Inequality Definition and 1000 Threads

Given: $$x>0,\, n\in\mathbb{N}$$ Prove: $$ (1+x)\times\left(1+x^2 \right)\times\left(1+x^3 \right)\times\cdots\times\left(1+x^n \right)\geq\left(1+x^{\large{\frac{n+1}{2}}} \right)^n$$

I'm beginning to read Spivak's Calculus 3ed, and everything is smooth until I reach page 12. My question is marked, between line 2 and 3. Why there's such sign change suddenly? In fact I tried with simple line 4 case and it's not in fact equal. I'm assuming that a and b is valid for all...

x>1,y>1 and z>1 prove :$\dfrac {x^4}{(y-1)^2}+\dfrac {y^4}{(z-1)^2}+\dfrac {z^4}{(x-1)^2}\geq 48$

I want to know that is it possible to show that $$ \int_{0}^{T}\Bigr(a(t )\Bigr)^{\frac{p+1}{2p}}dt\leq C\Bigr(\int_{0}^{T}a(t)dt\Bigr)^{\frac{p+1}{2p}} $$ for some ##C>0## where ##a(t)>0## and integrable on ##(0,T)## and ##p\in(\frac{1}{2},1)##. It is worth noting that this range for ##p##...

For what class of functions we have: $$ \int_{\Omega} [f(x)]^m dx \leq C\Bigr ( \int_{\Omega} f(x)dx\Bigr)^{m}, $$ where ##\Omega## is open bounded and ##f## is measurable on ##\Omega## and ##C,m>0##.

Find all $$x$$ in the interval $$[0, 2\pi] $$ which satisfies $$2\cos(x) \le \left|\sqrt{1+\sin (2x)}-\sqrt{1-\sin (2x)} \right|\le \sqrt{2}$$

When entangled photons are generated from a cascade of a Calciums' 6s level this inequality : n[y+z-] + n[x-y+] ≥ n[x-z-] is derived for what is equivalent to spin in photons. When the detectors at A and B are parallel the perfect anti correlations are due to conservation laws of angular...

Homework Statement Let ##x^2+y^2+xy+1 \geq a(x+y)## for all ##x,y \in R##. Find the possible integer(s) in the range of ##a##. Homework Equations The Attempt at a Solution I can rewrite this into ##(x+y)^2-xy+1 \geq a(x+y) \Rightarrow (x+y)(x+y-a)+1-xy \geq 0## but I don't think...

Homework Statement 3\sqrt{x}-\sqrt{x+3}>1 Homework Equations The Attempt at a Solution As obvious from the given inequality, x must be greater than zero. Rearranging and squaring both the sides, 9x>1+x+3+2\sqrt{x+3} \Rightarrow 4x-2>\sqrt{x+3} Squaring again, 16x^2+4-16x>x+3...

Show that : \left( {\sin x + a\cos x} \right)\left( {\sin x + b\cos x} \right) \leq 1 + \left( \frac{a + b}{2} \right)^2

Homework Statement x^2+x<0 [

I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.

$\pi\approx3.1416$ $A=\dfrac{1}{log_5 19}+\dfrac{2}{log_3 19}+\dfrac{3}{log_2 19}$ $B=\dfrac{1}{log_2\pi}+\dfrac{1}{log_3\pi}$ edit :$B=\dfrac{1}{log_2\pi}+\dfrac{1}{log_{\color{red}5}\pi}$ $Prove: \,\, A < B$

need help on this Show by induction that n^3 <= 3^n for all natural numbers n.

Homework Statement Find all numbers ##a## for each of which the least value of the quadratic trinomial ##4x^2-4ax+a^2-2a+2## on the interval ##0\leq x \leq 2## is equal to 3.Homework Equations The Attempt at a Solution I don't really know what should be the best way to start with this type of...

I guess the best way to start this is by admitting that my conceptual understanding of the Cauchy-Schwarz Inequality and the Lagrange Identity, as the title suggests, is not as deep as it could be. I'm working through Marsden's 3e "Basic Complex Analysis" and it contains a proof of the Cauchy...

i found this problem interesting in stack exchange unfortunately i will participate in discussion for 4 days(vacation) inequality - Prove $\sum_{i=1}^{n}\frac{a_{i}}{a_{i+1}}\ge\sum_{i=1}^{n}\frac{1-a_{i+1}}{1-a_{i}}$ if $a_{i}>0$ and $a_{1}+a_{2}+\cdots+a_{n}=1$ - Mathematics Stack Exchange...

IMO-2012: let $$a_2,a_3,...,a_n$$ [FONT=Arial]be positive real numbers that satisfy a2.a3...a​n=1 [FONT=Arial].Prove that, $$(a_2+1)^2.(a_3+1)^3...(a_n+1)^n>n^n$$ hint:

Let $a,b,c$ be positive real numbers with sum $3$. Prove that $√a+√b+√c≥ab+bc+ca$.

Homework Statement Let ##a,b,c## be positive real numbers with sum 3. Prove that \sqrt{a}+\sqrt{b}+\sqrt{c} \geq ab+bc+ca Homework Equations AM-GM inequalityThe Attempt at a Solution I don't really know how to start with. We are given ##a+b+c=3##. Also, ##2(ab+bc+ca)=(a+b+c)^2-(a^2+b^2+c^2)##...

Homework Statement 2x>3sinx-xcosx, 0<x<∏/2 Homework Equations The Attempt at a Solution One possible way is to draw the graph of the functions and compare but plotting a graph manually is not easy in this case. I want some other methods.

Homework Statement Find inverse of each. 1. y<x+1 2. y=2x/(x-2) Homework Equations Switch y and x? The Attempt at a Solution For 1. I switched y and x, so x<y+1. Do I have to switch the sign also? For 2. I switched y and x, so x=2y/(y-2). But I have to express the inverse...

Here's yet another assigned problem that I'm having difficulty with. I think I'm close to the end but am "nervous" (for lack of a better word) about whether or not I have used summation notation properly throughout the problem. Here it is: "Use the Schwarz inequality to establish that (...

Hello all, I am currently working through a proof in my Real Analysis book, by Royden/Fitzpatrick and I'm confused on a part. if f is a measurable function on E, f is integrable over E, and A is a measurable subset of E with measure less than δ, then ∫|f| < ε...

How many integers satisfy [SIZE="2"]{√n-√(23×24)}^2<1 I was able to solved this by trial and error method , but i want to know systematic step-wise solution.

Question: Find all numbers x for which \frac{1}{x}+\frac{1}{1-x}>0. Solution: If \frac{1}{x}+\frac{1}{1-x}>0, then \frac{1-x}{x(1-x)}+\frac{x}{x(1-x)}>0; hence \frac{1}{x(1-x)}>0. Now we note that \frac{1}{x(1-x)} \rightarrow ∞ as x \rightarrow 0 and \frac{1}{x(1-x)} \rightarrow 0 as x...

Homework Statement Use the triangle inequality to prove that \left| s_n - s \right| < 1 \implies \left| s_n \right| < \left| s \right| +1 Homework Equations The triangle inequality states that \left| a-b \right| \leq \left| a-c \right| + \left| c-b \right| The Attempt at a Solution...

Homework Statement If f and g are two distinct linear functions defined on R such that they map[-1,1] onto [0,2] and h:R-{-1,0,1}→R defined by h(x)=f(x)/g(x) then show that |h(h(x))+h(h(1/x))|>2 Homework Equations The Attempt at a Solution I assume f(x) to be ax+b and g(x) to be lx+m so...

Homework Statement Prove the following a>0, X is a non-negative function Ʃ_{n\in N} P(X>an)≥\frac{1}{a}(E[X]-a) Ʃ_{n\in N} P(X>an)≤\frac{E[X]}{a} The Attempt at a Solution I know that \sum_{n\in N} P(X>an)=\sum_{k \in N} kP((k+1)a≥X>ka)=\sum_{k \in N} E[k1_{[(k+1)a,ka)}(X)]...

Hi! I have to do this exercise: Define a finite probabilistic Space (Ω; Pr[ ]) and 2 events A,B⊆ Ω and Pr[A] ≠ Pr[B] so that we can verify that Pr[A∩B]>=9*Pr[A]*Pr[B] > 0. (1) ___________________________________________ I've been trying it but i have reached this conclusion: If Pr[A]>0...

in the clausius inequality is the temperature that of the system or of the surroundings? or is it temperature of the body receiving positive heat? (assuming the irreversibility is due to heat transfer with finite temperature difference) [borgnakke and sonntag-principle of entropy increase for...

I need a bit of help proving the following statement (n + 2)^n ≤ (n + 1)^n+1 where n is a positive integer. The (n+2) and (n+1) bases are making it hard for me solve this. I tried several time, I can't get the inductive step. Can someone lend me a little hand here? The base case is real...

Homework Statement x^2 \geq [x]^2 [] denotes Greatest Integer Function {} denotes Fractional Part Homework Equations The Attempt at a Solution x^2-[x]^2 \geq 0 \\ (x+[x])(x-[x]) \geq 0 \\ -[x] \leq x \leq [x] \\ Considering left inequality x \geq -[x] \\ \left\{x\right\}...

Homework Statement Prove that |a + b| ≤ |a| + |b|. Homework Equations |a| = √a2 The Attempt at a Solution Since |a| = √a2, then |a + b| = √(a + b)2 = √(a2 + 2ab + b2) = √a2 + √b2 + √(2ab) = |a| + |b| + √(2ab). And since the square root of a negative number is not defined...

Homework Statement I was discussing the proof for the Cauchy-Schwarz inequality used in our lectures, and another student suggested an easier way of doing it. It's really, really simple. But I haven't seen it anywhere online or in textbooks, so I'm wondering if it's either wrong or is only...

x>0 ,y>0 ,z>0 and xyz=1 ,prove : $ \dfrac{1}{(x+1)^2+y^2+1}+\dfrac{1}{(y+1)^2+z^2+1}+\dfrac{1}{(z+1)^2+x^2+1}\leq \dfrac{1}{2}$

Hi, I was hoping that someone might be able to please help me with this proof. Prove that var(x+y) ≤ 2(var(x) + var(y)). So far I have: var(x+y) = var(x) + var(y) + 2cov(x,y) where the cov(x,y) = E(xy) - E(x)E(y), but I'm not really sure to go from there. Any insight would be very...

Homework Statement lx/(x-2)l < 5 Homework Equations The Attempt at a Solution x/(x-2) < 5 x< 5x-10 10 < 4x 5/2 < x x/(x-2) > -5 x > -5x+10 6x > 10 x > 5/3 The answer is x < 5/3 and x > 5/2 so where did I go wrong on the second one?

For all positive integers $$m > n$$, prove that : $$\operatorname{lcm}(m,n)+\operatorname{lcm}(m+1,n+1)>\frac{2mn}{\sqrt{m-n}}$$

a,b,c >0 , prove that : $(1+\dfrac {a}{b})(1+\dfrac {b}{c})(1+\dfrac {c}{a})\geq 2(1+\dfrac {a+b+c}{\sqrt[3]{abc}})$

I have worked my way though the proof of the Cauchy Schwarz inequality in Rudin but I am struggling to understand how one could have arrived at that proof in the first place. The essence of the proof is that this sum: ##\sum |B a_j - C b_j|^2## is shown to be equivalent to the following...

I am stuck at the inequality proof of this convext set problem. $\Omega = \{ \textbf{x} \in \mathbb{R}^2 | x_1^2 - x_2 \leq 6 \}$ The set should be a convex set, meaning for $\textbf{x}, \textbf{y} \in \mathbb{R}^2$ and $\theta \in [0,1]$, $\theta \textbf{x} + (1-\theta)\textbf{y}$ also belong...

Hi, so here is my question that I am totally stumped on. for all real values of x and y, show that |x|+|y|≥ √(x^2+y^2 ) and find the real values of x and y in which equality holds. I sort of thought I could do the second part, but it confuses me with two pronumerals and how to get rid...

Deduce from the simple estimate that if $$1<\sqrt{3}<2$$, then $$6<3^{\sqrt{3}}<7$$. Hi members of the forum, This problem says the resulting inequality may be deduced from the simple estimate, but I was unable to do so; could anyone shed some light on how to deduce the intended result...

Let C[-pi,pi] be the set of continuous function from [-pi,pi] to C. Endow this with usual inner product (<f,g>= integral from -pi to pi of f multiplied by g conjugate, and let ||.|| be the corresponding norm). Let h(n) be Fourier coefficent of fNow, |h(n)|<_ 1/2pi( ||f||.||e^int||) by schwarz...

$ T_n=\left(1-\dfrac{1}{3^2} \right)+\left(1-\dfrac{1}{5^2} \right)+\left(1-\dfrac{1}{7^2} \right)+\cdots+\left[1-\dfrac{1}{(2n+1)^2} \right]$ prove: $ \sqrt{\dfrac{n+1}{2n+1}}<T_n<\sqrt{\dfrac{2n+3}{3n+3}}$

Homework Statement Find constraints on a,b,c \in \mathbb{R} such that \forall w_1,w_2,w_3 \in \mathbb{C} , (1) x = |w_1|^2(1-c) + a|w_2|^2 + c|w_1+w_3|^2 + |w_3|^2(b-c) \ge 0 and (2) x=0 \Rightarrow w_1=w_2=w_3=0 .Homework Equations The Attempt at a Solution I believe the solution is...

Homework Statement |4 + 2r - r^2| <1 Homework Equations 4 + 2r - r^2 = (r - (1+ √5) ) (r - (1 - √5)) The Attempt at a Solution I tried to use the roots but no use. How should I proceed?

What is the best way to prove it?

Homework Statement The task is to find all solutions of the following inequality: x+3^x <4 But I was trying to find a solution for this problem in general: x+a^x < b Homework Equations n/a The Attempt at a Solution a^x < b-x \text{log}_a(a^x) < \text{log}_a...