Prove that if $x,y \in R$, and $1 \leq p < \infty$ then $|x+y|^p \leq 2^p(|x|^p+|y|^p )$
44403
Prove that $|a+b|^p \leq 2^p \{ |a|^p +|b|^p \}$
46760
To show that $P(|X-Y| \leq 2) \leq 3P(|X-Y| \leq 1)$
94119
Prove that $xy \leq\frac{x^p}{p} + \frac{y^q}{q}$
95293
Proof that $|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|},\quad x,y \geq 0$
155232
Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$
49217
How do I show that $a^p + b^p > (a + b)^p$ for $a,b>0$ and $0<p<1$?
109778
Proving $(x-c)|(p(x)-p(c))$
34768
Proving $(p \oplus q) \oplus r=p \oplus (q \oplus r)$
106767
$|p(z)| \leq M$ for $|z| \leq 1$ Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$
88328
Proof of $1+x\leq e^x$ for all x?
72081
$\forall x,y>0, x^x+y^y \geq x^y + y^x$
110275
Let $p>0$. Show the $\lim_{n\to\infty}(a^n/n^p)$ is equal to:
12484
Prove that $\lfloor 2x \rfloor + \lfloor 2y \rfloor \geq \lfloor x \rfloor + \lfloor y \rfloor + \lfloor x+y \rfloor$ for all real $x$ and $y$.
142785
How can I prove Holder inequality for $0<p<1$?
102478
$X_n\leq Y_n$ implies $\liminf X_n \leq \liminf Y_n$ and $\limsup X_n \leq \limsup Y_n$
47895
Prove $\lim\limits_{n\to\infty}n^{-1/(p+1)}(1^{1^p}\cdot 2^{2^p}\cdots n^{n^p})^{n^{-p-1}}=e^{-1/(p+1)^2}$
13572
$x^y+y^x>1$ for all $(x, y)\in \mathbb{R_+^2}$