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This note is concerned with some new integral inequalities which are extensions of the results in [2]. The method by which these results are obtained is due to D. C. Benson [1]. Throughout the present note we shall assume 1<p<∞ and f(x) a non-negative measurable function.

Let p > l, r≠1, and let f(x) be a non-negative function defined in [0, ∞). The following inequality is due to G. H. Hardy [5, Ch. IX]:1.1where according as r>1 or r < l.

As products and processes become more and more complex, there is an increasing need in the industry to perform experiments with a large number of factors and a large number of levels for each factor. For such experiments, application of traditional designs such as factorial designs or orthogonal arrays is impractical because of the large number of runs required. As an alternative, a type of design, called the uniform design, can be used to solve such problems. The uniform design has been intensively studied by theoreticians for several decades and has many successful examples of application in industry. In this article, we report a successful application of uniform design in product formation in the cement manufacturing industry. Specifically, we investigate the effects of additives on bleeding and compressive strength of a cement mixture. This example illustrates how an experiment of 16 runs was performed to study three factors with 16 levels, 8 levels, and 8 levels, respectively.

Abstract. In this paper we give complete characterizations, in terms of Dini numbers and integrals, of positive functions $(u) defined in (0, oo) satisfying the conditions: (i) $(«)/«" is nondecreasing and (ii) ^(u)/ub is nonincreasing. By applying these results we obtain necessary and sufficient conditions for power series and trigonometric series to satisfy a certain Lipschitz condition, which include some known results of R. P. Boas, Jr.[1]. We also give complete characterizations of positive functions («) defined in (-oo, oo) satisfying the conditions: (i) ^(u)/eau is nondecreasing and (ii) *(«)/e'>" is nonincreasing.