Do someone know an elegant proof for the inequality: $(1+x)^r \le 2^{r-1} + 2^{r-1}x^r, x>0,r>1.$
2026-03-26 04:53:09.1774500789
$(1+x)^r \le 2^{r-1} + 2^{r-1}x^r, x>0,r>1.$
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Since $f(x)=x^r$ is a convex function in $(0,+\infty)$ for $r>1$, by Jensen's inequality, $$\left(\frac{1+x}{2}\right)^r\leq \frac{1+x^r}{2},$$ i.e. $$(1+x)^r\leq 2^{r-1}+2^{r-1}x^r.$$