Some aspects of convex functions and their applications.
In this paper we give some complete characterizations of the primitive of strongly Henstock-Kurzweil integrable functions which are defined on with values in a Banach space.
For nonquasianalytical Carleman classes conditions on the sequences and are investigated which guarantee the existence of a function in such that u(n)(a) = bn, bnKn+1Mn, n = 0,1,..., aJ. Conditions of coincidence of the sequences and are analysed. Some still unknown classes of such sequences are pointed out and a construction of the required function is suggested. The connection of this classical problem with the problem of the existence of a function with given trace at the boundary...
2000 Mathematics Subject Classification: 26C05, 26C10, 30A12, 30D15, 42A05, 42C05.In this paper we present some inequalities about the moduli of the coefficients of polynomials of the form f (x) : = еn = 0nan xn, where a0, ј, an О C. They can be seen as generalizations, refinements or analogues of the famous inequality of P. L. Chebyshev, according to which |an| Ј 2n-1 if | еn = 0n an xn | Ј 1 for -1 Ј x Ј 1.
MSC 2010: 33B15, 26A51, 26A48