Inequality regarding sum and root

Question

I need your expertise in order to solve the following problem:

Let $x \in \mathbb{R}$ and let $k \in (0,\infty)$ . It is known that $$ \sqrt{1} + \sqrt{e^x} \leq \sqrt{2} \cdot \sqrt{1 + e^x}.$$ How can this be generalized to any root, i.e. $$ \sqrt[k]{1} + \sqrt[k]{e^x} \leq \sqrt[k]{2} \cdot \sqrt[k]{1 + e^x} \ ?$$

Is it possible? If not what is the closest form of inequality that resembles the above inequality?

Please advise and thanks in advance.

Answer 1

No. Set $x=0$, then $2\sqrt[k]2\leq(\sqrt[k]2)^2$ so $k\leq1$. What is true is

$$\sqrt[k]1+\sqrt[k]{e^x}\leq\frac2{\sqrt[k]2}\sqrt[k]{1+e^x}$$

which can be seen as a case of the Generalized mean inequality/Power mean inequality or as a consequence of Jensen's inequality (they're essentially all the same).

Answer 2

for $k=10$ doesn't this inequality holds for example, set here $x=0$

Answer 3

You could differentiate the function $$f_k(x)=\dfrac{1+\sqrt[k]{e^x}}{\sqrt[k]{1+e^x}}$$ where $k\in\mathbb{N}$ is given. You obtain that $f_k$ always takes its maximum value when $x=0$ which is $$f_k(0)=\dfrac{2}{\sqrt[k]{2}}$$ hence $$1+\sqrt[k]{e^x}\leq\dfrac{2}{\sqrt[k]{2}}\sqrt[k]{1+e^x}$$