Weighted majorization theorems via generalization of Taylor’s formula
Aljinovic´ et al. Journal of Inequalities and Applications
Weighted majorization theorems via generalization of Taylor's formula
Andrea Aglic´ Aljinovic´ 0 1 3 6
Asif R Khan 0 1 2 3 5
Josip E Pecˇaric´ 0 1 3 4
0 Road, Karachi , 75270 , Pakistan
1 University of Karachi, University
2 Department of Mathematical
3 Sciences, Faculty of Science
4 Faculty of textile technology, University of Zagreb, Prilaz baruna Filipovic ́a 28A, Zagreb , 10000 , Croatia
5 Department of Mathematical Sciences, Faculty of Science, University of Karachi , University Road, Karachi, 75270 , Pakistan
6 Department of applied mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb , Unska 3, Zagreb, 10000 , Croatia
A new generalization of the weighted majorization theorem for n-convex functions is given, by using a generalization of Taylor's formula. Bounds for the remainders in new majorization identities are given by using the Cˇ ebyšev type inequalities. Mean value theorems and n-exponential convexity are discussed for functionals related to the new majorization identities.
majorization theorem; Taylor's formula; Cˇ ebyšev functional
1 Introduction
f λx + ( – λ)x ≤ λf (x) + ( – λ)f (x)
(b) If the inequality in (.) is reversed, then f is called concave. If it is strict for each
Proposition A function f : I → R is convex if the inequality
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The following result can be deduced from Proposition .
Proposition If a function f : I → R is convex, then the inequality
Definition The nth order divided difference of a function f : I → R at distinct points
xi, xi+, . . . , xi+n ∈ I = [a, b] ⊂ R for some i ∈ N is defined recursively by
[xj; f ] = f (xj), j ∈ {i, . . . , i + n},
[xi, . . . , xi+n; f ] =
[xi+, . . . , xi+n; f ] – [xi, . . . , xi+n–; f ] .
xi+n – xi
It may easily be verified that
[xi, . . . , xi+n; f ] =
Remark Let us denote [xi, . . . , xi+n; f ] by (n)f (xi). The value [xi, . . . , xi+n; f ] is
independent of the order of the points xi, xi+, . . . , xi+n. We can extend this definition by including
the cases in which two or more points coincide by taking respective limits.
Definition A function f : I → R is called convex of order n or n-convex if for all choices
of (n + ) distinct points xi, . . . , xi+n we have (n)f (xi) ≥ .
Remark For n = and i = , we get the second order divided difference of a function
f : I → R, which is defined recursively by
for arbitrary points x, x, x ∈ I. Now, we discuss some limiting cases as follows: taking
the limit as x → x in (.), we get
, x = x,
[xj; f ] = f (xj), j ∈ {, , },
[xj, xj+; f ] =
[x, x, x; f ] =
, j ∈ {, },
provided that f (x) exists.
For fixed m ≥ , let x = (x, . . . , xm) and y = (y, . . . , ym) denote two real m-tuples and
x[] ≥ x[] ≥ · · · ≥ x[m], y[] ≥ y[] ≥ · · · ≥ y[m] their ordered components.
Definition For x, y ∈ Rm,
This notion and notation of majorization was introduced by Hardy et al. []. Now, we
state the well-known majorization theorem from the same book [] as follows.
Proposition Let x, y ∈ [a, b]m. The inequality
f (xi) ≤
wixi ≤
wixi =
wif (xi) ≤
wif (yi).
The following weighted version of the majorization theorem was given by Fuchs in []
(see also [], p. and [], p.).
wiyi, k ∈ {, . . . , m – } and
Then for every continuous convex function f : [a, b] → R, the following inequality holds:
Remark Under the assumptions of Proposition , for every concave function f the
reverse inequality holds in (.).
w(t)x(t) dt ≤
w(t)y(t) dt, for each u ∈ (α, β), and
w(t)x(t) dt =
w(t)y(t) dt,
w(t)f x(t) dt ≤
w(t)f y(t) dt.
hold, then for every continuous convex function f : I → R the following inequality holds:
w(t)x(t) dt ≤
w(t)y(t) dt, for each u ∈ (α, β), and
w(t)x(t) dt =
w(t)y(t) dt,
The following proposition is a consequence of Theorem in [] (see also [], p.) and
represents an integral majorization result.
α
β
α
β
then again inequality (.) holds. In this paper we will state our results for decreasing x
and y satisfying the assumption of Proposition , but they are still valid for increasing x
and t satisfying the above condition; see for example [], p..
Proposition Let n ∈ N, f : I → R be such that f (n–) is absolutely continuous, I ⊂ R an
open interval, a, b ∈ I, a < b. Then the following identity holds:
f (x) =
f (t) dt +
k= k!(k + )
k= k!(k + )
Tn(x, s)f (n)(s) ds,
⎧
Tn(x, s) = ⎨ – n(x(b––s)an) + bx––aa (x – s)n–, a ≤ s ≤ x,
⎩ – n(x(b––s)an) + bx––ba (x – s)n–, x < s ≤ b.
In case n = the sum kn=– · · · is empty, so i (...truncated)