Chebyshev-Steffensen Inequality Involving the Inner Product

: In this paper, we prove the Chebyshev-Steffensen inequality involving the inner product on the real m -space. Some upper bounds for the weighted Chebyshev-Steffensen functional, as well as the Jensen-Steffensen functional involving the inner product under various conditions, are also given.


Introduction
Let f be a convex function defined on a real interval J ⊂ R. Jensen's inequality states that if x = (x 1 , . . ., for all nonnegative real n-tuples p = (p 1 , . . ., p n ), such that P n = p 1 + • • • + p n > 0. For f strictly convex (1) is strict unless all x i are equal [1] (p. 43).Jensen's inequality is, without any doubt, one of the most important inequalities, if not the most important inequality, in convex analysis with various applications in mathematics, statistics and engineering.
It is also known that the assumptions on p can be relaxed if we put more restrictions on x [2].Namely, if p is a real n-tuple such that 0 ≤ P i = p 1 + • • • + p i ≤ P n , i ∈ {1, . . . ,n − 1}, (2) and P n > 0, then for any monotonic n-tuple x = (x 1 , . . ., x n ) ∈ J n we get and for any function f convex on J inequality, (1) still holds.Note that (2) allows the occurrence of negative weights of p i , which usually complicate matters.Under such assumptions, inequality (1) is called the Jensen-Steffensen inequality for convex functions and (2) is called Steffensen's conditions by J. F. Steffensen.Again, for a strictly convex function f , inequality (1) remains strict under certain additional assumptions on x and p [3].The Jensen-Steffensen inequality is a proper generalization of the Jensen inequality since nonnegative weights of p satisfy condition (2) in every order, which means that for nonnegative weights the monotonicity condition on x becomes irrelevant.Another important inequality in mathematical analysis is the Chebyshev inequality ( Čebyšev inequality), as shown in [1] (p. 197) and [4]  Our goal is to prove the Chebyshev-Steffensen inequality (i.e., the Chebyshev inequality with weights p satisfying Steffensen's conditions (2)) involving the inner product on the real m-space R m and to establish some upper bounds for the weighted Chebyshev-Steffensen functional.The obtained results are used to find new Grüss-like upper bounds for the Jensen functional with weights of p satisfying (2).It is worth noting here that many interesting results of this type, but with nonnegative weights, can be found in [5].

Chebyshev-Steffensen Inequality
In the rest of the paper, for some n, m ∈ N, n ≥ 2, we denote •, • : R m × R m → R is the inner product on the real m-space R m , • norm related to •, • , and ≤ the coordinatewise partial order on R m , i.e., for we denote the weighted Chebyshev functional for the inner product on R m .Furthermore, To prove our main results we need the following lemma.
Proof.It can be easily proved (using summation by parts on the coordinates) that for k ∈ {2, . . . ,n − 1} and and in border cases In all of the cases we assume It could be checked directly that and also Using (5) with and next using (4) with a k = y k we obtain Pj (y j − y j−1 ) − P n y i ) Pj (y j − y j−1 )) where e = (1, . . ., n).
The next theorem states the Chebyshev-Steffensen inequality for the inner product on the real m-space R m with weights p satisfying (2). or Then for all real n-tuples p = (p 1 , . . ., p n ) ∈ R n satisfying (2) the following inequality holds C(X, Y; p) ≥ 0. If hence all products P i Pj are nonnegative.
Suppose that X and Y are such that (6) holds.Then and by Lemma 1 we immediately obtain (7).All other cases can be proven similarly.
The coordinatewise partial order is the most obvious choice of order on R m , and the conditions on X and Y in Theorem 1 are based on it, but it is possible to consider alternative conditions on X and Y.For instance, we can introduce a notion of monotonicity related to the inner product in the following way.Definition 1.We say that X = (x 1 , • • • , x n ) and Y = (y 1 , • • • , y n ), where x i , y i ∈ R m , i, j ∈ I n , are monotonic in the same direction with respect to the inner product if and we say that they are monotonic in opposite directions with respect to the inner product if the above inequality is reversed.
It is easy to see that Theorem 1 can be obtained as a simple consequence of the onedimensional version of the Chebyshev-Steffensen inequality using properties of the inner product.In the following theorem, we prove the Chebyshev-Steffensen inequality under slightly different conditions, which makes the use of Lemma 1 essential.
monotonic in the same direction with respect to the inner product.Then for all real n-tuples p = (p 1 , . . ., p n ) ∈ R n satisfying (2), inequality (7) holds.If X and Y are monotonic in opposite directions, (7) is reversed.

Proof. Directly from Lemma 1 and Definition 1.
A natural question to ask is this: Is there a connection between the conditions for X and Y in Theorem 1 and in Theorem 2? Obviously, monotonicity on the coordinates as in Theorem 1 implies monotonicity with respect to the inner product as in Definition 1 but not vice versa, as we show in the next example.
and Theorem 2 can be applied.On the other hand, x 1 and x 2 can not be compared in the coordinatewise partial on R 2 and Theorem 1 can not be applied.If we choose hence we can chose either Theorem 1 or Theorem 2.
The previous example points out that Theorem 2 is better than Theorem 1, but from the numerical point of view, it is good to have Theorem 1 too.Remark 1.In [6] (Theorem 4) the author considered some other conditions for weights of p, such as It can be easily seen that if the first assumption holds we get Pi ≤ 0, i ∈ {2, . . . ,n}, and if the second holds we get In both cases the products Pi+1 P j and P i Pj are nonpositive.From that we conclude that under such conditions on p and the same conditions on X and Y as in Theorem 1 or Theorem 2 in all of the cases, inequality (7) is reversed.

Bounds for the Chebyshev-Steffensen Functional
Our next goal is to establish upper bounds for the Chebyshev-Steffensen functional under various conditions of X and Y.
two n-tuples of elements from R m monotonic in the same direction with respect to the inner product.Then for all real n-tuples p = (p 1 , . . ., p n ) ∈ R n satisfying (2), the following inequalities hold where Proof.The left hand inequality in (8) follows from Theorem 2. For all i ∈ I n−1 we can write Since X and Y are monotonic in the same direction with respect to the inner product and the sum of the nonnegative summands is never smaller than any of its summands, we conclude that for all i, j ∈ I n−1 Then, by Lemma 1, we obtain In the rest of the paper we denote as in Theorem 3.
A simple way to bound the Chebyshev-Steffensen functional without monotonicity conditions is given in the following theorem.
Then for all real n-tuples p = (p 1 , . . ., p n ) ∈ R n satisfying (2), the following inequality holds Proof.Using Lemma 1 and the Cauchy-Bunyakovsky-Schwarz inequality for inner product spaces we obtain Observe that in the special case m = 1 conditions (9) become and by Theorem 4 we get which (with a slightly different notation) is a Grüss-like inequality obtained in [6] (Theorem 4).Some related results considering positive weights can be found in [7,8].
As in Remark 1, we can consider alternative conditions on weights p In either of those cases, (10) becomes (remember the nonpositivity of There is a way to bound the Chebyshev-Steffensen functional without bounding x i+1 − x i and y i+1 − y i : instead, we have to consider the max P i Pj . Then for all real n-tuples p = (p 1 , . . ., p n ) ∈ R n satisfying (2), the following inequalities hold Proof.Similarly, as in the proof of Theorem 4, we have In [9], the authors proved the following inequality Obviously, dealing with nonnegative weights gives more liberty because in that case we have max and since for such weights Pi ≥ Pj when i ≤ j we get max i∈I n−1 j∈{i+1,...,n} This means that for nonnegative weights, p inequalities (11) can be reformulated in the following way In the special case p = ( 1 n , . . ., 1 n ), we get max and At the end of this section we give a theorem that combines some of the previous approaches.
Then for all real n-tuples p = (p 1 , . . ., Proof.The conditions of this theorem imply that for all i, j ∈ {1, . . . ,n − 1} By Lemma 1 we get By the Cauchy-Bunyakovsky-Schwarz inequality for inner product spaces and the triangle inequality we obtain which completes the proof.

Steffensen-Grüss Inequality
In this section, we show how some of the results from Sections 2 and 3 can be used to obtain Grüss-like upper bounds for the Jensen-Steffensen functional.In this section, conv[S] denotes the convex hull of S ⊆ R m .Theorem 7. Let U be an open convex subset of R m and X = ( Let f : U → R be a continuously differentiable function and m, M ∈ R m such that Then for all p = (p 1 , . . ., p n ) ∈ R n satisfying (2), the following inequalities hold Proof.First, note that continuity of the partial derivatives on U implies the existence of some m, M ∈ R m such that (12) holds.Furthermore, under condition (2) on the weights p, we have From the mean-value theorem we know that for any x, y ∈ conv[x 1 , . . ., x n ] there exists some θ ∈ (0, 1) such that where z = y + θ(x − y).Applying this to x = x i , y = x and z = z Multiplying the above equality by p i and summing over i we obtain and therefore, after multiplication with P n , If in Theorem 6 we choose This implies which, after division by P 2 n , becomes (13).
Posing a stronger condition on ∇ f , namely the condition of Lipschitz continuity, we are able to remove the monotonicity condition for Then for all p = (p 1 , . . ., p n ) ∈ R n satisfying (2), the following inequality holds Consequently, for z i , i ∈ I n , defined as in the proof of Theorem 7, we have Definition 2. Let f be a real-valued function defined by J 1 × J 2 .We say that f is P-convex of order k if [x 0 , ..., x i ][y 0 , ..., y k−i ] f ≥ 0 , i ∈ {0, 1, ..., k}, for all x 0 , ..., x k ∈ J 1 and y 0 , ..., y k ∈ J 2 such that x 0 < ... < x k and y 0 < ... < y k , i.e., if f is convex of order (i, k − i) for all i ∈ {0, 1, ..., k}.We say that f is P-concave of order k if − f is P-convex of order k.
If a function f is P-convex of order 2, we simply say that the f is P-convex.Obviously, this definition can be extended for functions with more than two variables.In the same paper [11], the author proved several properties of P-convex functions of order k related to the properties of k-convex functions.
(i) A P-convex function of order k is not necessarily continuous on J 1 × J 2 .
(ii) If the kth partial derivatives of a function f : J 1 × J 2 → R exist, then f is P-convex of order k if these partial derivatives are nonnegative.(iii) If the (k − 1)th partial derivatives of a function f : J 1 × J 2 → R exist, then f is P-convex of order k if these partial derivatives are nondecreasing in each argument.
An interesting P-convex (but not convex) function is f : R whenever 0 ≤ h ∈ R m , x ≤ y, x, y + h ∈ J.
In the same paper [12] Brunk also proved that: (i) A function with nondecreasing increments is not necessarily continuous.
(ii) If the first partial derivatives of a function f : J → R exists for x ∈ J, then f has nondecreasing increments if each of these partial derivatives is nondecreasing in each argument.(iii) If the second partial derivatives of a function f : J → R exists for x ∈ J, then f has nondecreasing increments if each of these partial derivatives is nonnegative.
We may note here that if n = 2 and we consider only functions with partial derivatives of the second-order, then the class of P-convex functions and the class of functions with nondecreasing increments coincide.
P-convex functions and functions with nondecreasing increments have an important common property that ordinary convex functions of several variables do not have: the Jensen-Steffensen inequality holds for them (see [11,13] and [1] (p. 62)).
In the next theorem, we show how Theorem 7 can be used to establish a new upper bound for the Jensen-Steffensen functional for functions with nondecreasing increments.
(p. 240), which states that n i whenever a = (a 1 , . . ., a n ), b = (a 1 , . . ., b n ) are real n-tuples monotonic in the same direction, and p = (p 1 , . . ., p n ) a positive n-tuple [1] (p. 43).It is also useful to consider the Chebyshev functional (sometimes also called the Chebyshev difference) C defined by C(a, b; p) positive and a, b are monotonic in the same direction.In the special case a = b, we immediately get C(a, a; p) ≥ 0.

2 →Definition 3 .
R defined by f (x, y) = xy which provides a beautiful connection between the Chebyshev-Steffensen inequality and the Jensen-Steffensen inequality.Wright-convex functions have an important generalization for functions of several variables introduced in [12] and [1] (p. 14).An interval [a, b] in R m , where a, b ∈ R m and a ≤ b, is the set [a, b] = {x ∈ R m : a ≤ x ≤ b}.A real-valued function f defined on an interval J ⊂ R m is said to have nondecreasing increments if f (x + h) − f (x) ≤ f (y + h) − f (y)