Abstract

In this paper generalized Steffensen type inequalities related to the class of functions that are “convex at point ” are derived and as a consequence inequalities involving the class of convex functions are obtained. Moreover, linear functionals from the difference of the right- and left-hand side of the obtained generalized inequalities are constructed and new families of exponentially convex functions related to constructed functionals are derived.

1. Introduction

The well-known Steffensen inequality [1] reads as follows.

Theorem 1. Suppose that is nonincreasing and is integrable on with and . Then one has The inequalities are reversed for is nondecreasing.

Since its appearance in 1918 numerous papers have been devoted to generalizations and refinements of Steffensen’s inequality. In [2] Pečarić proved the following generalization.

Theorem 2. Let be a positive integrable function on and an integrable function such that is nondecreasing on . If is a real-valued integrable function on such that , then holds, where is the solution of the equation If is a nonincreasing function, then the reverse inequality in (2) holds.

By substitutions and , Theorem 2 becomes as follows.

Theorem 3. Let the conditions of Theorem 2 be fulfilled. Then holds, where is the solution of the equation If is a nonincreasing function, then the reverse inequality in (4) holds.

In 2000 Mercer [3] gave a generalization that contains various already known generalizations, one of which is the aforementioned generalization given by Pečarić. In 2007 Wu and Srivastava [4] noted that Mercer’s result is incorrect and they have corrected it and gave a refinement of Steffensen’s inequality. Liu [5] also noted that Mercer’s result is incorrect as stated.

Motivated by refinement of Steffensen’s inequality given in [4], Pečarić et al. [6] obtained the following refined version of results given in Theorems 2 and 3.

Corollary 4. Let be a positive integrable function on and , integrable functions on such that is nonincreasing and . Then where is given by (3).
If is a nondecreasing function, then the reverse inequality in (6) holds.

Corollary 5. Let be a positive integrable function on and integrable functions on such that is nonincreasing and . Then where is given by (5).
If is a nondecreasing function, then the reverse inequality in (7) holds.

In this paper we obtain generalized Steffensen type inequalities, related to the aforementioned generalizations and refinements of Steffensen’s inequality, for the class of functions that are convex at point . Moreover, we construct linear functionals from the difference of the right- and left-hand side of obtained generalized inequalities and derive new families of exponentially convex functions related to constructed functionals.

2. Main Results

Let us begin by introducing a class of functions that extends the class of convex functions.

Definition 6. Let be a positive function, a function, and . We say that belongs to the class () if there exists a constant such that the function is nonincreasing (nondecreasing) on and nondecreasing (nonincreasing) on .

As noted in [7] we can describe the property from Definition 6 as “convexity at point .” In [7] Pečarić and Smoljak also proved that there is a connection between the class of functions and the class of convex functions. This connection is given in the following theorem.

Theorem 7. The function is convex (concave) on if and only if for every .

Applying the generalizations of Steffensen’s inequality given in the Introduction to functions that are convex at point we obtain the following results.

Theorem 8. Let be a positive integrable function, let be an integrable function, and let . Let be an integrable function such that . Let be the solution of the equation and let be the solution of the equation If and then If and (10) holds, the inequality in (11) is reversed.

Proof. Let and let , where is the constant from Definition 6. Since is nonincreasing, from Theorem 2 we obtain Since is nondecreasing, from Theorem 3 we obtain Now from (12) and (13) we obtain Hence, if (10) is satisfied, then (11) holds.
It is similar for .

Remark 9. From the proof we deduce that condition (10) can be weakened. So, for inequality (11) still holds if (10) is replaced by the weaker condition where is the constant from Definition 6. For the reverse inequality in (11) holds if (10) is replaced by (15) with the reverse inequality.
Moreover, condition (15) can be further weakened if the function is monotonic.
First, let us show that for we have Since is nonincreasing on and nondecreasing on for all distinct points and we have Therefore, if and exist, letting and , we get (16). Similarly, for we have (16) with the reverse inequality.
Hence, if the function is nondecreasing or is nonincreasing, from (15) we obtain that (10) can be weakened to Further, if is nonincreasing or is nondecreasing, (10) can be weakened to (18) with the reverse inequality.

Theorem 10. Let be a positive integrable function, let be an integrable function, and let . Let be an integrable function such that . Let be the solution of the equation and let be the solution of the equation If and then If and (21) holds, the inequality in (22) is reversed.

Proof. Let and let , where is the constant from Definition 6. is nonincreasing, so from Theorem 3 we obtain is nondecreasing, so from Theorem 2 we obtain Hence, from (23) and (24) we conclude Hence, if , then (22) holds.
It is similar for .

Remark 11. For inequality (22) still holds if condition (21) is replaced by the weaker condition where is the constant from Definition 6. Also, for the reverse inequality in (22) holds if (21) is replaced by (26) with the reverse inequality.
Additionally, condition (26) can be further weakened if the function is monotonic. Similarly, as in Remark 9, if the function is nondecreasing or is nonincreasing, from (26) we obtain that (21) can be weakened to Further, if is nonincreasing or is nondecreasing, (21) can be weakened to (27) with the reverse inequality.

As a consequence of Theorems 8 and 10 we obtain generalized Steffensen type inequalities that involve convex functions.

Corollary 12. Let be a positive integrable function, let be an integrable function, and let . Let be an integrable function such that . Let be the solution of (8) and the solution of (9). If is convex and (10) holds, then the inequality in (11) holds.
If is concave, the inequality in (11) is reversed.

Proof. Since is convex, from Theorem 7, we have that for every . Hence, we can apply Theorem 8.

Corollary 13. Let be a positive integrable function, let be an integrable function, and let . Let be an integrable function such that . Let be the solution of (19) and the solution of (20). If is convex and (21) holds, then the inequality in (22) holds.
If is concave, the inequality in (22) is reversed.

Proof. The proof is similar to that of Corollary 12 applying Theorem 10.

Remark 14. Similarly, as in Remarks 9 and 11, we obtain that conditions (10) and (21) in Corollaries 12 and 13 can be weakened if, additionally, the function is monotonic.

Remark 15. For in Theorems 8 and 10 and Corollaries 12 and 13 we obtain the results given in [7].

In [6, Theorems 2.4 and 2.5] Pečarić et al. gave a corrected version of Mercer’s result [3, Theorem 2] which follows from Theorems 2 and 3. In the following theorems we obtain generalizations of these results for functions from the class .

Theorem 16. Let be a positive integrable function, let be an integrable function, and let . Let be an integrable function such that . Let be the solution of the equation and the solution of the equation . If and then If and (28) holds, the inequality in (29) is reversed.

Proof. Take and in Theorem 8.

Theorem 17. Let be a positive integrable function, let be an integrable function, and let . Let be an integrable function such that . Let be the solution of the equation and the solution of the equation . If and then If and (30) holds, the inequality in (31) is reversed.

Proof. Take and in Theorem 10.

In [6, Theorem 2.6] Pečarić et al. showed that Mercer’s generalization [3, Theorem 3] is equivalent to Theorem 2. Further, in [6, Theorem 2.7] they obtained analogue theorem equivalent to Theorem 3. Motivated by the mentioned generalizations in the following theorems we obtain generalizations for functions from class .

Theorem 18. Let be a positive integrable function, let be an integrable function, and let . Let be integrable functions such that . Let be the solution of the equation and let be the solution of the equation If and then If and (34) holds, the inequality in (35) is reversed.

Proof. Take , and in Theorem 8.

Theorem 19. Let be a positive integrable function, let be an integrable function, and let . Let be integrable functions such that . Let be the solution of the equation and let be the solution of the equation If and then If and (38) holds, the inequality in (39) is reversed.

Proof. Take , and in Theorem 10.

Remark 20. Taking in Theorems 18 and 19 we obtain Theorems 8 and 10, respectively.

In the following theorems we obtain refined version of results given in Theorems 8 and 10.

Theorem 21. Let be a positive integrable function, let be an integrable function, and let . Let be an integrable function such that . Let be the solution of (8) and the solution of (9). If and then If and (40) holds, the inequality in (41) is reversed.

Proof. The proof is similar to that of Theorem 8 applying Corollary 4 for nonincreasing and Corollary 5 for nondecreasing.

Theorem 22. Let be a positive integrable function, let be an integrable function, and let . Let be an integrable function such that . Let be the solution of (19) and the solution of (20). If and then If and (42) holds, the inequality in (43) is reversed.

Proof. The proof is similar to that of Theorem 10 applying Corollary 5 for nonincreasing and Corollary 4 for nondecreasing.

Motivated by sharpened and generalized versions of Theorems 2 and 3 obtained by Pečarić at al. in [6, Corollaries 2.4 and 2.5] we obtain the following results.

Theorem 23. Let be a positive integrable function, let be an integrable function, and let . Let be integrable functions such that . Let be the solution of (8) and the solution of (9). If and then If and (44) holds, the inequality in (45) is reversed.

Proof. The proof is similar to that of Theorem 8 applying [6, Corollary 2.3] for nonincreasing and [6, Corollary 2.4] for nondecreasing.

Theorem 24. Let be a positive integrable function, let be an integrable function, and let . Let be integrable functions such that . Let be the solution of (19) and the solution of (20). If and then If and (46) holds, the inequality in (47) is reversed.

Proof. The proof is similar to that of Theorem 10 applying [6, Corollary 2.4] for nonincreasing and [6, Corollary 2.3] for nondecreasing.

Remark 25. Generalized Steffensen type inequalities obtained in Theorems 18–24 also hold if the function is convex (concave). This follows from Theorem 7; that is, if is a convex function, then for every .

Remark 26. Similarly, as in Remarks 9 and 11, we obtain that conditions (28), (30), (34), (38), (40), (42), (44), and (46) can be weakened, but here we omit the details.

3. Mean Value Theorems

Generalizations of Steffensen type inequalities given by (11), (22), (35), (39), (41), and (43) are linear in . Hence, we can define the following linear functionals:

Under the assumptions of Theorems 8, 10, 18, 19, 21, and 22 we have that , , for . Further, Corollaries 12 and 13 and Remark 25 assure that , , for any convex function .

Let us begin by showing a Lagrange type mean value theorem for the functional .

Theorem 27. Let be a positive integrable function and let . Let be an integrable function such that . Let be the solution of (8) and the solution of (9). Let be such that . If (10) holds, then there exists such that where is defined by (48).

Proof. Since , there exist Let The functions and are convex since , . Hence, , , and we obtain where . Since is convex we have .
If , then (57) implies and (54) holds for every . Otherwise, multiplying (57) by we obtain so continuinity of ensures the existence of satisfying (54).

We continue with a Cauchy type mean value theorem for the functional .

Theorem 28. Let be a positive integrable function and let . Let be an integrable function such that . Let be the solution of (8) and the solution of (9). Let the functions and be such that . If (10) holds and , then there exists such that where is defined by (48).

Proof. Define . Due to linearity of we have . Now by Theorem 27 there exist such that where . Therefore, and which gives the claim of the theorem.

Remark 29. As in Theorem 27 we can obtain Lagrange type mean value theorems for the functionals , . Similarly, Cauchy type mean value theorems can be obtained for the functionals , . Hence, we can obtain that there exist , , such that

4. Exponential Convexity

We begin by recalling some definitions and results on exponential convexity; see [8, 9].

Definition 30. A function is -exponentially convex in the Jensen sense on if holds for all choices and all choices .
A function is -exponentially convex on if it is -exponentially convex in the Jensen sense and continuous on .

Remark 31. It is clear from the definition that -exponentially convex functions in the Jensen sense are nonnegative functions.
Also, -exponentially convex functions in the Jensen sense are -exponentially convex in the Jensen sense for all and .

Definition 32. A function is exponentially convex in the Jensen sense on if it is -exponentially convex in the Jensen sense on for every .
A function is exponentially convex on if it is exponentially convex in the Jensen sense and continuous on .

Remark 33. A function is log-convex in the Jensen sense, that is, if and only if holds for all and , that is, if and only if is -exponentially convex in the Jensen sense. By induction from (64) we have Therefore, if is continuous and for some , then from the last inequality and nonnegativity of (see Remark 31) we get Hence, either -exponentially convex function is identically equal to zero or it is strictly positive and log-convex.

We also use the following well known results for convex functions.

Lemma 34. A function is convex if and only if the inequality holds for all such that .

Proposition 35. If is a convex function on and if , , , , then the following inequality holds: If the function is concave, the inequality is reversed.

Definition 36. Let be a function defined on . The th order divided difference of at distinct points in is defined recursively by

Remark 37. The value is independent of the order of the points . Previous definition can be extended to include the case in which some or all of the points coincide by assuming that and letting provided that exists.

Next, we construct exponentially convex functions using the previously defined functionals , . In the sequel the notation denotes the natural logarithm function and , denote intervals in .

Theorem 38. Let be a family of functions such that for all mutually different points the mapping is -exponentially convex in the Jensen sense on . Let , , be linear functionals defined by (48)–(53). Then the mapping is -exponentially convex in the Jensen sense on .
If the mapping is continuous on , then it is -exponentially convex on .

Proof. For and , , we define the function Since the mapping is -exponentially convex in the Jensen sense we have So is a convex function and Therefore, the mapping is -exponentially convex on in the Jensen sense.
If the mapping is also continuous on , then is -exponentially convex by definition.

If the assumptions of Theorem 38 hold for all , then we have the following corollary.

Corollary 39. Let be a family of functions such that for all mutually different points the mapping is exponentially convex in the Jensen sense on . Let , , be linear functionals defined by (48)–(53). Then the mapping is exponentially convex in the Jensen sense on .
If the mapping is continuous on , then it is exponentially convex on .

Corollary 40. Let be a family of functions such that for all mutually different points the mapping is -exponentially convex in the Jensen sense on . Let , , be linear functionals defined by (48)–(53). Then the following statements hold. (i)If the mapping is continuous on , then, for , such that , we have (ii)If the mapping is positive and differentiable on J, then for all such that and we have where

Proof. (i) By Theorem 38 the mapping is -exponentially convex. Hence, by Remark 33, either this mapping is identically equal to zero, in which case inequality (75) holds trivially with zeros on both sides, or it is strictly positive and log-convex. Therefore, for such that Lemma 34 gives which is equivalent to inequality (75).
(ii) By (i) we have that the mapping is log-convex on ; that is, the function is convex on . Applying Proposition 35 with , , , , we obtain that is
The limit cases and are obtained by taking the limits and .

Remark 41. The results stated in Theorem 38 and Corollaries 39 and 40 still hold when some or all of the points coincide. The proofs are obtained by recalling Remark 37 and a suitable characterization of convexity.
We continue with an example of a family of functions which satisfies the previous conditions.
Let be a positive integrable function and let be a family of functions where is defined by We have , so is convex on for every and is exponentially convex by definition. Using analogous arguing as in the proof of Theorem 38 we have that is exponentially convex (and so exponentially convex in the Jensen sense). We see that the family satisfies the assumptions of Corollary 39, so mappings are exponentially convex in the Jensen sense. It is easy to verify that these mappings are continuous, so they are exponentially convex.

For this family of functions , , from (77) becomes

Explicitly for we have the following:(i)for , (ii)for , (iii)for ,

Theorem 28 applied on functions implies that satisfies . Hence is a monotonic mean by (76).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The research of the authors was supported by the Croatian Ministry of Science, Education and Sports under the Research Grant 117-1170889-0888.