General inequalities, consequences and applications

We present a variety of sharp inequalities of integral, polynomial, coefficient, binomial, exponential, and other types. In particular, we prove inequalities for convolutions with weights generated by the gamma and beta functions. Applications include hypergeometric series, fractional integrals, bi-hermitian forms, and univalent functions. A family of positive definite kernels and related transformations naturally arise in the study. The formulation in terms of positive definite matrices is discussed as well. The research is associated with author's recent result on general inequalities with binomial weights.