Analytic inequalities with applications to special functions

AbstractSharp inequalities are derived for certain (polynomial-like) functions of the real variables pi (i = 1(1)σ) by interpreting pi as the probabilities that various switches be thrown in certain directions. Parameters mv in the inequalities are at first taken to be integers; later the inequalities are established when mv are arbitrary real numbers. The side condition ∑pi = 1 occurs throughout analysis, so there are many corollaries. Examples of the inequalities established are ∑i=1σ (1−pim)m>K−1, valid ifm>1∑j=0r njpjm(1−pm)m−j+1−∑j=0rnjpj(1−p−s)n−jm > 1+smax[m,n] valid if m > 1, n > r + 1, 0 < p, s, p + s ⩽ 1, and also valid if 0 < m < 1, 0 < n < r + 1 (1 − x)u + x1u < 1, if 12 < x < 1, u > 1. (1.03