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The work is related to the problem by P. Erdős about the estimation of the numbers $$A_N = _{\left| z \right| = 1}^{\max } \left| {\left( {z - z_1 } \right)\left( {z - z_2 } \right) \cdots \left( {z - z_N } \right)} \right|,{\text{ where }}\left| {z_j } \right| \equiv 1,$$ as N→∞.We shall deal with the case where the z j are all possible roots of the unity ordered in such a way that all the roots of degree n follow the roots of degree n-1. Within the group of roots of degree n, the enumeration can vary. We prove that ln A N grows like √N, and we get estimates of possible values of the lower limit of the ratio (ln A N /√N as well as exact bounds of the upper limit of this ratio.

This chapter is mainly devoted to studying those properties of Banach lattices and operators on them whose description requires the vector lattice structure as well as the pure Banach space structure. In particular, we study the properties of the operators acting between vector lattices and remaining to be order bounded under multiplication by arbitrary Banach endomorphisms in these lattices. In spite of being natural, such classes of operators have attracted very little attention of mathematicians. At present, the theory of p-absolutely summing operators plays a key role in solving the arising problems; in Sections 3.1–3.3 and 3.5, we briefly expose those results of the theory which are most important for us. The reader interested in a more complete presentation of the theory will refer to the monographs [42, 32, 45] and the articles [20, 22, 39].

It is shown that a Banach space X with the Orlicz property is isomorphic to a Hilbert space if and only if in the class C2(X) of twice continuously Frechet differentiable functions one can find a nonzero function with a bounded support.