In what follows, is an arbitrary time scale, denotes the set of rd-continuous functions, denotes the set of all regressive and rd-continuous functions, for all, denotes the set of real numbers, , and denotes the set of nonnegative integers. We use the usual conventions that empty sums and products are taken to be 0 and 1, respectively. Throughout this paper, we always assume that and are time scales, , , , , , and we write for the partial delta derivatives of with respect to , and for the partial delta derivatives of with respect to .

The following two lemmas are useful in our main results.

Lemma 2.1 (see [18]).

If , and with , then

with equality holding if and only if .

Lemma 2.2 (Comparison Theorem [1]).

Suppose , . Then,

implies

Next, we establish our main results.

Theorem 2.3.

Assume that , , , , and are nonnegative functions defined for that are right-dense continuous for , and is a real constant. Then,

implies

where

Proof.

Define a function by

Then, (2.4) can be written as

From (2.9), by Lemma 2.1, we have

It follows from (2.8)–(2.10) that

where is defined by (2.6). It is easy to see that is nonnegative, right-dense continuous, and nondecreasing for . Let be given, and from (2.11), we obtain

Define a function by

It follows from (2.12) and (2.13) that

From (2.13), a delta derivative with respect to yields

where is defined by (2.7). Noting that , , and using Lemma 2.2, from (2.15), we obtain

It follows from (2.9), (2.14), and (2.16) that

Letting in (2.17), we immediately obtain the required (2.5). The proof of Theorem 2.3 is complete.

Remark 2.4.

Letting and , respectively, we easily see that Theorem 2.3 reduces to Theorem 2.3.3 and Theorem 5.2.4 in [19].

Theorem 2.5.

Assume that all assumptions of Theorem 2.3 hold. If and is nondecreasing for , then

implies

where

Proof.

Noting that and is nondecreasing for , from (2.18), we have

By Theorem 2.3, from (2.21), we easily obtain the desired (2.19). This completes the proof of Theorem 2.5.

Remark 2.6.

If in Theorem 2.5, then we easily obtain Theorem 2.3.3 in [19].

Theorem 2.7.

Assume that , , and are nonnegative functions defined for that are right-dense continuous for , and is a real constant. If is right-dense continuous on and continuous on such that

for , , where is right-dense continuous on and continuous on , then

implies

where

Proof.

Define a function by

As in the proof of Theorem 2.3, from (2.23), we easily see that (2.9) and (2.10) hold. Combining (2.10), (2.27) and noting the assumptions on , we have

where is defined by (2.25). It is easy to see that is nonnegative, right-dense continuous, and nondecreasing for . The remainder of the proof is similar to that of Theorem 2.3 and we omit it.

Remark 2.8.

Letting and in Theorem 2.7, respectively, we can obtain Theorem 2.3.4 and Theorem 5.2.4 in [19].

Theorem 2.9.

Assume that , , and are nonnegative functions defined for that are right-dense continuous for , and is a real constant. If is right-dense continuous on and continuous on , and such that

for , , where is right-dense continuous on and continuous on , is the inverse function of , and

then

implies

where is defined by (2.25), and

Proof.

Define a function by (2.27). Similar to the proof of Theorem 2.3, we have

From (2.27), (2.35) and the assumptions on and , we obtain

where is defined by (2.25). Obviously, is nonnegative, right-dense continuous, and nondecreasing for . The remainder of the proof is similar to that of Theorem 2.3, and we omit it here. This completes the proof of Theorem 2.9.

Remark 2.10.

We note that when , Theorem 2.9 reduces to Theorem 2.3.4 in [19].

Remark 2.11.

Using our main results, we can obtain many integral inequalities for some peculiar time scales. For example, letting , , from Theorem 2.3, we easily obtain the following result.

Corollary 2.12.

Assume that , , , and are nonnegative functions defined for , that are continuous for , and is a real constant. Then,

implies

where