Inequalities of Jensen-pečarić-svrtan-fan Type
نویسندگان
چکیده
By using the theory of majorization, the following inequalities of Jensen-PečarićSvrtan-Fan type are established: Let I be an interval, f : I → R and t ∈ I, x, a, b ∈ I. If a1 ≤ · · · ≤ an ≤ bn ≤ · · · ≤ b1, a1 +b1 ≤ · · · ≤ an +bn; f(t) > 0, f ′(t) > 0, f ′′(t) > 0, f ′′′(t) < 0 for any t ∈ I, then f(A(a)) f(A(b)) = fn,n(a) fn,n(b) ≤ · · · ≤ fk+1,n(a) fk+1,n(b) ≤ fk,n(a) fk,n(b) ≤ · · · ≤ f1,n(a) f1,n(b) = A(f(a)) A(f(b)) , the inequalities are reversed for f ′′(t) < 0, f ′′′(t) > 0,∀t ∈ I , where A(·) is the arithmetic mean and fk,n(x) := 1 ( n k ) ∑ 1≤i1<···<ik≤n f ( xi1 + · · ·+ xik k ) , k = 1, . . . , n.
منابع مشابه
Inequalities of J-P-S-F type
By means of the theory of majorization and under the proper hypotheses, the following inequalities of Jensen-Pečarić-Svrtan-Fan (Abbreviated as J-P-S-F) type are established: f (A(x)) g(A(x)) · · · fk+1,n(x) gk+1,n(x) fk,n(x) gk,n(x) · · · A( f (x)) A(g(x)) , where fk,n(x) := 1 (k) ∑ 1 i1<···<ik n f ( xi1 + · · ·+ xik k ) , x ∈ [a,b] .
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