Advanced Algebra?

I only studied retarded/subdued/garden-variety Algebra. So I'll attack your question here with my crude tools, and then I'll leave it hanging for others who were/are into Advanced Algebra.

{1/sqrt(b + a/2 + 1/2)} + {1/sqrt(c + b/2 + 1/2)} + {1/sqrt(a + c/2 + 1/2)} >= sqrt(2)

{1/sqrt[(2b+a+1)/2]} + {1/sqrt[(2c+b+1)/2]} + {1/sqrt[(2a+c+1)/2]} >= sqrt(2)

{sqrt(2)/sqrt(2b+a+1)} + {sqrt(2)/sqrt(2c+b+1)} + {sqrt(2)/sqrt(2a+c+1)} >= sqrt(2)

{1/sqrt(2b+a+1)} + {1/sqrt(2c+b+1)} + {1/sqrt(2a+c+1)} >= 1

I'm up to there only.

Maybe I'd expand that by eliminating all the radical signs, but I'm not yet too crazy.

{1/sqrt(2b+a+1)} + {1/sqrt(2c+b+1)} + {1/sqrt(2a+c+1)} >= abc ?

I don't know that too.