I need also your expertise in finding a solution to the following problem:
It is known that $\forall x \in \mathbb{R}$ and $\forall c \in [1,\infty)$ that $$\sqrt[c]{1} + \sqrt[c]{e^x} \geq \sqrt[c]{1 + e^x} .$$
Let $k \in (0,1)$, Is there an inequality that regards the following problem $$ \sqrt[k]{1} + \sqrt[k]{e^x} \geq \sqrt[k]{1 + e^x},$$ i.e. let $k^\prime \in (1,\infty)$ such that $$ 1 + e^{k^\prime \cdot x} \geq (1 + e^x)^{k^\prime}??$$
I Think that Jensen equality would help but i want to make sure that maybe there is other inequality regarding this manner.
Please advice and and thanks in advance.
Take $k'=2$, and you get
$$1+e^{2x}\geq 1+e^{2x}+2e^x$$
which is clearly false because $e^x>0$.
So there is no such inequality.