# Show inequality using the Jensen inequality

X is a non-negative random variable, with $$\mathbb{E}(X) < \infty$$.

My goal is to show this inequality: $$\sqrt{1+(\mathbb{E}(X))^2} \leq \mathbb{E}(\sqrt{1+X^2})$$

x² is a convex function, so with the Jensen inequality I get that:

$$\sqrt{1+(\mathbb{E}(X))^2} \leq \sqrt{1+\mathbb{E}(X^2)} = \sqrt{\mathbb{E}(1+X^2)}$$

But when I use the Jensen inequality a second time, for the concave $$\sqrt{ }$$ function, I get that:

$$\sqrt{1+(\mathbb{E}(X))^2} \leq \sqrt{1+\mathbb{E}(X^2)} = \sqrt{\mathbb{E}(1+X^2)} \geq \mathbb{E}(\sqrt{1+X^2})$$

where the inequality is in the wrong direction.

Did I make a mistake? Or is more than the Jensen inequality needed to show this?

Try to show that the function $$x\mapsto\sqrt{1+x^2}$$ is convex.