Weighted inequalities and negative binomials

The following topics and their interconnection are discussed: 1. a general product inequality for the weighted seminorms on the vector space of formal power series and its special cases and applications; 2. the properties and applications of the binomial coefficients d n ( α ) that arise in the expansion ( 1 − z ) − α = ∑ n = 0 ∞ d n ( α ) z n with α > 0 . The recursive methods of proof and the new product inequalities (4-parameter generalizations of the classical Hölder inequality) are presented. It is shown that the product inequality with the binomial weights constructed of coefficients d n ( α ) is of particular importance as it leads to a variety of applications. The applications include the sharp weighted norm inequalities for complex-valued functions, exponential inequalities of binomial type, coefficient inequalities for univalent functions, some properties of generalized hypergeometric series, entire functions, convolutions, Bernstein polynomials, and Laplace–Borel transforms, as well as the various pure binomial and matrix results.