In this paper, we present a Selberg type inequality in a Hilbert C*- module, which ia simultaneous extensions of the Cauchy-Schwarz inequality and the Bessel inequality in a Hibert C*-module. As an application, we give a generalization of the Selberg inequality in a Hilbert C*-module.

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We discuss the condition that the centre of a ring is an ideal. We also show that some classical commutativity results of Jacobson and Herstein have elementary proofs under the added assumption that the centre is an ideal.

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In the previous paper we gave elementary proofs of operator monotonicity of the representing function of the weighted arithmetic mean and some other related functions. In this note, we show some extensions and applications of those results.

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In previous paper we defined the derivative of mappings from a vector lattice into a complete vector lattice. In this paper we define an approximately derivative of mappings from a vector lattice into a complete vector lattice. Moreover we consider a relation between these two derivatives.

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In managing service provider businesses, it is important to decentralize consumers at peak time and increase sales not at peak time as well. Shy and Stanbacka[5] have dealt with this problem to explore optimal service hours under a specific ideal time distribution, and discussed the existence of optimal opening and closing times. In the actual environments, however, service providers strategically introduce a wide variety of special offers such as discounted price to collect more consumers. In this study, we deal with optimal service hours with a special offer of price discount immediately after the opening time and just before the closing time with the view to attracting extra consumers whose ideal and convenient service times are before the opening time and after the closing time. Under the ideal service time distribution by Shy and Stanbacka[5], the provider’s profit is first formulated as an objective function to be maximized and then clarified is the condition under which the service provider can earn more profit by special offers than without special offers. An optimal business hours is also explored to clarify the conditions where there exist optimal opening and closing times. Numerical examples are also presented to illustrate the proposed model formulation.

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Based on a curved exponential family, there is a regularity condition that the score function with random variables is the linear independence, which is commonly used in the information geometry. An equivalence relation to the regularity condition is that the Fisher information is positive definite under the curved exponential family. We investigate a key condition for two regularity conditions and we recognize it as the linear independence for the first derivative of natural parameter with respect to the parameter.

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In this paper we prove a mean value property for polycaloric functions in one space dimensional case. The proof given here is a slight modification of that of the recent paper by F.Da Lio and L.Rodino [3] and seems more straightforward.

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