It is proven that for any system of n points z1, . . . , zn on the (complex) unit circle, there exists another point z of norm 1, such that ∑ 1 |zk − z|2 6 n 4 . Two proofs are presented: one uses a characterisation of equioscillating rational functions, while the other is based on Bernstein’s inequality.

Equivalent conditions are given for the nonnegativity of the coefficients of both the Chebyshev expansions and inversions of the first n polynomials defined by a certain recursion relation. Consequences include sufficient conditions for the coefficients to be positive, bounds on the derivatives of the polynomials, and rates of uniform convergence for the polynomial expansions of power series.

In real-world optimization problems, a wide range of uncertainties have to be taken into account. The presence of uncertainty leads to different results for repeated evaluations of the same solution. Therefore, users may not always be interested in the so-called best solutions. In order to find the robust solutions which are evaluated based on the predicted worst case, Worst Case Optimization P...

This paper studies the asymptotic behavior for a class of delayed reaction-diffusion Hopfield neural networks driven by finite-dimensional Wiener processes. Some new sufficient conditions are established to guarantee the mean square exponential stability of this system by using Poincaré’s inequality and stochastic analysis technique. The proof of the almost surely exponential stability for this...

The theory of fuzzy measures and fuzzy integrals was introduced by Sugeno [24] as a tool for modeling nondeterministic problems. Sugeno’s integral is analogous to Lebesgue integral which has been studied by many authors, including Pap [18], Ralescu and Adams [19] and, Wang and Klir [25], among others. RománFlores et al [9, 20–23], started the studies of inequalities for Sugeno integral, and the...

For a fixed unit vector a= (a1, a2, . . . , an) ∈ Sn−1, that is, ∑i=1 ai = 1, we consider the 2n signed vectors ε = (ε1, ε2, . . . , εn) ∈ {−1, 1}n and the corresponding scalar products a · ε = ∑i=1 aiεi. In [3] the following old conjecture has been reformulated. It states that among the 2n sums of the form ∑±ai there are not more with ∣ ∣∑i=1±ai ∣ ∣ > 1 than there are with ∣ ∣∑i=1±ai ∣ ∣≤ 1. T...

Rough random theory, generally applied to statistics, decision-making, and so on, is an extension of rough set theory and probability theory, in which a rough random variable is described as a random variable taking “rough variable” values. In order to extend and enrich the research area of rough random theory, in this paper, the well-known probabilistic inequalities (Markov inequality, Chebysh...