 views

what is the area of the polygon with given constraints?

What is the area of the polygon formed by all points $(x, y)$ in the plane satisfying the inequality $ ||x| – 2 | + | |y| – 2 | ≤ 4 $ ?
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0answers
61 views

Trigonometric sum evaluation

Let $q$ a prime number and $1 \leq a<q$ a positive integer. We know from Ramanujan identity that $$\underset{h=1,\left(h,q\right)=1}{\overset{q}{\sum}}e^{2\pi ih\frac{a}{q}}=\underset{h=1}{\overset{...
3
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5answers
95 views

How prove this $xyzw>0$

let $x,y,z,w\neq 0$ are real numbers,and such $$x+y+z+w=\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}+\dfrac{1}{w}=0$$ show that $$xyzw>0$$ My idea: let $$x+y+z+w=p=0,xy+xz+xw+yz+yw+zw=q,xyz+xyw+yzw+...
0
votes
2answers
61 views

Sum of modulus of complex numbers : $|\sin(z)|+|\cos(z)| \geq 1$

I'm trying to establish if $|\sin(z)|+|\cos(z)|$ is greater than or equal to $1$. I have tried to write out the expression in exponential form, but I don't really arrive at anything useful. I ...
5
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4answers
221 views

How prove $\pi^2>2^\pi$

show that $$\pi^2>2^\pi$$ I use computer found $$\pi^2-2^\pi\approx 1.044\cdots,$$ can see this I know $$\Longleftrightarrow \dfrac{\ln{\pi}}{\pi}>\dfrac{\ln{2}}{2}$$ so let $$f(x)=\dfrac{\...
2
votes
4answers
79 views

Prove $\left|\sum_{i=1}^n x_i y_i \right| \le \dfrac{1}{a} \sum_{i=1}^n {x_i}^2 + \dfrac{a}{4}\sum_{i=1}^n {y_i}^2$

If $X,Y$ are vectors in $\mathbb{R}^n$ and $a>0$ show that: $$\left|\sum_{i=1}^n x_i y_i \right| \le \dfrac{1}{a} \sum_{i=1}^n {x_i}^2 + \dfrac{a}{4}\sum_{i=1}^n {y_i}^2 (*)$$ I started with ...
1
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1answer
32 views

Why this inequality is true?

Why this following two inequalities are true? Where $c$ is a constant. $P(X_n+Y_n\le x) \ge P(X_n\le x-c-\epsilon) -P(Y_n>c+\epsilon)$ $P(X_n+Y_n\le x) \le P(X_n\le x-c+\epsilon) +P(Y_n<c-\...
0
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1answer
39 views

For an entire function $|f(z)| \le \sup_{\xi \in \partial D} |f(\xi)|$ for all $z \in D$

I was presented with a fact that for any entire function $f$ and any open disk $D$ it is true that $\forall z \in D$ $$ |f(z)| \le \sup_{\xi \in \partial D} |f(\xi)|. $$ But why is it true? Maybe it ...
0
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2answers
59 views

Can multiple inequality symbols be used in an inequality?

About these "inequalities" (I don't know if I'm being sarcastic): $$1 \lt 4 \gt 3$$$$6 \lt 8 \lt 15 \gt -1$$$$(6\cdot9)\gt(5\cdot10)\gt(2\cdot7)\lt(3\cdot5)\gt(-1\cdot3)$$ can multiple inequality ...