Bounds for the normalized Jensen-Mercer functional
نویسندگان
چکیده
منابع مشابه
Bounds for the Normalized Jensen – Mercer Functional
We introduce the normalized Jensen-Mercer functional Mn( f ,x, p) = f (a)+ f (b)− n ∑ i=1 pi f (xi)− f ( a+b− n ∑ i=1 pixi ) and establish the inequalities of type MMn( f ,x,q) Mn( f ,x, p) mMn( f ,x,q) , where f is a convex function, x = (x1, . . . ,xn) and m and M are real numbers satisfying certain conditions. We prove them for the case when p and q are nonnegative n -tuples and when p and q...
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In this paper we generalize the inequality MJn (f,x,q) ≥ Jn (f,x,p) ≥ mJn (f,x,q) where Jn (f,x,p) = n ∑ i=1 pif (xi)− f ( n ∑ i=1 pixi ) , obtained by S.S. Dragomir for convex functions. We provide cases where we can improve the bounds m and M for convex functions, and also, we show that for the class of superquadratic functions nonzero lower bounds of Jn (f,x,p)− mJn (f,x,q) and nonzero upper...
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and Applied Analysis 3 where Pj j ∑ i 1 pi, j 1, . . . , n. 1.7 Lemma 1.6. Let f be a convex function on I, p a positive n-tuple such that Pn ∑n i 1 pi 1 and x1, x2, . . . , xn ∈ I, n ≥ 3 such that x1 ≤ x2 ≤ · · · ≤ xn. For fixed x1, x2, . . . , xk, where k 2, 3, . . . , n− 1, the Jensen functional J x,p, f defined in 1.2 is minimal when xk xk 1 · · · xn−1 xn, that is, J ( x,p, f ) ≥ k−1 ∑ i 1 ...
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ژورنال
عنوان ژورنال: Journal of Mathematical Inequalities
سال: 2009
ISSN: 1846-579X
DOI: 10.7153/jmi-03-52