Dynamic Inequalities for Convex Functions Harmonized on Time Scales ()
1. Introduction
In the following, we present a result proved by Mitrinović and Pečarić as given in [1] and ( [2] , p. 235).
Theorem 1. Let
for
be a class, where
for
are continuous functions and
implies
for every
and
are represented by
where
is nonnegative arbitrary kernel. Consider
for every
. Let
be a convex and increasing function, then the following inequality holds
(1)
where,
Next we present a result on diamond-α calculus, as given in [3] .
Theorem 2. Let
,
be two time scales, and
;
;
is a kernel function with
,
; k is continuous function from
into
. Consider
We assume that
,
. Consider
continuous, and the
-integral operator function
Consider also the weight function
, which is continuous.
Define further the function
. Let I denote
any of
or
, and
be a convex and increasing function. In particular, we assume that
Then
(2)
We extend these results on time scale calculus. In this paper, it is assumed that all considerable integrals exist and are finite and
is a time scale,
with
and an interval
means the intersection of a real interval with the given time scale.
2. Preliminaries
We need here basic concepts of delta calculus. The results of delta calculus are adapted from [4] [5] [6] .
Time scale calculus was initiated by Stefan Hilger as given in [7] . A time scale is an arbitrary nonempty closed subset of the real numbers. It is denoted by
. For
, forward jump operator
is defined by
The mapping
such that
is called the forward graininess function. The backward jump operator
is defined by
The mapping
such that
is called the backward graininess function. If
, we say that t is right-scattered, while if
, we say that t is left-scattered. Also, if
and
, then t is called right-dense, and if
and
, then t is called left-dense. Points that are right-dense and left-dense at the same time are called dense. If
has a left-scattered maximum M, then
. Otherwise
.
For a function
, the derivative
is defined as follows. Let
, if there exists
such that for all
, there exists a neighborhood U of t, such that
for all
, then
is said to be delta differentiable at t, and
is called the delta derivative of
at t.
A function
is said to be right-dense continuous (rd-continuous), if it is continuous at each right-dense point and there exists a finite left limit in every left-dense point. The set of all rd-continuous functions is denoted by
.
The next definition is given in [4] [5] [6] .
Definition 1. A function
is called a delta antiderivative of
, provided that
holds for all
, then the delta integral of
is defined by
The following results of nabla calculus are taken from [4] [5] [6] [8] .
If
has a right-scattered minimum m, then
. Otherwise
. The function
is called nabla differentiable at
, if there exists
such that for any
, there exists a neighborhood V of t, such that
for all
.
A function
is left-dense continuous (ld-continuous), provided it is continuous at left-dense points in
and its right-sided limits exist (finite) at right-dense points in
. The set of all ld-continuous functions is denoted by
.
The next definition is given in [4] [5] [6] [8] .
Definition 2. A function
is called a nabla antiderivative of
, provided that
holds for all
, then the nabla integral of
is defined by
Now we present short introduction of diamond-α derivative as given in [4] [9] .
Let
be a time scale and
be differentiable on
in the
and
senses. For
, where
, diamond-α dynamic derivative
is defined by
Thus
is diamond-α differentiable if and only if
is
and
differentiable.
The diamond-α derivative reduces to the standard
-derivative for
, or the standard
-derivative for
. It represents a weighted dynamic derivative for
.
Theorem 3. [9] : Let
be diamond-α differentiable at
. Then
1)
is diamond-α differentiable at
, with
2)
is diamond-α differentiable at
, with
3) For
,
is diamond-α differentiable at
, with
Theorem 4. [9] : Let
and
. Then the diamond-α integral from
to
of h is defined by
provided that there exist delta and nabla integrals of
on
.
Theorem 5. [9] : Let
,
. Assume that
and
are
-integrable functions on
, then
1)
;
2)
;
3)
;
4)
;
5)
.
We need the following results.
Theorem 6. [4] : Let
and
. Suppose that
and
with
. If
is convex, then generalized Jensen’s inequality is
(3)
If F is strictly convex, then the inequality
can be replaced by
.
Theorem 7. [3] [10] : Let
. Let
,
are
-
integrable functions and
such that
. Then
(4)
which is generalized Rogers-Hölder’s Inequality.
Definition 3. [11] : A function
is called convex on
, where
is an interval of
(open or closed), if
(5)
for all
and all
such that
.
The function
is strictly convex on
if (5) is strict for distinct
and
.
The function
is concave (respectively, strictly concave) on
, if
is convex (respectively, strictly convex).
3. Main Results
First we present
-integral general fractional Schlömilch’s type inequalities on time scales, which is an extension of Schlömilch’s inequality given in [12] .
Theorem 8. Let
and
be two time scales;
is continuous kernel function with
and
. Let
-integral operator functions
belonging to a class
for
are represented by
where
are continuous functions. Continuous weight function is defined by
with
. Define
and
, where
implies
. Let
be a convex and increasing function.
If
, then the following inequality holds
(6)
Proof. In order to prove this Theorem, we need Bernoulli’s inequality, that is, if
, then
Since
, we have
. Thus, by Bernoulli’s inequality, we have
that is,
Let
be replaced by
and taking power
, we get
where we used the generalized Jensen’s inequality and Fubini’s theorem.
This proves the claim. □
Remark. If we set
and
be a convex and increasing function, then (6) takes the form
(7)
If
, where
, then (7) takes the form of (1).
Corollary 1. If
,
be a convex and increasing function and
, then delta version form of (6) is
(8)
If
,
be a convex and increasing function and
, then nabla version form of (6) is
(9)
Remark. Now we take that
is not necessarily increasing and is taken from
into
and
has fixed and strict sign, then according to
Theorem 8, we get
Corollary 2. If we apply for
,
, then (6) takes the form
(10)
Corollary 3. If we apply for
, then (6) takes the form
(11)
Corollary 4. If
,
be a convex and not necessarily
increasing function,
has fixed and strict sign and we apply for
, then (6) takes the form
(12)
Remark. If we set
,
,
,
and
be a convex and increasing function, then
We assume that
, and define
Then (6) takes the form of (2), as proved in [3] .
Corollary 5. If we take
,
, where
is the set of nonnegative integers and
.
Then
for
,
, where
.
And
When
and
be a convex and increasing function, then (6) can be written as
We can generalize Theorem 8 for convex functions of several variables on time scales in the upcoming theorem.
Theorem 9. Let
and
be two time scales;
is continuous kernel function with
and
. Let
-integral operator functions
belonging to a class
for
are represented by
where
are continuous functions. Continuous weight function is defined by
with
. Define
and
, where
implies
. Let
be a convex and increasing function.
If
, then the following inequality holds
(13)
Proof. Proof is similar to Theorem 8. □
Remark. If we set
,
be a convex and increasing function and
, where
, then (13) reduces to
as given in ( [2] , p. 236).
Now we present
-integral general fractional Rogers-Holder’s type inequalities.
Upcoming result is an application of general fractional Schlömilch’s type dynamic inequality.
Theorem 10. Let
and
be two time scales;
for
are continuous kernel functions with
and
. Let
-integral operator functions
for
are represented by
and
where
are continuous functions for
. Continuous weight function is defined by
with
. Define
, and
for
, where
implies
for
. Let
for
are convex and increasing functions.
If
with
. Then the following inequality holds
(14)
Proof. Let
and
for
. Then
,
where
for
. We use here generalized Rogers-Hölder’s inequality, Schlömilch’s inequality, generalized Jensen’s inequality and Fubini’s theorem, as
This proves the claim. □
Corollary 6. If we apply for
,
,
and let
,
. Then (14) takes the form
(15)
4. Conclusion and Future Work
The study of dynamic inequalities on time scales has a lot of scope. This research article is devoted to some general fractional Schlömilch’s type and Rogers-Hölder’s type dynamic inequalities for convex functions harmonized on diamond-α calculus and their delta and nabla versions are similar cases. Similarly, in future, we can present such inequalities by using Riemann-Liouville type fractional integrals and fractional derivatives on time scales. It will also be very interesting to present such inequalities on quantum calculus.