Note on an open problem of Feng Qi.

ABSTRACT. In this paper, an integral inequality is studied. An answer to an open problem proposed by Feng Qi is given. Key words and phrases: Integral inequality, Cauchy’s Mean Value Theorem. 2000 Mathematics Subject Classification. 26D15. In [5], Qi studied a very interesting integral inequality and proved the following result Theorem 1. Let f(x) be continuous on [a, b], differentiable on (a, b) and f(a) = 0. If f ′ (x) ≥ 1 for x ∈ (a, b), then � b (1) [f(x)] 3 � � b �2 dx ≥ f(x) dx. If 0 ≤ f ′ (x) ≤ 1, then the inequality (1) reverses. a Qi extended this result to a more general case [5], and obtained the following inequality (2). Theorem 2. Let n be a positive integer. Suppose f(x) has continuous derivative of the n-th order on the interval [a, b] such that f (i) (a) ≥ 0 where 0 ≤ i ≤ n − 1, and f (n) (x) ≥ n!, then (2) � b [f(x)] n+2 � � b dx ≥ f(x) dx a Qi then proposed an open problem: Under what condition is the inequality (2) still true if n is replaced by any positive real number r? Some new results on this subject can be found in [1], [2], [3], and [4]. We now give an answer to Qi’s open problem. The following result is a generalization of