In this paper we prove that for any modulus of continuity on $[0,\infty)$ there exists a concave majorant that is infinitely differentiable on $(0,\infty)$ and satisfies an additional inequality. This extends the results of Stechkin and Korneychuk obtained previously without the requirement that majorants be differentiable.
"On a Concave Differentiable Majorant of a Modulus of Continuity." Real Anal. Exchange 27 (1) 123 - 130, 2001/2002.