We define the least r-concave majorant for the modulus of continuity of order r on \mathbf{C}[a,b], denoted by \overset{- _r}{\omega}_r and we establish the inequality

    \[\overset{- _r}{\omega}_r(f,2t)\leq 2^rK_r^1\pr{f,t;\mathbf{C}[a,b], \mathbf{C^r}[a,b]}, 0<t\leq \dfrac{b-a}{2r}.\]

Additional Information

Author(s)

Talpău Dimitriu, Maria