Good evening, I was wondering how to prove easily this inequality: \begin{equation} py^{p-1}(x-y)\leq x^p-y^p\leq px^{p-1}(x-y) \end{equation} for all $x,y\geq0$ and $p\in[1,+\infty)$. I tried to study both the functions $\frac{x^p-y^p}{y^{p-1}(x-y)}$ and $\frac{x^p-y^p}{x^{p-1}(x-y)}$, but I'm not getting anywhere.
I was trying to solve the following:  Given: $0 &lt; a &lt; b$ and $n&gt;1$, prove: $$na^{n-1}(b-a) &lt; b^n-a^n &lt; nb^{n-1}(b-a)$$  I managed to get this far using the mean value theorem: $$a^n(b-a)&lt;b^n -a^n&lt;b^n (b-a)$$ Any idea how to continue?
In an urn there are $n$ red balls and $m$ blue balls. I extract them without replacement. Let $X$=time of first blue. What is $E(X)$?  I found PMF of $X$ and it is, if $k &gt; n+1$  $$P(X=k) =\frac{n(n-1) \dots (n-k+1)m}{(m+n)(m+n-1) \dots (n+m-k+1)}$$  How could I evaluate $E(X)$?  Edit: I'm looking for an explicit form of $E(X)$