Abstract
We investigate a weighted Simpson-type identity and obtain new estimation-type results related to the weighted Simpson-like type inequality for the first-order differentiable mappings. We also present some applications to f-divergence measures and to higher moments of continuous random variables.
Highlights
Corollary 2.1 If we take q = 1 in Theorem 2.1, we have the following inequality for (α, m, h)-convex functions: a+b b
Introduction and preliminariesThe following inequality is named the Simpson integral inequality:1 6 f (r1) + 4f r1 + r2 2 + f (r2) ≤1 2880 f (4)∞(r2 – r1)4, r2 – r1 r2 f (x) dx r1 (1.1)where f : [r1, r2] → R is a four times continuously differentiable mapping on (r1, r2), and f (4) ∞ = supt∈(r1,r2)|f (4)(t)| < ∞
Let us recall that Miheşan [20] presented a class of mappings, called (α, m)-convex functions, as follows: A mapping f : [0, b∗] → R, b∗ > 0, is said to be (α, m)-convex if f λx + m(1 – λ)y ≤ λαf (x) + m 1 – λα f (y) for all x, y ∈ [0, b∗] and λ ∈ [0, 1] with some fixed (α, m) ∈
Summary
Corollary 2.1 If we take q = 1 in Theorem 2.1, we have the following inequality for (α, m, h)-convex functions: a+b b (ii) Putting h(t) = t(1 – t) and α = 1, we have the following inequality for (m, tgs)-convex functions: a+b b
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