Generalized Steffensen's inequality by Lidstone interpolation and Montogomery's identity.

By using a Lidstone interpolation, Green's function and Montogomery's identity, we prove a new generalization of Steffensen's inequality. Some related inequalities providing generalizations of certain results given in (J. Math. Inequal. 9(2):481-487, 2015) have also been obtained. Moreover, from these inequalities, we formulate linear functionals and describe their properties.

Introduction

Steffensen [12] proved the following inequality: if , , and f is decreasing, then

Since then, a generalization and an improvement of Steffensen’s inequality has been a topic of interest of several mathematicians, for example see [10] and the references therein. One generalization of Steffensen’s inequality is given by Pečarić [8].

Theorem 1

Consider any non-decreasing real valued functions g on and f on interval I (where I is such that it contains all a, b, and ) such that g is differentiable. Consider the conditions (i) and (ii) .

If (i) holds, then

If (ii) holds, then inequality in part (a) is reversed.

Remark 1

In Theorem 1 one may take g as an absolutely continuous function instead of differentiable function because if f is non-decreasing then the function is well defined and holds almost everywhere on I. Then if g is any absolutely continuous and non-decreasing function then the substitution in the integral is justified (see [5, Corollary 20.5]), so where the last inequality holds when g satisfies condition (i).

Recently, Fahad, Pečarić and Praljak proved a generalization [4] of (1) by improving the results given in [8] and [11]. The following is a consequence of a result given in [4].

Corollary 1

Consider any non-decreasing, differentiable and real valued functions g and f on and interval I, respectively, (where I is such that it contains all a, b, and ). If f is convex, then:

If g satisfies condition (i) given in Theorem 1, then

If g satisfies condition (ii) given in Theorem 1, then the reverse of the above inequality holds.

The above corollary gives (3) and consequently Steffensen’s inequality. Now, we present some other consequences from results in [4].

Corollary 2

Let be a differentiable convex function with and let be another function.

If for every , then

If for every , then the reverse of the above inequality holds.

Corollary 3

Let f and h be as in Corollary 2 and let and denote .

If for every , then

If for every , then the reverse of the above inequality holds.

The goal of this paper is to obtain generalized Steffensen’s inequality by proving generalization of (4). Moreover, inequalities (5) and (6) can be used to obtain classical Hardy-type inequalities; see [4]. Keeping in view the importance of (5) and (6) we obtain their generalizations as well. In our construction, we use Green’s function, Montogomery’s identity and Lidstone interpolation. The following lemma is given in [7].

Lemma 1

For a function , see [13], we have where and

Consider the following functions on : Clearly, Throughout the calculations in the main results, we will use corresponding to for and for , and , and , respectively.

Now, we consider the following simple lemma.

Lemma 2

Let and , for be as defined above then and

Proof

For fix consider By replacing specific value of and simplifying we get the required identities. □

We conclude the section by recalling the Lidstone interpolation and some of its properties. The details related to the Lidstone interpolation can be found in [1]. We start with the following lemma given in [1].

Lemma 3

If , then where is a Lidstone polynomial; see [1]. Moreover, is a homogeneous Green’s function of the differential operator on , and with the successive iterates of

From the lemma it is not difficult to conclude that a function can be represented by using a Lidstone interpolation in the following way:

The following remark describes the positivity of the Green’s function of the Lidstone interpolation.

Remark 2

Clearly and (26) yields for even n and for odd n for every .

The next section contains the main results of the paper.

Main results

Throughout this section we use the following notations: , and . We start the section with the following theorem which enables us to obtain a generalization of (4).

Theorem 2

Let with and be 2n times differentiable function. Let be a non-decreasing function with then: where is given by (12)–(16).

For , we have

If then

□

We prove it for the case when , the other cases are similar to this proof. By using (7) and (21) for f and , respectively, we have By simplifying and using Fubini’s theorem, we have Further, by substituting n with in (27) for , we get which upon simplification gives the required identities.

The proof is similar to part (a) except the use of the assumption .

The following theorem gives a generalized Steffensen’s inequality.

Theorem 3

Let with and let be 2n times differentiable and be a non-decreasing with .

If f is 2n-convex and n is odd, then for , where for .

If f is 2n-convex and n is even, then the reverse of the inequality in part (a) holds.

If −f is 2n-convex and n is odd, then the reverse of the inequality in part (a) holds.

If −f is 2n-convex and n is even, then inequality in part (a) holds.

□

For fix s and any , the function is convex and differentiable and since g is non-decreasing, therefore Corollary 1(a) gives . Moreover, 2n-convexity of f implies for . Since is even, Remark 2 implies . Thus, we have Using this fact in Theorem 2 we get the desired inequality.

The proof is similar to the part (a) except the fact that f is 2n-convex, therefore, , odd implies . Therefore, the reverse of (28) holds, which proves part (b).

It follows from the facts that, under the assumptions, and , which give the reverse of (28) and prove (c).

It follows from the facts that, under the assumptions, and , which yield (28) and complete the proof.

In particular, the above theorem gives and , which give (4) and its reverse. Consequently, Theorem 3 produces a generalization of Steffensen’s inequality and its reverse. Now, we prove the following theorem, which enables us to prove the generalization of (5).

Theorem 4

Let with and let be 2n times differentiable function with . If be an integrable function then:

for .

If then

If then

First, we prove for , the proof of other cases are similar. By using (7) and (21) for f and , respectively, and using the assumption that , we have Further, by substituting n with in (27) for and simplifying we get the required identities. □

In the next theorem, we prove a generalization of (5) and its reverse.

Theorem 5

Let with and let be 2n times differentiable function with and h be as in Corollary 2(a).

If f is 2n-convex and n is odd, then:

For , we have

If then

If then

If f is 2n-convex and n is even, then for each j the reverse of the inequality in part (a) holds.

If −f is 2n-convex and n is odd, then for each j the reverse of the inequality in part (a) holds.

If -f is 2n-convex and n is even, then for each j inequality in part (a) holds.

The proof can be obtained from Theorem 4 and Corollary 2(a) along the same lines as Theorem 3 has been proved by using Theorem 2 and Corollary 1(a). □

Now, we prove identities to obtain a generalization of (6).

Theorem 6

Let with and let be 2n times differentiable function with and k and K be as in Corollary 3. If is integrable then:

For , we have

If then

If then

We prove the result for . The proofs of the other parts are similar. By using (7) and (21) for f and , respectively, we have Since , The rest follows from (27). □

The following theorem gives a generalization of (6) and its reverse.

Theorem 7

Let with and let be 2n times differentiable function with and k, K and h be as in Corollary 3(a).

If f is 2n-convex and n is odd, then:

For , we have

If then

If then

If f is 2n-convex and n is even, then for each j the reverse of the inequality in part (a) holds.

If −f is 2n-convex and n is odd, then for each j the reverse of the inequality in part (a) holds.

If -f is 2n-convex and n is even, then for each j inequality in part (a) holds.

The proof follows from Theorem 6 and Corollary 3(a) in a similar way to the proof of Theorem 3 by using Theorem 2 and Corollary 1(a). □

Application to -convex function at a point

In the present section we prove results related to the following -convex function at a point introduced in [9]. Let be an interval, and . A function is said to be -convex at point c if there exists a constant such that the function is n-concave on and n-convex on . A function f is said to be -concave at point c if the function −f is -convex at point c. Pečarić, Praljak and Witkowski in [9] studied necessary and sufficient conditions on two linear functionals and so that the inequality holds for every function f that is -convex at point c. In this section, we define linear functionals and obtain such inequalities for defined functionals. Let be odd with , be 2n times differentiable function, and , where . Let and be non-decreasing with for . For , we define and

Similarly let , where and let and be as defined in Corollary 2(a) (w.l.o.g. we may assume on by taking when ). We define the following pair of functionals: Lastly, we define: where k is as defined in Corollary 3. If f is 2n-convex then Theorem 3(a), Theorem 5(a) and Theorem 7(a) implies , and for (and for ), respectively. Moreover, if −f is 2n-convex then Theorem 3(c), Theorem 5(c) and Theorem 7(c) imply , and for (and for ), respectively.

and where .

and

and

and

and where .

and

and

and

Theorem 8

Let with be odd and let be -convex at a point c in . Let and , where , be non-decreasing and differentiable functions. If , for all and for , where then for .

Since f is -convex at c, there exists a such that is 2n-concave (or −F is 2n-convex) on and 2n-convex on . Therefore, for each , we have . Moreover, since F is 2n-convex on , . Since , , which completes the proof. □

Theorem 9

Let with be odd, let and be as defined in Corollary 2(a) and be as in Corollary 3. If is -convex at a point c in then:

If then for , where .

If and then .

If then .

If then .

If then for .

If and then .

If then .

If then .

□

Since f is -convex at c, there exists a such that is 2n-concave (or −F is 2n-convex) on and 2n-convex on . Therefore, On the other hand, since F is 2n-convex on , So under the given assumption, we have for , which completes the proof of part (i). The proofs of the other parts are similar.

The proof is similar to the proof of part (a).

Further refinements

Theorem 3 can be refined further for some classes of functions, using exponential convexity (for details see [2] and [3]). First, we use linear functional define in previous section. Under assumptions of Theorem 3(a), we conclude that, for any odd n and for any , acts non-negatively on the class of 2n-convex functions.

Further, let us introduce a family of 2n-convex functions on with This is indeed a family of 2n-convex functions since .

Since is an exponentially convex function, the quadratic form is positively semi-definite. According to Theorem 3(a), is also positively semi-definite, for any , and , showing the exponential convexity of the mapping . Specially, if we take in (33) we see additionally that is also a log-convex mapping, a property that we will need in the next theorem.

Theorem 10

Under the assumptions of Theorem 3(a) the following statements hold:

The mapping is exponentially convex on .

For such that , we have for .

Remark 3

We have outlined the proof of the theorem above. The second part of Theorem 10 is known as the Lyapunov inequality, it follows from log-convexity, and it refines the lower (upper) bound for the action of the functional on the class of functions given in (31). This conclusion is a simple consequence of the fact that exponentially convex mappings are non-negative and if an exponentially convex mapping attains zero value at some point it is zero everywhere (see [6]).

A similar estimation technique can be applied for classes of 2n-convex functions given in [6]. Lastly, a similar construction can be made for the linear functionals and to obtain the inequalities given in Theorem 10 for these functionals.

Results and discussion

Steffensen’s inequality first appeared in 1918 and has remained of interest for several mathematicians. This paper is devoted to proving a generalization of Steffensen’s inequality which is related to recent and older developments. We also provide applications of the obtained results. In the primary step we prove a simple lemma which approximates a continuously differentiable function in different forms. By using these approximations, we prove such identities as under the condition of 2n-convexity and 2n-concavity give a generalization of Steffensen’s inequality and its reverse. Subsequently, we provide applications of the results to the theory of -convex functions at a point and exponentially convex functions. We also prove the Lyapunov inequality.

Conclusion

Several identities related to recent generalization of Steffensen’s inequality have been proved. Under the assumption of 2n-convexity and 2n-concavity, a generalization of Steffensen’s inequality and its reverse has been obtained from the identities. Applications of the results have been presented of the theory of -convex functions at a point and exponentially convex functions.