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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.

Graph of $(x + 1)\ln(x)$

Thank you for your testimonial

insipidintegrator
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Barry Alen
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  • second derivative is $(x-1)/x^2$ so concave on interval $(1,\infty).$ – coffeemath Feb 11 '23 at 04:21
  • It is because the concavity of this problem is segmented, while the domain of definition is > 0. I am now thinking whether concavity is the right entry point. But the construction of a, b, $\frac{a+b}{2}$, is too attractive. – Barry Alen Feb 11 '23 at 04:37
  • Does it answer your question https://math.stackexchange.com/questions/4277352/prove-2x-y-ln-fracx-y2-x-1-ln-x-y-1-ln-y-ge-0 – Miss and Mister cassoulet char Feb 11 '23 at 09:36

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