Let and be two positive decreasing -tuples, let be a real -tuple and let (2.18).
Let and be two positive decreasing -tuples, let be a real -tuple such that conditions (1.10) are satisfied and is positive defined as in Theorem 2.6, then for there exists, with being defined as in (3.4), such that (3.24).
Let and be two positive decreasing -tuples, let be a real -tuple such that conditions (1.10) are satisfied, is positive defined as in Theorem 2.6, and with being defined as in (3.4), then there exists such that (3.22).
Let and be two positive decreasing -tuples, be a real -tuple such that conditions (1.10) are satisfied, is positive defined as in Theorem 2.6 and, is defined as in (3.4).
Let x and y be two decreasing real n-tuples, let p = (p1,..., p n ) be a real n-tuple such that ∑ i = 1 k p i x i ≤ ∑ i = 1 k p i y i for k = 1, …, n - 1 ; (1).
Let (Lambda =(lambda_{1},ldots,lambda_{n})) be a real n-tuple.
Let be two decreasing real n-tuples, and let be a real -tuple such that (1.10).
Let be a real linear space, a non empty convex set in, a convex function, and also let be positive -tuples such that and.
where functions F k and are as in (5) and (10), respectively, x = (x1,..., x n ) ∈ I n is a monotonic n-tuple and p = (p1,..., p n ) is a real n-tuple such that (3) holds.
In the real case, the set of real numbers is replaced by the vector space Rn of all n-tuples of real numbers x = (x1, …, xn) where each xj is a real number.
In the real case, the set of real numbers is replaced by the vector space Rn of all n-tuples of real numbers x = (x1, …, xn) where each xj is a real number.
