Abstract

We give refinements and generalizations of the Dresher and Bellman inequalities for positive linear functionals. We also give reverse of the new obtained generalized version of these inequalities. Finally, we apply our results on time scales integrals to obtain refinements and generalizations of time scales Dresher’s and Bellman’s inequalities.

1. Introduction

Dresher’s and Bellman’s inequalities are obtained from the classical Hölder and Minkowski inequalities and are well known in the theory of inequalities and means. For introduction and some applications of these classical inequalities, we refer to [1–5]. Due to the importance of Dresher’s and Bellman’s inequalities, there are various generalizations, refinements, and variants that appeared in the literature. In this paper, we give refinements and further generalizations of the generalized version of these inequalities for positive linear functionals and time scales integrals. For this, first we recall the definition of positive linear functionals and the two functional inequalities from [5].

Definition 1 (positive linear functional). Let be a nonempty set and let be a linear class of real valued functions having the following properties:(L₁)If and , then .(L₂)If for all , then .

An isotonic positive linear functional is a functional having the following properties:(A₁)If and , then .(A₂)If and for all , then .

Remark 2. Sums and Lebesgue integrals are the most familiar examples of positive linear functionals. In [6] it is shown that time scales integrals including the Cauchy, Riemann, Lebesgue, multiple Riemann, multiple Lebesgue delta, nabla, and diamond- also satisfy the properties of positive linear functionals.

Theorem 3 (Dresher’s inequality [5, Theorem ]). Let and be such that (L₁) and (L₂) are satisfied and suppose that both and satisfy (A₁) and (A₂). If with where and for , then

Theorem 4 (Bellman’s inequality [5, Theorem ]). Let , , and be such that (L₁), (L₂), (A₁), and (A₂) are satisfied. For , assume that are such that . Suppose that are such that Then the following inequality holds:

Remark 5. The inequality in Theorem 4 also holds if . Moreover, it holds in reverse order if .

Recently, the author, Rabia Bibi, gives the following refinement of Bellman’s inequality.

Theorem 6 (refinement of Bellman’s inequality [7, Theorem ]). Let satisfy conditions (L₁) and (L₂) and let satisfy conditions (A₁) and (A₂) on a nonempty set . For , assume that are nonnegative functions on such that and . Suppose that are such that Then the following inequality holds:

In Section 2, we give refinements of Dresher’s inequality for positive linear functionals (Theorem 3). In Section 3, we generalize Dresher’s inequality and new refinements obtained in Section 2. Further, we obtain the reverse of new generalized Dresher’s inequality. Section 4 provides the generalizations of Bellman’s inequality and its refinement (Theorem 18) and reverse of the new generalized Bellman’s inequality. Finally, in Section 5, we apply the obtained results to time scales integrals and obtain improvements of Dresher’s and Bellman’s inequalities on time scales.

In order to prove our main results, we use the classical Hölder and Minkowski inequalities, and the following refinements of them (see [8]).

Theorem 7 (see [8, Theorem ]). Let satisfy conditions (L₁) and (L₂) and let satisfy conditions (A₁) and (A₂) on a nonempty set . Let and define by . Then, for all nonnegative functions such that the inequality holds. In the case , inequality in (8) is reversed.

Theorem 8 (see [8, Theorem ]). Let and be as in Theorem 7. If , then for all nonnegative functions on such that , and , the inequality holds.

Remark 9. From the proof of Theorem 8 (see [8]), it is obvious that inequality (9) holds in reverse order for or .

We also need the following classical Radon’s inequality in order to prove the reverse of Dresher’s inequality.

Theorem 10 (see [1, Theorem , page 181]). Let and let for . If , then If , then inequality (10) holds in reverse order.

2. Refinements of Dresher’s Inequality

Our first result gives the refinement of Theorem 3, for .

Theorem 11. Let and be such that (L₁) and (L₂) are satisfied and suppose that both and satisfy (A₁) and (A₂). Suppose that are such that where ,   such that and for . Then holds, where

Proof. By using the Minkowski inequality for positive linear functionals, we get By using the discrete case of Theorem 7, we get Now by combining (14) and (15), we obtain inequality (12).

In next theorem, we give a new inequality of the Dresher type for positive linear functionals.

Theorem 12. Let and be such that (L₁) and (L₂) are satisfied and suppose that both and satisfy (A₁) and (A₂). Suppose that are such that where ,   or and for . Then holds, provided that the denominator of the right-hand side is positive.

Proof. By using Theorem 8, we get By using discrete Hölder’s inequality, we get Now by combining (18) and (19), we obtain inequality (17).

3. Generalizations of Dresher’s Inequality

Let , , , , and be real valued functions of ,  ,  ,  , and variables, respectively. Then, throughout this section, we use the following notations: where , , , , and are real valued functions defined on .

The following results present the generalizations of Theorems 3, 11, and 12, respectively.

Theorem 13. Let be defined as in (20) such that and . Suppose that and satisfy (A₁) and (A₂) and where and . Then

Proof. By using the Minkowski inequality for positive linear functionals, we get By using discrete Hölder’s inequality, we get Now by combining (23) and (24), we obtain inequality (22).

Theorem 14. Let be defined as in (20) such that and . Suppose that and satisfy (A₁) and (A₂) and where ,   such that and . Then holds, where