Miscellaneous sharp inequalities and Korovkin-type convergence theorems involving sequences of probability measures (Q1063196)

We generalize a theorem due to \textit{P. P. Korovkin} (see [Usp. Mat. Nauk 13, No.6(84), 99-103 (1958; Zbl 0090.283)]) to sequences of arbitrary probability measures on [0,\(\pi\) ]. Korovkin's result is concerned with the convergence of certain ratios of the Fourier coefficients of a sequence of density functions. Earlier, \textit{E. L. Stark} (see [Nederl. Akad. Wet. Proc., Ser. A 75, 227-235 (1972; Zbl 0238.42006)]) gave a different generalization of this Korovkin theorem. Analogous characterizations are given for the same type of ratios of the hyperbolic coefficients (respectively, the Laplace transforms) of a sequence of probability measures on \({\mathbb{R}}\) (respectively, on \({\mathbb{R}}^+)\). In the course of the proofs we establish various inequalities, on subsets of \({\mathbb{R}}\), leading to several sharp estimates. A number of related applications are given.