Journal of Applied Mathematics and Physics
Vol.05 No.12(2017), Article ID:81389,11 pages
10.4236/jamp.2017.512193
Dynamic Inequalities for Convex Functions Harmonized on Time Scales
Muhammad Jibril Shahab Sahir
University of Sargodha, Sub-Campus Bhakkar, Bhakkar, Pakistan
Copyright © 2017 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: November 1, 2017; Accepted: December 26, 2017; Published: December 29, 2017
ABSTRACT
We present here some general fractional Schlömilch’s type and Rogers-Hölder’s type dynamic inequalities for convex functions harmonized on time scales. First we present general fractional Schlömilch’s type dynamic inequalities and generalize it for convex functions of several variables by using Bernoulli’s inequality, generalized Jensen’s inequality and Fubini’s theorem on diamond-α calculus. To conclude our main results, we present general fractional Rogers-Hölder’s type dynamic inequalities for convex functions by using general fractional Schlömilch’s type dynamic inequality on diamond-α calculus for with .
Keywords:
Delta, Nabla and Diamond-α Time Scales, Fractional Integral Inequalities
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.
Cite this paper
Sahir, M.J.S. (2017) Dynamic Inequalities for Convex Functions Harmonized on Time Scales. Journal of Applied Mathematics and Physics, 5, 2360-2370. https://doi.org/10.4236/jamp.2017.512193
References
- 1. Mitrinovic, D.S. and Pecaric, J.E. (1988) Generalizations of Two Inequalities of Godunova and Levin. Bulletin of the Polish Academy of Sciences, 36, 645-648.
- 2. Pecaric, J.E., Proschan, F. and Tong, Y.L. (1992) Convex Functions, Partial Orderings, and Statistical Applications, vol. 187 of Mathematics in Science and Engineering. Academic Press, Boston.
- 3. Anastassiou, G.A. (2012) Integral Operator Inequalities on Time Scales. International Journal of Difference Equations, 7, 111-137.
- 4. Agarwal, R.P., O’Regan, D. and Saker, S. (2014) Dynamic Inequalities on Time Scales. Springer International Publishing, Cham, Switzerland. https://doi.org/10.1007/978-3-319-11002-8
- 5. Bohner, M. and Peterson, A. (2001) Dynamic Equations on Time Scales. Birkhauser Boston Inc., Boston. https://doi.org/10.1007/978-1-4612-0201-1
- 6. Bohner, M. and Peterson, A. (2003) Advances in Dynamic Equations on Time Scales. Birkhauser Boston, Boston. https://doi.org/10.1007/978-0-8176-8230-9
- 7. Hilger, S. (1988) Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. PhD Thesis, Universitßt Würzburg, Würzburg.
- 8. Anderson, D., Bullock, J., Erbe, L., Peterson, A. and Tran, H. (2003) Nabla Dynamic Equations on Time Scales. Pan-American Mathematical Journal, 13, 1-48.
- 9. Sheng, Q., Fadag, M., Henderson, J. and Davis, J.M. (2006) An Exploration of Combined Dynamic Derivatives on Time Scales and Their Applications. Nonlinear Analysis: Real World Applications, 7, 395-413. https://doi.org/10.1016/j.nonrwa.2005.03.008
- 10. Chen, G.S., Huang, F.L. and Liao, L.F. (2014) Generalizations of Holder Inequality and Some Related Results on Time Scales. Journal of Inequalities and Applications, 2014, 207. https://doi.org/10.1186/1029-242X-2014-207
- 11. Dinu, C. (2008) Convex Functions on Time Scales, Annals of the University of Craiova. Mathematics and Computer Science Series, 35, 87-96.
- 12. Hardy, G.H., Littlewood, J.E. and Polya, G. (1952) Inequalities. 2nd Edition, Cambridge University Press, Cambridge.