Abstract
In this study, we generalize and sharpen an integral inequality raised in theory for convex and star-shaped sets and relax the conditions on the integrand.Mathematics Subject Classification (2000): 26D15
Highlights
In the study [1], which investigated convex and star-shaped sets, the following interesting result was obtained.Theorem 1. ([1, Lemma 2.1]) Let p : [0, T ] ® R be a nonnegative convex function such that p(0) = 0
We shall show that the convexity of the function p(t) may be replaced by the condition that p(t) t is increasing, sharpen inequality
Competing interests The author declares that they have no competing interests
Summary
In the study [1], which investigated convex and star-shaped sets, the following interesting result was obtained. ([1, Lemma 2.1]) Let p : [0, T ] ® R be a nonnegative convex function such that p(0) = 0. We shall show that the convexity of the function p(t) may be replaced by the condition that p(t) t is increasing, sharpen inequality (1), and obtain the following a general result using a monotone form of l’Hospital’s rule, a elementary method, and Mitrinović-Pečarić inequality, respectively. Let p : [0, T ] ® R be a nonnegative continuous function such that p(0). These paired numbers a and b defined in (i) and (ii) are the best constants in (2)
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