I had to prove the following inequality
$$a^{3/5}b^{2/5}\leq 3a/5+2b/5$$
For real and non negative a and b
I was able to prove it using Holder’s inequality and also AM-GM inequality. I need to know if can be done just by using the binomial theorem. Any help will be appreciated.
Per your question, the answer is no. Here is another approach which is already mentioned in the comment line. But I still post it here to remind you about the concavity of log. The function $f(x) = \ln x $ in concave $ \implies \dfrac{3}{5}\ln a + \dfrac{2}{5}\ln b \le \ln\left(\dfrac{3}{5}a+\dfrac{2}{5}b\right)$. Can you finish it ?