Analytic Inequalities and Other Tools

Here we present analytic inequalities and other useful results which are applied in the following chapters of the book. Mainly, the inequalities evaluate sharply either finite sums of products of elements of two sequences or integrals of products of two functions. These represent linear continuous functionals on Euclidean spaces and various function spaces. We frequently consider weighted sums, when each term of the sum is multiplied by a positive weight coefficient, and weighted integrals, when the standard Lebesgue integral is replaced by the Lebesgue-Stieltjes one. Sometimes we also present classic unweighted versions of inequalities, which are commonly known, and referred to by the names of inventors. For each inequality, we provide the necessary and sufficient conditions for attainability. We do not try to present the most general result first, and deduce simpler cases from them. Therefore we first present, mostly elementary proofs of inequalities for sums, and then we prove analogous results for integrals.