General inequalities, consequences and applications
The author proves a variety of sharp inequalities of integral, polynomial, coefficient, binomial, exponential and other types. This is a continuation of author's recent work [``Inequalities generated by the gamma function'', Adv. Appl. Math. 29, No. 1, 115--125 (2002; Zbl 1026.26007)] on inequalities involving the binomial coefficients \(d_{n}(\alpha)\), defined by \((1-z)^{-\alpha}=\sum_{n=0}^{\infty}d_{n}(\alpha)z^n\). The author proves far reaching extensions of his earlier results. He focuses on a convolution inequality with weights generated by the gamma and beta functions, which can also be viewed as an inequality for fractional integrals. Then the author gives its bi-Hermitian equivalent and considers applications of this result in hypergeometric series, fractional integrals and univalent functions. A family of positive definite kernels and related transformations naturally arise in this study. The formulation in terms of positive definite matrices is also discussed.
scientific article; zbMATH DE number 2164931