Prove: $2\left(a+b\right)\ln{\left(\frac{a+b}{2}\right)}\geq\left(a+1\right)\ln{\left(a\right)}+\left(b+1\right)\ln{\left(b\right)}$
Graph of the two-variable function
A diagram was drawn to show that this inequality is correct, but I don't know how to prove it.
At first I considered this problem as a deformation of
$\left(a+b\right)\ln{\left(\frac{a+b}{2}\right)}\le\left(a\right)\ln{\left(a\right)}+\left(b\right)\ln{\left(b\right)}$
Constructing $x\ln(x)$ in this way, using concavity, solves.
Although the signs of greater than and less than are reversed.It can be solved if the concavity of the two sides is also opposite.
But the concavity of the present problem $(x + 1)\ln(x)$ is uncertain.
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