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new-tag deleted-tag The tag bernoulli-inequality was created, but removed soon after that: math.stackexchange.com/posts/3580873/revisions Prove that $a,b>0$ and $a+b=1$ imply $a^{\frac{1}{(2a)^n}}+b^{\frac{1}{(2b)^n}}\leq 1$

Q: Prove that $a,b>0$ and $a+b=1$ imply $a^{\frac{1}{(2a)^n}}+b^{\frac{1}{(2b)^n}}\leq 1$

Playing with exponent I found this : Let $a,b>0$ such that $a+b=1$ then we have : $$a^{\frac{1}{(2a)^n}}+b^{\frac{1}{(2b)^n}}\leq 1$$ Where $n\geq 1$ a natural number . First of all I have tested until $n=30$ . The equality cases are $a=b=0.5$ or $a=0$ and $b=1$ . I have tr...