Conformable fractional approximations by max-product operators using convexity

Here, we consider the approximation of functions by a large variety of max-product operators under conformable fractional differentiability and using convexity. These are positive sublinear operators. Our study relies on our general results about positive sublinear operators. We derive Jackson-type inequalities under conformable fractional initial conditions and convexity. So our approach is quantitative by obtaining inequalities where their right hand sides involve the modulus of continuity of a high-order conformable fractional derivative of the function under approximation. Due to the convexity assumptions, our inequalities are compact and elegant with small constants.

Assume that D α f is continuous on [0, ∞). Then, We need Definition 1. 7 Let f ∈ C( [a, b]). We define the first modulus of continuity of f as:

Main results
We give Therefore it holds that By (11) and (12), we get: In particular, it holds that By assumption here D n+1 Next, we estimate (20).
We have proved that (case of t ≥ x 0 ) 2. We observe that (t ≤ x 0 ) In conclusion, we have established that By (6), we have Thus by (29) and (30), the claim is proved.
We rewrite the statement of Theorem 2.1 in a convenient way as follows: We need We call {L N } N ∈N positive sublinear operators.
We need a Hölder's type inequality; see next: , be a positive sublinear operator and f, g We make Then Furthermore, we also have that From now on, we assume that L N (1) = 1. Hence, it holds that We give Proof By (31) and (38).
We give Theorem 2.7 All as in Theorem 2.6. Additionally assume that Then, Proof By (39) and Theorem 2.4: we have proving the claim.

Applications
(I) Here we apply Theorem 2.7 to well known max-product operators.
We make Remark 3.1 In [5, p. 10], the authors introduced the basic max-product Bernstein operators where ∨ stands for maximum, and p N , These are nonlinear and piecewise rational operators.
see [5, p. 31]. B (M) N are positive sublinear operators and thus they possess the monotonicity property, also since |· − x| ≤ 1, then Furthermore, clearly it holds that The operator B (M) Proof By (47), we get that B , proving the claim.
We continue with  [5, p. 11]. By [5, p. 178-179], we get that Clearly, it holds that We give Proof By (53), we have T proving the claim.
We make Remark 3.5 Next, we study the truncated max-product Baskakov operators (see [5, p. 11]) Let λ ≥ 1, clearly then it holds that Also, it holds that U Proof By (60), we have that U (40) and (59), we get , proving the claim.
We give Proof By (74) proving the claim.
(II) Here, we apply Theorem 2.6 to the well-known max-product operators in the case of (n + 1) α ≥ 1, that is, when 1 n+1 ≤ α ≤ 1, where n ∈ N. We give proving the claim.
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