In this paper generalizations of Steffensen's inequality obtained by Pečarić, Mercer and Wu-Srivastava are further extended in a measure theoretic sense. Motivated by Wu and Srivastava's refined and sharpened version of Mercer's result related inequalities for positive Borel measures are obtained.
We prove weaker conditions for Steffensen type inequalities obtained by Masjed-Jamei, Qi and Srivastava. Moreover, we extend these inequalities to the class of convex functions. Further, we give an application of new inequalities to obtain Stolarsky type means.
Mathematics SubjectClassification. Primary 26D15; Secondary 26A51. The well-known Steffensen inequality reads, [10]: Theorem 0.1. Suppose that f is nonincreasing and g is integrable on [a, b] with 0 ≤ g ≤ 1 and λ = b a g(t)dt. Then we haveThe inequalities are reversed for f nondecreasing.In [5] Masjed-Jamei, Qi and Srivastava obtained the following Steffensen type inequalities:Theorem 0.2. If f and g are integrable functions such that f is nonincreasing andon (a, b), where q = 0 and
Abstract. We find necessary and sufficient conditions for Steffensen's and inverse Steffensen's inequality in measure theoretic settings. We also explain natural transition from the Riemann integral in Steffensen's inequality to an integral with measure.Mathematics subject classification (2010): Primary 26D15; Secondary 26D20.
Abstract. The object is to obtain weaker conditions for the parameter λ in Steffensen's inequality and its generalizations and refinements additionally assuming nonnegativity of the function f . Furthermore, we contribute to the investigation of the Bellman-type inequalites establishing better bounds for the parameter λ.
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