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Let $x, \ y, \ p$ be any real numbers with $x>0$, $y>0$, and $p>1$.

The question is about (most probably) an elementary inequality:

- Proof of Nested Interval Theorem: insufficient
- Why is the Cantor function not absolutely continuous?
- Prove that if $S$ is a finite set then $S$ has no limit points.
- Does $\mu_k(U \cap \mathbb{R}^k)=0$ for all affine embeddings of $\mathbb{R}^k$ in $\mathbb{R}^n$ imply $\mu_n(U)=0$?
- Is the space $C$ complete?
- Uniform convergence of difference quotients to the derivative

Is it always true that $x^p+y^p\leq (x+y)^p$ ?

Note that if $p$ is any positive integer, then the above inequality is obviously correct. What about if the number $p \ (\text{with} \ p>1)$ is any non-integer real number?

I guess that (by my intuition) the answer should be positive. But how can we proceed to prove this inequality?

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- Prove: If $\sum \limits_{n=1}^{\infty}|f_n|$ converges uniformly, so does $\sum \limits_{n=1}^{\infty}f_n$.
- Prove that every function that verifies $|f(x)-f(y)|\leq(x-y)^2$ for all $x,y$ is constant.

Write $x=r^2 \cos^2{\theta}$, $y = r^2 \sin^2{\theta}$, for $0 \leqslant \theta \leqslant \pi/2$ (i.e. polar coordinates on the first quadrant). Then you have

$$ \frac{x^p+y^p}{(x+y)^p} = \frac{r^{2p}\cos^{2p}{\theta}+r^{2p}\sin^{2p}{\theta}}{r^{2p} (1)} = \cos^{2p}{\theta} + \sin^{2p}{\theta}. $$

We clearly want to show that this is less than $1$.

Now, $0 \leqslant \cos^2{\theta},\sin^2{\theta} \leqslant 1$, and for $0 \leqslant x \leqslant 1$, we have $x^p \leqslant x$ since $p$ is at least $1$, so $x^{p-1}\leqslant 1$. Hence

$$ \cos^{2p}{\theta} + \sin^{2p}{\theta} \leqslant \cos^{2}{\theta} + \sin^{2}{\theta} = 1, $$

as required.

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