We introduce left general fractional Caputo style derivatives with respect to an absolutely continuous strictly increasing function g. We give various examples of such fractional derivatives for different g. Let f be a p-times continuously differentiable function on [a, b], and let L be a linear left general fractional differential operator such that L(f ) is non-negative over a closed subinterval I of [a, b]. We find a sequence of polynomials Q n of degree ≤ n such that L(Q n ) is non-negative

more »

... over I, and furthermore f is approximated uniformly by Q n over [a, b]. The degree of this constrained approximation is given by an inequality using the first modulus of continuity of f (p) . We finish with applications of the main fractional monotone approximation theorem for different g. On the way to proving the main theorem we establish useful related general results. Definition 1 ([4, p. 50]). Let α > 0 with α = m ( · is the ceiling of the number). Consider f ∈ C m ([−1, 1]). We define the left Caputo fractional derivative of f of order α as follows: for any x ∈ [−1, 1], where Γ is the gamma function, Γ (ν) = ∞ 0 e −t t ν−1 dt, ν > 0. We set