Improvements of Jensen-Type Inequalities for Diamond- Integrals

Abstract

We give further improvements of the Jensen inequality and its converse on time scales, allowing also negative weights. These results generalize the Jensen inequality and its converse for both discrete and continuous cases. Further, we investigate the exponential and logarithmic convexity of the differences between the left-hand side and the right-hand side of these inequalities and present several families of functions for which these results can be applied.

1. Preliminaries

The combined dynamic derivative, also called diamond- dynamic derivative , was introduced as a linear convex combination of the well-known delta and nabla dynamic derivatives on time scales. By a time scale we mean any nonempty closed subset of real numbers. Using the delta and nabla derivatives, the notions of delta and nabla integrals were defined (see [1]). We assume, throughout this paper, that the basic notions of the time scales are well known and understood.

Definition 1 (diamond- integral [1, Definition 3.2]). Let be continuous function and let . Then, the diamond- integral of from to is defined by

Remark 2. From the above definition it is clear that for the diamond- integral reduces to the standard delta integral and for the diamond- integral reduces to the standard nabla integral.
Moreover, if , then if , then if , where , then if , where , then

Recently, the Jensen inequality, the improvement of the Jensen inequality, and their converses are given for time scale integrals (see [2–4]).

Let be a time scale, and let . Then the time scale interval is denoted and defined by .

Theorem 3 (see [2]). Let . If with and is convex, then

Note that in this Jensen inequality we have nonnegative weights. In order to give a better version of this Jensen inequality on time scales, Dinu [4] gives the definition of -Steffensen-Popoviciu (-SP) weight.

Definition 4 (-SP weight [4, Definition 1]). Let . Then, is an -Steffensen-Popoviciu (-SP) weight for on if for every convex function , where

In the following lemma he gives a characterization for -SP weight for a nondecreasing function on time scales.

Lemma 5 (see [4, Lemma 2]). Let such that . Then, is an -SP weight for a nondecreasing function if and only if it verifies the following condition: for every .
If the following stronger (but more suitable) condition holds then is also an -SP weight for the nondecreasing continuous function .

As given in [4], all positive weights are -SP weights, for any continuous function and every . But there are some -SP weights that are allowed to take the negative values. The Jensen inequality on time scales, where it is allowed that the weight function takes some negative values, is given in the following theorem.

Theorem 6 (see [4, Theorem 2]). Let and let such that . Then, the following two statements are equivalent: (i) is an -SP weight for on ;(ii) for every convex function , it holds

Remark 7. Let be nondecreasing function. If , then Theorem 6 is equivalent to the Jensen-Steffensen inequality given by Steffensen in [5] (see also [6, page 57]). On the other hand if we take in Theorem 6, we obtain the integral version of the Jensen-Steffensen inequality given by Boas [7] (see also [6, page 59]).

Considering the converse of the Jensen inequality, Dinu [4] gives the following definition of -Hermite-Hadamard (-HH) weight, its characterization for a nondecreasing function on time scales, and the improvement of the converse of the Jensen inequality for some negative weights.

Definition 8 (-HH weight [4, Definition 2]). Let . Then is an -Hermite-Hadamard (-HH) weight for on if for every convex function , where

Lemma 9 (see [4, Lemma 3]). Let be such that . Then is an -HH weight for a nondecreasing function if and only if it verifies the following condition: for every .

In the next result Dinu [4] gives the connection between these two classes of weights on a time scale.

Theorem 10 (see [4, Theorem 3]). Let . Then every -SP weight for on is an -HH weight for on , for all .

In the following two sections of our paper we give some further generalizations of the Jensen-type inequalities on time scales allowing negative weights, and we also give the mean-value theorems of the Lagrange and Cauchy type for the functionals obtained by taking the difference of the left-hand side and right-hand side of these new inequalities. These results also generalize the results given in [8] for continuous and discrete cases. Section 4 in our paper deals with exponential convexity and logarithmic convexity of the functionals obtained in two previous sections. Finally, in Section 5 we present several families of exponentially convex functions which fulfil the conditions of our results. The results from Sections 4 and 5 generalize the results given in [9] for continuous and discrete cases.

2. Improvement of the Jensen Inequality on Time Scales

Let , where . Consider the Green function defined by The function is convex and continuous with respect to both and .

It is well known that (see, e.g., [8–11]) any function can be represented by where the function is defined in (15). Using (16), we now derive several interesting results concerning the Jensen-type inequalities.

In the following theorem, we give the generalization of the Jensen inequality on time scales, where negative weights are also allowed.

Theorem 11. Let be such that . Let be such that and . Then, the following two statements are equivalent: (i)for every convex function holds;(ii)for all holds, where is defined in (15).Furthermore, the statements (i) and (ii) are also equivalent if we change the sign of inequality in both (17) and (18).

Proof. (i) (ii): let (i) hold. As the function is continuous and convex, it follows that (18) holds.
(ii) (i): let (ii) hold. Let . Then by using (16) we get If the function is also convex, then for all , and hence it follows that for every convex function inequality (17) holds. Moreover, it is not necessary to demand the existence of the second derivative of the function (see [6, page 172]). The differentiability condition can be directly eliminated by using the fact that it is possible to approximate uniformly a continuous convex function by convex polynomials.
The last part of our theorem can be proved analogously.

Remark 12. Let the conditions of Theorem 11 hold. Then the following two statements are equivalent. ()For every concave function the reverse inequality in (17) holds.()For all inequality (18) holds.Moreover, statements () and () are also equivalent if we change the sign of inequality in both statements () and ().

Remark 13. Consider (19). Suppose that is nondecreasing and that it has the first derivative. Let and , and make the substitution . Then we get Since is nondecreasing, we have that for all . If is convex, then for all . Hence, if and only if holds for all , then for every continuous convex function inequality (17) holds.

Combining the result from Theorem 11 with Theorem 6 and Lemma 5, we get the following two corollaries.

Corollary 14. Let and let be such that . Then is an -SP weight for on if and only if holds for all , where is defined in (15).

Corollary 15. Let be nondecreasing function and such that . Then hold for all , if and only if holds for all , where is defined in (15).

To shorten the notation, in the sequel we will use the following notation:

Under the assumptions of Theorem 11, we define the following functional : where the function is defined on . Clearly, if is continuous and convex, then is nonnegative.

Theorem 16. Let and . Let be defined as in (26). Then there exists such that holds, where .

Proof. Since the function is continuous and does not change its positivity on , applying the integral mean-value theorem on (19) we get that there exists such that holds. As in [12], it can be easily checked that it holds Calculating the integral on the right-hand side of (29), we get and the proof is completed.

Remark 17. Theorem 16 can also be proved by using the following two convex functions: where Since and are continuous and convex, we have This implies that Hence, as the function is continuous, there exists such that (27) holds.

Theorem 18. Let and . Let be defined as in (26). Then there exists such that holds, provided that the denominator on the left-hand side of (36) is nonzero.

Proof. Let be defined as the linear combination of functions and by Then . By applying Theorem 16 on , it follows that there exists such that After a short calculation we get that . By hypothesis (otherwise, we have a contradiction with ), so it follows that which is equivalent to (36).

Remark 19. In Theorem 18, if the inverse of the function exists, then (36) gives

Remark 20. Note that setting the function as in Theorem 18, we get the statement of Theorem 16.

As a consequence of the above two mean-value theorems, the following corollaries easily follow.

Corollary 21. Let , , and let be an -SP weight for . Let be defined as in (26). Then the following two statements hold. (i)If , then there exists such that (27) holds.(ii)If , then there exists such that (36) holds.

Proof. The statement (i) (statement (ii), resp.,) directly follows from Theorem 16 (Theorem 18, resp.,) and Corollary 14.

Corollary 22. Let be monotone function, , and such that . Let (9) hold for all . Let be defined as in (26). Then the following two statements hold. (i)If , then there exists such that (27) holds.(ii)If , then there exists such that (36) holds.

Proof. The statement (i) (statement (ii), resp.,) directly follows from Theorem 16 (Theorem 18, resp.,) and Corollary 15.

3. Improvement of the Converse of the Jensen Inequality on Time Scales

Using the similar method as in previous section, in the following theorem we obtain the generalization of the converse of the Jensen inequality on time scales, where negative weights are also allowed.

Theorem 23. Let be such that and let    be such that for all . Let be such that . Then, the following two statements are equivalent. (i)For every convex function holds.(ii)For all holds, where the function is defined in (15).Furthermore, the statements (i) and (ii) are also equivalent if we change the sign of inequality in both (41) and (42).

Proof. The idea of the proof is very similar to the proof of Theorem 11.
(i) ⇒ (ii): let (i) hold. As the function is continuous and convex, it follows that (42) holds.
(ii) ⇒ (i): let (ii) hold. Let . Then by using (16) we get If the function is also convex, then for all , and hence it follows that for every convex function the inequality (41) holds. Moreover, it is not necessary to demand the existence of the second derivative of the function (see [6, page 172]). The differentiability condition can be directly eliminated by using the fact that it is possible to approximate uniformly a continuous convex function by convex polynomials.
The last part of our theorem can be proved analogously.

Remark 24. Let the conditions of Theorem 23 hold. Then the following two statements are equivalent. ()For every concave function , the reverse inequality in (41) holds.()For all inequality (42) holds.Moreover, statements () and () are also equivalent if we change the sign of inequality in both statements () and ().

Remark 25. Note that in all the results in this section we allow that the mean value goes out of the interval , while in the results from the previous section we demanded that .

Setting and in Theorem 23, we get the following result.

Corollary 26. Let be such that . Let be such that . Then, the following two statements are equivalent. (i)For every convex function holds.(ii)For all holds, where the function is defined as in (15).Furthermore, statements (i) and (ii) are also equivalent if we change the sign of inequality in both (44) and (45).

Remark 27. As a consequence of Corollary 26, we get the result from Lemma 9.
Let and . Then (43) transforms into
Let , and suppose that is nondecreasing and that it has the first derivative. Now, similarly as in [4], we derive the result from Lemma 9. Let and , and make the substitution . Then we get Since is nondecreasing, we have that for all . If is convex, then for all . Hence, if and only if holds for all , then for every continuous convex function inequality (44) holds. Note that (48) is equivalent to condition (14).

Corollary 28. Let and let be an -SP weight for on . Then holds for all .

Proof. The proof follows directly from Theorem 10 and Corollary 26.

Under the assumptions of Theorem 23, we define the following functional : where the function is defined on . Clearly, if is continuous and convex, then is nonnegative.

Theorem 29. Let and . Let be defined as in (50). Then there exists such that holds, where .

Proof. The idea of the proof is very similar to the proof of Theorem 16.
Since the function is continuous and does not change its positivity on , applying the integral mean-value theorem on (43) we get that there exists such that Calculating the integral on the right-hand side of (53), we get and we get statement (51) of our theorem.

Remark 30. Note that (54) can also be expressed as

Theorem 31. Let and . Let be defined as in (50). Then there exists such that holds, provided that the denominator on the left-hand side of (56) is nonzero.

Proof. The proof is very similar to the proof of Theorem 18.

4. Exponential and Logarithmic Convexity

First we recall some definitions and facts about exponentially convex and logarithmically convex functions (see, e.g., [13, 14] or [9]) which we need for our results.

Definition 32. A function is -exponentially convex in the Jensen sense on , if holds for all choices and , .
A function is -exponentially convex if it is -exponentially convex in the Jensen sense and continuous on .

Remark 33. It is clear from the definition that 1-exponentially convex functions in the Jensen sense are in fact nonnegative functions. Also, -exponentially convex functions in the Jensen sense are -exponentially convex in the Jensen sense for every .

By definition of positive semidefinite matrices and some basic linear algebra, we have the following proposition.

Proposition 34. If is an -exponentially convex function in the Jensen sense, then the matrix is positive, semidefinite for all . Particularly, for all .

Definition 35. A function is exponentially convex in the Jensen sense on , if it is -exponentially convex in the Jensen sense for all .
A function is exponentially convex if it is exponentially convex in the Jensen sense and continuous.

Remark 36. Some examples of exponentially convex functions are (see [15]) as follows: (i) defined by , where and ;(ii) defined by , where ;(iii) defined by , where .

Remark 37. It is known (and easy to show) that is log-convex in the Jensen sense on , if and only if holds for every , and . It follows that a positive function is log-convex in the Jensen sense if and only if it is -exponentially convex in the Jensen sense. Also, using basic convexity theory, it follows that a positive function is log-convex if and only if it is -exponentially convex.

The following lemma is equivalent to the definition of convex function (see [6, page 2]).

Lemma 38. If are such that , then the function is convex if and only if the following inequality holds:

We will also need the following result (see, e.g., [6]).

Proposition 39. If is a convex function and are such that , , , and , then the following inequality is valid:

When dealing with functions with different degree of smoothness, divided differences are found to be very useful.

Definition 40. The second order divided difference of a function at mutually different points is defined recursively by

Remark 41. The value is independent of the order of the points , , and . This definition may be extended to include the case in which some or all the points coincide (see [6, page 16]). Namely, taking the limit in (61), we obtain provided that exists, and furthermore, taking the limits , , in (61), we obtain provided that exists.
A function is convex if and only if for every choice of three mutually different points holds.

Now, we use an idea from [15] to give an elegant method of producing an -exponentially convex and exponentially convex functions applying the functionals and on a given family of functions with the same property.

Theorem 42. Let , , be linear functionals defined in (26) and (50), respectively. Let , where is an interval in , be a family of functions such that the function is -exponentially convex in the Jensen sense on for every choice of three mutually different points . Then is an -exponentially convex function in the Jensen sense on . If the function is also continuous on , then it is -exponentially convex on .

Proof. Define the function by where , , ,