On the Partial Product of Series and Related Basic Inequalities

The notation and terminology used in this paper are introduced in the following articles: [1], [9], [10], [5], [2], [4], [6], [7], [8], and [3]. For simplicity, we adopt the following convention: a, b, c are positive real numbers, m, x, y, z are real numbers, n is a natural number, and s, s1, s2, s3, s4, s5 are sequences of real numbers. Let us consider x. Note that |x| is non negative. We now state a number of propositions: (1) If y > x and x ≥ 0 and m ≥ 0, then x y ≤ x+m y+m .