In a paper that I'm reading, the authors use the following "elementary inequality" to further derive other inequalities. For $ \sigma > 0, \gamma \in [0,1], x \geq y$, and $\langle \cdot \rangle$ is the Japanese bracket, i.e. $\ \langle x \rangle = \sqrt{1+x^2}$, we have $$ e^{\langle x \rangle^{\gamma}} \langle x \rangle^{\sigma} - e^{\langle y \rangle^{\gamma}} \langle y \rangle^{\sigma} \lesssim \frac{|x-y|}{\langle x\rangle^{1 - \gamma} + \langle y \rangle^{1-\gamma}} \langle x \rangle^{\sigma} e^{\langle x \rangle^{\gamma}}$$
I am unable to prove this inequality. I have tried using the mean-value theorem or convexity arguments but it doesn not help with having both $\langle x \rangle^{1-\gamma}$ and $\langle y \rangle^{1- \gamma}$ in the denominator. Any hint would be appreciated.
