Weighted averages of n-convex functions via extension of Montgomery’s identity


              <jats:p>Using an extension of Montgomery’s identity and the Green’s function, we obtain new identities and related inequalities for weighted averages of <jats:italic>n</jats:italic>-convex functions, i.e. the sum <jats:inline-formula><jats:alternatives><jats:tex-math>$$\sum _{i=1}^m \rho _i h(\lambda _i)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
                    <mml:msubsup>
                      <mml:mo>∑</mml:mo>
                      <mml:mrow>
                        <mml:mi>i</mml:mi>
                        <mml:mo>=</mml:mo>
                        <mml:mn>1</mml:mn>
                      </mml:mrow>
                      <mml:mi>m</mml:mi>
                    </mml:msubsup>
                    <mml:msub>
                      <mml:mi>ρ</mml:mi>
                      <mml:mi>i</mml:mi>
                    </mml:msub>
                    <mml:mi>h</mml:mi>
                    <mml:mrow>
                      <mml:mo>(</mml:mo>
                      <mml:msub>
                        <mml:mi>λ</mml:mi>
                        <mml:mi>i</mml:mi>
                      </mml:msub>
                      <mml:mo>)</mml:mo>
                    </mml:mrow>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula> and the integral <jats:inline-formula><jats:alternatives><jats:tex-math>$$\int ^{b}_{a} \rho (\lambda ) h(\gamma (\lambda ))d\lambda $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
                    <mml:msubsup>
                      <mml:mo>∫</mml:mo>
                      <mml:mi>a</mml:mi>
                      <mml:mi>b</mml:mi>
                    </mml:msubsup>
                    <mml:mi>ρ</mml:mi>
                    <mml:mrow>
                      <mml:mo>(</mml:mo>
                      <mml:mi>λ</mml:mi>
                      <mml:mo>)</mml:mo>
                    </mml:mrow>
                    <mml:mi>h</mml:mi>
                    <mml:mrow>
                      <mml:mo>(</mml:mo>
                      <mml:mi>γ</mml:mi>
                      <mml:mrow>
                        <mml:mo>(</mml:mo>
                        <mml:mi>λ</mml:mi>
                        <mml:mo>)</mml:mo>
                      </mml:mrow>
                      <mml:mo>)</mml:mo>
                    </mml:mrow>
                    <mml:mi>d</mml:mi>
                    <mml:mi>λ</mml:mi>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula> where <jats:italic>h</jats:italic> is an <jats:italic>n</jats:italic>-convex function.</jats:p>


Introduction
In this paper we will give some identities and related inequalities involving weighted discrete or integral averages of functions, i. e., containing sums m i=1 ρ i h(λ i ) or integrals b a ρ(λ)h(γ (λ)) dλ. As a consequence we will give conditions on numbers λ 1 , . . . , λ m , ρ 1 , . . . , ρ m under which the inequality m i=1 ρ i h(λ i ) ≥ 0 holds for every function h from a particular class of functions. For example, for the class of convex functions such results were studied in [5], while Popoviciu [7][8][9] gave results for the class of n-convex functions (see [6, Chapter 9] also). We will extend the results of Popoviciu and we would start first with some basic definitions and properties of n-convex functions.
A function h is said to be convex of order n or n-convex if for all choices of distinct points λ i , . . . , λ i+n , we have λ i , . . . , λ i+n ; h ≥ 0.
From the definition, it is easy to see that 1-convex functions are nondecreasing functions, while 2-convex functions are the classical convex functions, so n-convex functions are a generalization of the notion of convexity. If the nth order derivative h (n) exists, then h is n-convex iff h (n) ≥ 0. For 1 ≤ k ≤ n − 2, a function h is n-convex iff h (k) exists and is (n − k)-convex.

Proposition 1.2 The inequality
holds for all n-convex functions h :
In fact, Popoviciu proved a stronger result that it is enough to assume that the inequality in (1.3) holds for every t ∈ [λ (1) , λ (m−n+1) ], where λ (1) ≤ · · · ≤ λ (m) is the ordered m-tuple λ, since this, together with (1.2), implies that it holds for every t ∈ [a, b] (see [9]). In the case of convex functions, i.e., n = 2, Pečarić [5] proved the result with the conditions (1.2) and (1. (1.4) The integral analogue of Proposition 1.2 is given in the next proposition. (1.6) We will also need the following extension of Montgomery's identity given in [1] which was derived using Taylor's formula. (1.8) In case n = 1 the sum n−2 k=0 · · · is empty, so identity (1.7) reduces to the well-known Montgomery identity (see for instance [4]) where P (λ, s) is the Peano kernel defined by Let us denote by G : [a, b] × [a, b] → R the Green's function of the boundary value problem The function G is given by and integration by parts easily yields that for any function h ∈ C 2 [a, b] the following identity holds (1.10) The function G is continuous, symmetric and convex with respect to both variables t and s. The paper is organized as follows: in Sect. 2, we obtain new identities involving discrete and integral weighted averages of n-convex functions by appropriate use of the extended Montgomery's identity and the Green's function. Related Popoviciu type inequalities are also derived. In Sect. 3, we obtain new Grüss-and Ostrowski-type inequalities by obtaining bounds for the remainders of the identities from Sect. 2.

Identities and related Popoviciu-type inequalities for n-convex functions
We will first prove couple of identities which will have a key role in the rest of the paper.
Moreover, the following identity holds where T n is as defined in (1.8). (2.5) and then using Fubini's theorem in the last term we get (2.1).
Moreover, by applying formula (1.7) with h and n replaced by h and n − 2, respectively, and rearranging the indices, we get Similarly, using (2.6) in (2.4) and applying Fubini's Theorem, we get (2.3).
Next we will state some inequalities that can be derived from the obtained identities.

Theorem 2.2 Let all the assumptions of Theorem 2.1 hold with the additional condition
where G andT n−2 are defined in (1.9) and (2.2). If h is n-convex, then the following inequality holds Proof Since the function h is n-convex, we have h (n) ≥ 0. Using this fact and (2.7) in (2.1) we easily arrive at our required result.

9)
where G and T n are defined in (1.9) and (1.8). If h is n-convex, then the following inequality holds (2.11) Also note that for even n bothT n−2 (s, t) ≥ 0 and T n−2 (s, t) ≥ 0. Therefore, combining this fact with (2.11) we get inequalities (2.7) and (2.9). As h is n-convex, the results follow from Theorems 2.2 and 2.3.
We will next state the integral versions of our main results. Since the proofs are of similar nature we will omit the details.  where G is defined in (1.9) andT n is defined in (2.2). If h is n-convex, then the following inequality holds:

Theorem 2.7 Let all the assumptions of Theorem 2.5 hold with the additional condition
where G is defined in (1.9) and T n is defined in (1.8). If h is n-convex, then the following inequality holds

Bounds for the remainders
In these sections, we will give bounds for the remainders which occur in certain representations of the sum m i=1 ρ i h(λ i ) and the integral β α ρ(λ)h(γ (λ)) dλ. Namely, we will give some Grüss-and Ostrowski-type inequalities.
Let h, γ : [a, b] → R be two Lebesgue integrable functions. We consider theČebyšev functional The following results can be found in [3]: The constant 1 √ 2 in (3.2) is the best possible.

Proposition 3.2 Let g : [a, b] → R be a monotonic nondecreasing function and let h : [a, b] → R be an absolutely continuous function such that h ∈ L ∞ [a, b]. Then we have the inequality
The constant 1 2 in (3.3) is the best possible. For the ease of notation, throughout this section j , j ∈ {1, 2, 3, 4}, will denote the following functions: under the assumption of Theorems 2.1 and 2.5 we define h; a, b), (3.5) where the remainders R j n (h; a, b), j = 1, 2, satisfy the bounds

4)
Proof We will prove the claim for j = 1, while the proof for j = 2 is analogous. Proposition 3.1 with h → 1 and g → h (n) yields the bound (3.6) for the remainder R 1 n (h; a, b) follows from (3.7).
Using Proposition 3.2, we obtain the following Grüss-type inequality.
Here, the symbol L p [a, b] (1 ≤ p < ∞) denotes the space of p-power integrable functions on the interval [a, b] equipped with the norm Now we state some Ostrowski-type inequalities related to the generalized linear inequalities.
The constant on the right-hand sides of (3.10) and (3.11) is sharp for 1 < q ≤ ∞ and the best possible for q = 1.
Using identities (2.1) and (2.3) from Theorem 2.1 and Hölder's inequality, we obtain inequalities (3.10) and (3.11), i.e. that the left-hand sides of these inequalities are less than or equal to L.H.S. ≤ h (n) q μ j r . (3.12) For the proof of the sharpness of the constant b a μ j (t) r dt 1/r , let us find a function h for which the equality in (3.12) is obtained.
Finally, for q = 1, we prove that is the best possible inequality. Suppose that μ j (t) attains its maximum at t 0 ∈ [a, b]. First we consider the case μ j (t 0 ) > 0. For δ small enough we define h δ (t) by Therefore, we have