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- W996039869 abstract "Abstract. In this paper, we established some new inequalities via s−convexand s−concave functions. 1. introductionThe following inequality is well known in the literature as Simpson’s inequality: Z ba f(x)dx −b−a3f(a) +f(b)2+2fa+b2 (1.1)≤12880f (4)∞ (b− a) 5 ,where the mapping f : [a,b] → Ris assumed to be four times continuously differ-entiable on the interval and f (4) to be bounded on (a,b), that is,f (4)∞ = sup t∈(a,b) f (4) (t) < ∞.For some results which generalize, improve and extend the inequality (1.1), seethe papers [9]-[13].In [3], Hudzik and Maligranda considered among others the class of functionswhich are s−convex in the second sense. This class is defined in the following way:A function f : R + → R, where R + = [0,∞), is said to be s-convex in the secondsense iff(αx +βy) ≤ α s f(x) +β f(y)for all x,y ∈ [0,∞), α,β ≥ 0 with α + β = 1 and for some fixed s ∈ (0,1]. Thisclass of s−convex functions in the second sense is usually denoted by K 2s .It can be easily seen that for s = 1, s−convexity reduces to ordinary convexityof functions defined on [0,1).Some interesting and important inequalities for s−convex functions can be foundin [3]-[8].In [4], Dragomirand Fitzpatrick proveda variant ofHadamard’sinequality whichholds for s-convex functions in the second sense." @default.
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- W996039869 date "2011-12-12" @default.
- W996039869 modified "2023-10-18" @default.
- W996039869 title "Simpson type inequalities for functions whose third derivatives in the absolute value are s-convex and s-concave" @default.
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