2. Background
Let
, the left and right Riemann-Liouville fractional integrals of order
(
) are defined by
; where
stands for the gamma function,
The Riemann-Liouville left and right fractional derivatives of order
(
) are defined by
(
,
means ceiling of the number;
)
(
;
), respectively, where
is the real part of
.
In particular, when
, then
see [
6].
Let
,
,
f an integrable function defined on
I and
an increasing function such that
, for all
. Left fractional integrals and left Riemann-Liouville fractional derivatives of a function
f with respect to another function
are defined as ([
6,
7])
and
respectively, where
.
Similarly, we define the right ones:
and
The following semigroup property holds; if
,
, then
Next let again
,
,
,
is increasing and
, for all
. The left
-Caputo fractional derivative of
f of order
is given by ([
8])
and the right
-Caputo fractional derivative ([
8])
Clearly, when
we have
and if
, then
and
If
, then we get the usual left and right Caputo fractional derivatives
for
, and (
)
Next we will deal with the -Hilfer fractional derivative.
Definition 1. ([9]) Let , , and , ψ is increasing and , for all . The ψ-Hilfer fractional derivative (left-sided and right-sided) of order α and type , respectively, are defined byand The original Hilfer fractional derivatives ([10]) come from , and are denoted by and . When , we get Riemann-Liouville fractional derivatives, while when we have Caputo type fractional derivatives.
We define . We notice that , hence . We can easily write that ([9])and In particular, when and ; , we have thatand Remark 1. ([9]) Let , then Assume that , we have that Assume that . Hence We mention the simplified -Hilfer fractional Taylor formulae:
Theorem 2. (see also [9]) Let , with ψ being increasing such that over , where , , and , . Thenand Here notice that .
We also mention the following alternative -Hilfer fractional Taylor formulae:
Theorem 3. ([11]) Let , with ψ being increasing, over , , . Assume that . Then
Next we list two Hilfer fractional derivatives representation formulae:
Theorem 4. ([11]) Let , , , ; , ; and set . Assume further that , for . Let also , with , and assume that and . Then∀ Furthermore, (absolutely continuous functions) if and if .
Theorem 5. ([11]) Let , , , ; , ; and set . Assume further that , . Let also , with , and assume that and . Then∀ Furthermore, if and if .
3. Main Results
We present the following Hilfer-Polya type fractional inequalities:
Theorem 6. Let , , , ; , ; and set . Assume further that for and , . Let also , with , and assume that and .
Proof. From (
33) we have
∀
By (
34), we get
∀
We derive that
∀
and similarly,
∀
We continue with the -variant:
Theorem 7. All as in Theorem 6 with (i.e., ). Call Proof. By (
38) we have
∀
Similarly, from (
39) we find that
∀
Next comes the -variant of Hilfer-Polya fractional inequality:
Theorem 8. All as in Theorem 6 with , where . Call Proof. By (
38) we have
∀
, with
And, by (
39), similarly we derive
∀
, with
Consequently, we obtain that
Therefore, we obtain
proving the claim. □
Next come -Hilfer-Ostrowski type inequalities for several functions involved.
For basic
-Hilfer-Ostrowski type inequalities involving one function see [
11].
We make
Remark 2. Our setting here follows: Let , , , ; , . Assume that and , for all
Notice that if , we get , all
In general, for we have Hence
Similalry, we haveThat is So when , by the above we obtain , for all
Thus, it is always true that ,
We present
Theorem 9. Let , , , ; , . Here ψ is increasing, over , , . Assume that and , for all and is as in (61). Assume also that , for all Then
(1)and in case of , we have that (2) furthermore, it holds It follows the -variant.
Theorem 10. All as in Theorem 9, with . Then Next we have the -variant.
Theorem 11. All as in Theorem 9. Let also with . Then Proof of Theorems 9–11. By Theorem 3 we have
for all
That is
for all
Multiplying (
70) by
we get, respectively,
∀
And
∀
for all
Adding (
71) and (
72), separately, we obtain
∀
In addition,
∀
Next integrate (
73) and (
74) with respect to
We have
and
Finally adding (
75) and (
76) we obtain the useful and nice identity (
64).
If
and
, then
, and
. Hence
and
. So we have
proving (
67).
Let with , and let , with . Clearly . Let , then , furthermore . That is
From (
81), by using Hölder’s inequality twice, we have
proving (
68). □
Next we present a -Hilfer-Hilbert-Pachpatte left fractional inequality:
Theorem 12. Let , with being strictly increasing over , where , , and , . Assume that , for . Let also , such that and . Then Proof. By Theorem 2 we have
∀
,
Then
∀
By Hölder’s inequality we obtain
∀
and
∀
Hence we have
(using Young’s inequality for
,
)
∀
;
So far we have
∀
;
The denominator in (
94) can be zero only when
and
.
Therefore we obtain (
87), by integrating (
94) over
□
It follows the right side analog of last theorem.
Theorem 13. Let , with being strictly increasing over , where , , and , . Let also , such that and . Assume that , for . Then Proof. Similar to Theorem 12, by the use of (
30). □
We continue with other Hilfer-Hilbert-Pachpatte fractional inequalities.
Theorem 14. Let , , , and set . Assume further that , for . Let also , with , and assume that and . Furthermore, let , such that and . Then Proof. Similar to Theorem 12, by the use of Theorem 4. □
It follows
Theorem 15. Let , , , and set . Assume further that , for . Let also , with , and assume that and . Furthermore, let , such that and . Then Proof. Similar to Theorem 12, by the use of Theorem 5. □
We finish with two applications:
Corollary 1. All as in Theorem 12, with , . Then Corollary 2. All as in Theorem 13, with , and , . Then