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a b s t r a c tIn this paper, splitting finite sums, refinements of the inequalities of Aczél, Popoviciu and Bellman are obtained.

In this note a procedure to deduce new identities involving elementary rational sums is presented and applying this technique some sums including binomial coefficients and harmonic numbers are obtained.

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Analyzing data type produced, stored, and aggregated in Big Data environments is a challenge in understanding data quality and represents a crucial support for decisionmaking. Big Data application modeling requires meta-data modeling, interaction modeling, and execution modeling. Entropy, relative entropy, and mutual information play important roles in information theory. Our purpose within this paper is to present a new upper bound for the classical Shannon's entropy. The new bound is derived from a refinement of a recent result from the literature, the inequality of S. S. Dragomir (2010). The reasoning is based on splitting the considered interval into the mentioned inequality. The upper bound can be considered in understanding the potential information that each data type may have in a Big Data environment.Then summating all m C 1 previous inequalities, we obtain D.f; p; xI J; J 1 ; J 2 ; : : : ; J m / 6 m X i D1