# inequality with constant power

Let $a$ and $b$ be two real numbers which are $\geq 1$.

I am wondering if one can get the following upper bound $$\frac{(a+b)^{a+b-1/2}}{a^{a-1/2}b^{b-1/2}}\leq (a+b)^{c},$$ where $c$ is some constant, i.e. independent of $a$ and $b$.

Thank you.

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Set $a=b=x$. On the left, we get $\dfrac{2^{2x}}{\sqrt{2x}}$.
On the right we get $(2x)^c$, which for any constant $c$ grows in the long run much more slowly than $\dfrac{2^{2x}}{\sqrt{2x}}$. For any constant $c$, and large enough $x$, the desired inequality does not hold.