# Tagged Questions

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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### 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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### 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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### 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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### 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, ...
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### 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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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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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### 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 ...