Prove an inequality by a convexity argument

Question

I need to prove the inequality $|x^{2k-1} -y^{2k-1}|\leq \frac{2k-1}{2}(x^{2k-2}+y^{2k-2})|x-y|, \,\,\forall x, y \in \mathbb{R},$ by using a convexity argument. I would like to refer to the following, $$|\frac{e^x-e^y}{x-y}| = \int_0^1 e^{tx+(1-t)y}dt \leq\int_0^1(te^x + (1-t)e^y) dt = \frac{1}{2}(e^x + e^y), \,\,\forall x, y \in \mathbb{R}.$$ Given that $x^k$ is convex in $\mathbb{R}$ for only $k$ even, I do not know how to handle this. Can somebody provide a solution or some hints ? Thanks.

Answer 1

You can use a similar argument as the one you are referring to. The inequality you want to prove only has odd exponents on the left side and even exponents on the right side. So you can use the convexity of $x^k$ for even $k$ and the fact that $\frac d {dx} x^{2k-1} = (2k-1)x^{2k-2}$ as follows: \begin{align*} |\frac{x^{2k-1}-y^{2k-1}}{x-y}|&=(2k-1)\int_0^1(tx+(1-t)y)^{2k-2}dt\\ &\leq(2k-1)\int_0^1(tx^{2k-2}+(1-t)y^{2k-2})dt \\ &=\frac {2k-1} 2(x^{2k-2}+y^{2k-2}) \end{align*}