Shahab Sahir, Cogent Mathematics & Statistics (2018), 5: 1438030https://doi.org/10.1080/23311835.2018.1438030

APPLIED & INTERDISCIPLINARY MATHEMATICS | RESEARCH ARTICLE

Fractional dynamic inequalities harmonized on time scalesMuhammad Jibril Shahab Sahir1*

Abstract: We present here some symmetric fractional Rogers-Hölder’s inequalities using Riemann–Liouville integral on time scales. We consider and impose different conditions on three non-zero real numbers p, q and r when 1

p+

1

q+

1

r= 0.

Subjects: Discrete Mathematics; Pure Mathematics; Applied Mathematics

Keywords: Riemann–Liouville integral; Rogers-Hölder’s inequality; time scales

1. IntroductionRogers-Hölder’s inequality is an important and well-known inequality. This inequality has a lot of applications. We give its symmetric form here. When fjgjhj = 1 for all j = 1, 2, ..., m, where fj, gj, hj are sets of positive values, then,

as given in Aczél and Beckenbach (1978, p. 147) for three non-zero real numbers p, q and r, where all but one of p, q and r are positive. Inequality (1.1) is reversed if all but one of p, q and r are negative for sets of positive values of fj, gj and hj for all j = 1, 2, ..., m.

We unify and extend inequality (1.1) on time scales. Time-scale calculus was initiated by Stefan Hilger as in Hilger (1990). This inequality and its reverse form can be proved in weighted form by us-ing Riemann–Liouville integral on time scales. Their different versions can be expressed in same manner.

(1.1)

(m∑

j=1

fp

j

) 1

p(

m∑

j=1

gq

j

) 1

q(

m∑

j=1

hrj

) 1

r

≥ 1,

*Corresponding author: Muhammad Jibril Shahab Sahir, Department of Mathematics, University of Sargodha, Sub-Campus Bhakkar, Bhakkar, PakistanE-mail: jibrielshahab@gmail.com

Reviewing editor:Feng Qi, Tianjin Polytechnic University, China

Additional information is available at the end of the article

ABOUT THE AUTHORMuhammad Jibril Shahab Sahir has recently earned his degree of M Phil from University of Sargodha, Sub-Campus Bhakkar, Pakistan. He is working as a subject specialist at GHSS, 67ML, Bhakkar and also teaching at University of Sargodha, Sub-Campus Bhakkar. His research article “Dynamic Inequalities for Convex Functions Harmonized on Time Scales” is recently published. His fields of interest are fractional calculus, convex functions, quantum calculus, time-scale calculus and dynamic inequalities on time scales.

PUBLIC INTEREST STATEMENTThe theory of time-scale calculus is applied to reveal the symmetry of continuous and discrete and to combine them in one comprehensive form. In time-scale calculus, results are unified and extended. It is studied as Delta calculus, Nabla calculus and Diamond-α calculus. This hybrid theory is also widely applied on dynamic inequalities. Riemann–Liouville integral is denoted by Iαf, where α is a positive integer and is called order of Riemann–Liouville integral, is actually a generalization of the repeated antiderivative of f. Riemann–Liouville integral is named for Bernhard Riemann and Joseph Liouville. The possibility of fractional calculus was first considered by Joseph Liouville in 1832. Dynamic inequalities on time scales using Riemann–Liouville integrals are presented in more generalized form.

�

I�f�

Received: 18 September 2017Accepted: 31 January 2018First Published: 09 February 2018

© 2018 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license.

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Muhammad Jibril Shahab Sahir

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Shahab Sahir, Cogent Mathematics & Statistics (2018), 5: 1438030https://doi.org/10.1080/23311835.2018.1438030

Throughout the work, we suppose that � is a time scale, a, b ∈ � with a < b and an interval [a, b]�

means the intersection of a real interval with the given time scale.

2. PreliminariesWe need here basic concepts of delta calculus. The results of delta calculus are adapted from (Bohner & Peterson, 2001, 2003).

A time scale is an arbitrary non-empty closed subset of the real numbers. For t ∈ � , forward jump operator �:� → � is defined by:

The mapping �:𝕋 → ℝ+

0 = [0,∞) such that �(t): = �(t) − t is called the forward graininess func-tion. When 𝕋 = ℝ, we have �(t) = t and �(t) ≡ 0 for all t ∈ � and when 𝕋 = ℕ, we have �(t) = t + 1 and �(t) ≡ 1 for all t ∈ � . The backward jump operator �:� → � is defined by:

The mapping �:𝕋 → ℝ+

0 such that �(t): = t − �(t) is called the backward graininess function. If 𝜎(t) > t, we say that t is right-scattered, while if 𝜌(t) < t, we say that t is left-scattered. Also, if t < sup � and �(t) = t, then t is called right-dense, and if t > inf � and �(t) = t, then t is called left-dense. If � has a left-scattered maximum M, then � k = � − {M}, otherwise � k = � .

For a function f :𝕋 → ℝ, the derivative f Δ is defined as follows. Let t ∈ �k, if there exists f Δ(t) ∈ ℝ

such that for all 𝜖 > 0, there exists a neighborhood U of t with

for all s ∈ U, then f is said to be differentiable at t, and f Δ(t) is called the delta derivative of f at t.

A function f :𝕋 → ℝ 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 at every left-dense point. The set of all rd-contin-uous functions is denoted by Crd(𝕋 ,ℝ).

The next definition is given in (Bohner & Peterson, 2001, 2003).

Definition 1 A function F:𝕋 → ℝ is called a delta antiderivative of f :𝕋 → ℝ, provided that FΔ(t) = f (t) holds for all t ∈ �

k, then the delta integral of f is defined by:

The following results of nabla calculus are taken from (Anderson, Bullock, Erbe, Peterson, & Tran, 2003; Bohner & Peterson, 2001, 2003).

If � has a right-scattered minimum m, then �k = � − {m}, otherwise �k = � . The function f :𝕋 → ℝ is called nabla differentiable at t ∈ �k, if there exists f ∇(t) ∈ ℝ , such that for all 𝜖 > 0, there exists a neighborhood V of t, such that

for all s ∈ V. For 𝕋 = ℝ, we have f ∇(t) = f �(t) and for 𝕋 = ℤ, the backward difference operator is defined by f ∇(t) = ∇f (t) = f (t) − f (t − 1).

𝜎(t): = inf{s ∈ � :s > t}.

𝜌(t): = sup{s ∈ � :s < t}.

|f (�(t)) − f (s) − f Δ(t)(�(t) − s)| ≤ �|�(t) − s|,

b

∫a

f (t)Δt = F(b) − F(a).

|f (�(t)) − f (s) − f ∇(t)(�(t) − s)| ≤ �|�(t) − s|,

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Shahab Sahir, Cogent Mathematics & Statistics (2018), 5: 1438030https://doi.org/10.1080/23311835.2018.1438030

A function f :𝕋 → ℝ is said to be left-dense continuous (ld-continuous), provided it is continuous at all 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 Cld(𝕋 ,ℝ). If 𝕋 = ℝ, then f is ld-continuous if and only if f is continuous. If 𝕋 = ℤ, then any function is ld-continuous.

The next definition is given in (Anderson et al., 2003; Bohner & Peterson, 2001, 2003).

Definition 2 A function G:𝕋 → ℝ is called a nabla antiderivative of g:𝕋 → ℝ, provided that G∇(t) = g(t)

holds for all t ∈ �k, then the nabla integral of g is defined by

We need the following definitions of Δ-Riemann–Liouville-type fractional integral and ∇-Riemann–Liouville-type fractional integral.

The following definition is taken from (Anastassiou, 2010a, 2012).

Definition 3 For � ≥ 1, the time scale Δ-Riemann–Liouville-type fractional integral is defined by:

where t ∈ [a, b]�. Notice

where h�:𝕋 × 𝕋 → ℝ, � ≥ 0 are coordinate wise rd-continuous functions, such that h

0(t, s) = 1,

The following definition is taken from (Anastassiou, 2010b, 2012).

Definition 4 For � ≥ 1, the time scale ∇-Riemann–Liouville-type fractional integral is defined by:

where t ∈ [a, b]�. Notice

where h𝛼:𝕋 × 𝕋 → ℝ, � ≥ 0 are coordinate wise ld-continuous functions, such that h

0(t, s) = 1,

3. Main ResultsThe following result can be proved for different conditions imposed on p, q and r as p > 0, q > 0 but r < 0; p > 0, r > 0 but q < 0 or q > 0, r > 0 but p < 0.

b

∫a

g(t)∇t = G(b) − G(a).

(2.1)I�af (t) =

t

∫a

h�−1

(t, �(�))f (�)Δ�,

I1af (t) =

t

∫a

f (�)Δ�,

(2.2)h�+1

(t, s) =

t

∫s

h�(�, s)Δ�, ∀s, t ∈ � .

(2.3)J𝛼af (t) =

t

∫a

h𝛼−1

(t, 𝜌(𝜏))f (𝜏)∇𝜏,

J1af (t) =

t

∫a

f (�)∇�,

(2.4)h𝛼+1

(t, s) =

t

∫s

h𝛼(𝜏, s)∇𝜏, ∀s, t ∈ � .

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Shahab Sahir, Cogent Mathematics & Statistics (2018), 5: 1438030https://doi.org/10.1080/23311835.2018.1438030

Theorem 1 Let w, fi ∈ Crd([a, b]𝕋 ,ℝ − {0}) for i = 1, 2, 3 and p, q and r be three non-zero real numbers

with 1p+

1

q+

1

r= 0. Further assume that

3∏i=1

fi(t) = 1, where t ∈ [a, b]�. If p > 0, q > 0 but r < 0, then for

� ≥ 1,

Proof Given condition 1p+

1

q+

1

r= 0 can be rearranged as:

Set P = −p

r> 1 and Q = −

q

r> 1. Now we apply Rogers-Hölder’s inequality for F(�) and G(�), as:

So,

Replacing |F(�)| by: (h�−1

(t, �(�)))− r

p |f1(�)|−r and |G(�)| by:

(h�−1

(t, �(�)))− r

q|f2(�)|−r, where h

𝛼−1(t, 𝜎(𝜏)) > 0

for � ∈ [a, t)� and taking power − 1

r> 0, then (3.2) takes the form:

As f1(�)f

2(�)f

3(�) = 1, then (3.3) takes the form:

Then the proof is clear. □

Remark 1 If we take � = 1 and w(�) = 1 and also 𝕋 = ℤ in Theorem 1, then, we get discrete ver-sion of Rogers-Hölder’s inequality as given in (1.1) , where f

1(�) = fj, f2(�) = gj and f

3(�) = hj for all

j = 1, 2, ...,m are sets of positive values.

Remark 2 If we take � = 1, w(�) = 1, r = −1, p > 1 and also 𝕋 = ℤ, where f1(�) = fj, f2(�) = gj and

f3(�) = hj for all j = 1, 2, ...,m are sets of positive values, then, we get discrete version of Rogers-

Hölder’s inequality from Theorem 1, as:

Hölder’s inequality was first found by Hölder as given in Hölder, (1889).

(3.1)(I�a(|w(t)||f

1(t)|p)

) 1

p(I�a(|w(t)||f

2(t)|q)

) 1

q(I�a(|w(t)||f

3(t)|r)

) 1

r ≥ 1.

1

(−p

r)+

1

(−q

r)= 1.

�

t

a

|w(�)||F(�)G(�)|Δ� ≤

(

�

t

a

|w(�)||F(�)|P�) 1

P(

�

t

a

|w(�)||G(�)|Q�) 1

Q

.

(3.2)�

t

a

|w(�)||F(�)G(�)|Δ�

≤

(

�

t

a

|w(�)||F(�)|−p

r Δ�

)−r

p(

�

t

a

|w(�)||G(�)|−q

r Δ�

)−r

q

.

(3.3)

(

�

t

a

h�−1

(t, �(�))|w(�)||f1(�)f

2(�)|−rΔ�

)−1

r

≤

(

�

t

a

h�−1

(t, �(�))|w(�)||f1(�)|p�

) 1

p(

�

t

a

h�−1

(t, �(�))|w(�)||f2(�)|q�

) 1

q

.

(

�

t

a

h�−1

(t, �(�))|w(�)||f3(�)|r�

)−1

r

≤

(

�

t

a

h�−1

(t, �(�))|w(�)||f1(�)|p�

) 1

p(

�

t

a

h�−1

(t, �(�))|w(�)||f2(�)|q�

) 1

q

.

m∑

j=1

fjgj ≤

(m∑

j=1

fp

j

) 1

p(

m∑

j=1

gq

j

) 1

q

.

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Shahab Sahir, Cogent Mathematics & Statistics (2018), 5: 1438030https://doi.org/10.1080/23311835.2018.1438030

Remark 3 If we take � = 1, w(�) = 1, r = −1, p > 1, where f1(�)f

2(�)f

3(�) = 1, then we get integral ver-

sion of Rogers-Hölder’s inequality from Theorem 1

as given in Agarwal et al. (2001, 2014), Bohner and Peterson (2001).

When w(�) = 1, r = −1, p > 1 and also 𝕋 = ℝ, where f1(�)f

2(�)f

3(�) = 1, then (3.1) takes the form:

as given in Dahmani (2012).

Similarly nabla version of Theorem 1 can be written as:

Theorem 2 Let w, fi ∈ Cld([a, b]𝕋 ,ℝ − {0}) for i = 1, 2, 3 and p, q and r be three non-zero real numbers

with 1p+

1

q+

1

r= 0. Further assume that

3∏i=1

fi(t) = 1. If p > 0, q > 0 but r < 0, then for � ≥ 1

Proof Similar to the proof of Theorem 1. □

Result given in (3.1) is reversed if q < 0, r < 0 but p > 0; p < 0, r < 0 but q > 0 or p < 0, q < 0 but r > 0.

Corollary 1 Let w, fi ∈ Crd([a, b]𝕋 ,ℝ − {0}) for i = 1, 2, 3 and p, q and r be three non-zero real num-

bers with 1p+

1

q+

1

r= 0. Further assume that

3∏i=1

fi(t) = 1. If q < 0, r < 0 but p > 0; p < 0, r < 0 but q > 0 or p < 0, q < 0 but r > 0, then for � ≥ 1

Proof Similar to the proof of Theorem 1. □

Similarly reverse of nabla version of inequality (3.4) can be written as:

Corollary 2 Let w, fi ∈ Cld([a, b]𝕋 ,ℝ − {0}) for i = 1, 2, 3 and p, q and r be three non-zero real num-

bers with 1p+

1

q+

1

r= 0. Further assume that

3∏i=1

fi(t) = 1. If q < 0, r < 0 but p > 0; p < 0, r < 0 but q > 0 or p < 0, q < 0 but r > 0, then for � ≥ 1

Proof Similar to the proof of Theorem 1. □

Now, we generalize the result given in inequality (3.1) on delta calculus.

Theorem 3 Let w, fi ∈ Crd([a, b]𝕋 ,ℝ − {0}) for i = 1, 2, ...,n and pi be non-zero real numbers with n∑i=1

1

pi= 0. Further assume that

n∏i=1

fi(t) = 1. If all pi for i = 1, 2, ...,n but one are positive, then for � ≥ 1,

�

t

a

|f1(�)||f

2(�)|Δ� ≤

(

�

t

a

|f1(�)|p�

) 1

p(

�

t

a

|f2(�)|q�

) 1

q

,

I�a(|f1(t)f

2(t)|

)≤(I�a|f1(t)|

p) 1

p(I�a|f2(t)|

q) 1

q ,

(3.4)(J�a(|w(t)||f

1(t)|p)

) 1

p(J�a(|w(t)||f

2(t)|q)

) 1

q(J�a(|w(t)||f

3(t)|r)

) 1

r ≥ 1.

(3.5)(I�a(|w(t)||f

1(t)|p)

) 1

p(I�a(|w(t)||f

2(t)|q)

) 1

q(I�a(|w(t)||f

3(t)|r)

) 1

r ≤ 1.

(3.6)(J�a(|w(t)||f

1(t)|p)

) 1

p(J�a(|w(t)||f

2(t)|q)

) 1

q(J�a(|w(t)||f

3(t)|r)

) 1

r ≤ 1.

(3.7)n∏

i=1

(I�a(|w(t)||fi(t)|

pi )) 1

pi ≥ 1.

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Shahab Sahir, Cogent Mathematics & Statistics (2018), 5: 1438030https://doi.org/10.1080/23311835.2018.1438030

Proof Given condition becomes n−1∑i=1

1

pi+

1

pn= 0 and can be rearranged as:

n−1∑i=1

1

(−pi

pn)= 1, where all pi are

positive for i = 1, ...,n − 1 but pn is negative. Set Pi = −pi

pn> 1 for i = 1, ...,n − 1. Applying generalized

Rogers-Hölder’s inequality, we have

and

Now, we replace |fi(�)| by (h�−1

(t, �(�)))− p

npi |fi(�)|

−pn for i = 1, ...,n − 1, where h𝛼−1

(t, 𝜎(𝜏)) > 0 for � ∈ [a, t)

� and taking power − 1

pn> 0, inequality (3.8) takes the form

Applying condition n∏i=1

fi(�) = 1, inequality (3.9) takes the form:

So claim is proved. □

Now, we generalize our result (3.4) on nabla calculus.

Theorem 4 Let w, fi ∈ Cld([a, b]𝕋 ,ℝ − {0}) for i = 1, 2, ...,n and pi be non-zero real numbers with n∑i=1

1

pi= 0. Further assume that

n∏i=1

fi(t) = 1. If all pi for i = 1, 2, ...,n but one are positive, then for � ≥ 1,

Proof Similar to the proof of Theorem 3. □

Upcoming result is the reverse version of inequality (3.7) on delta calculus.

Corollary 3 Let w, fi ∈ Crd([a, b]𝕋 ,ℝ − {0}) for i = 1, 2, ...,n and pi be non-zero real numbers with n∑i=1

1

pi= 0. Further assume that

n∏i=1

fi(t) = 1. If all pi for i = 1, 2, ...,n but one are negative, then for � ≥ 1,

Proof Similar to the proof of Theorem 3. □

To conclude our results, we present reverse of inequality (3.10) on nabla calculus.

�

t

a

|w(�)|n−1∏

i=1

|fi(�)|Δ� ≤

n−1∏

i=1

(

�

t

a

|w(�)||fi(�)|Pi�

) 1

Pi

,

(3.8)�

t

a

|w(�)|n−1∏

i=1

|fi(�)|Δ� ≤

n−1∏

i=1

(

�

t

a

|w(�)||fi(�)|−

pi

pn Δ�

)−pnpi

.

(3.9)

(

�

t

a

h�−1

(t, �(�))|w(�)|n−1∏

i=1

|fi(�)|−pnΔ�

)−1

pn

≤

n−1∏

i=1

(

�

t

a

h�−1

(t, �(�))|w(�)||fi(�)|pi�

) 1

pi

.

(

�

t

a

h�−1

(t, �(�))|w(�)||fn(�)|pn�

)−1

pn

≤

n−1∏

i=1

(

�

t

a

h�−1

(t, �(�))|w(�)||fi(�)|pi�

) 1

pi

.

(3.10)n∏

i=1

(J�a(|w(t)||fi(t)|

pi )) 1

pi ≥ 1.

(3.11)n∏

i=1

(I�a(|w(t)||fi(t)|

pi )) 1

pi ≤ 1.

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Shahab Sahir, Cogent Mathematics & Statistics (2018), 5: 1438030https://doi.org/10.1080/23311835.2018.1438030

© 2018 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license.You are free to: Share — copy and redistribute the material in any medium or format Adapt — remix, transform, and build upon the material for any purpose, even commercially.The licensor cannot revoke these freedoms as long as you follow the license terms.

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Corollary 4 Let w, fi ∈ Cld([a, b]𝕋 ,ℝ − {0}) for i = 1, 2, ...,n and pi be non-zero real numbers with n∑i=1

1

pi= 0. Further assume that

n∏i=1

fi(t) = 1. If all pi for i = 1, 2, ...,n but one are negative, then for � ≥ 1,

Proof Similar to the proof of Theorem 3. □

4. Conclusion and future workIn this research article, we have presented delta and nabla versions of some fractional dynamic in-equalities of Rogers-Hölder’s type using Riemann–Liouville integral. In future, using Riemann–Liouville integral, we can find dynamic inequalities for convex functions on time scales as given in Sahir (2017). We can also find such inequalities using Riemann–Liouville fractional derivatives. We can find such type of inequalities using functional generalization. It will be interesting to present these inequalities on quantum calculus.

FundingThe author received no direct funding for this research.

Author detailsMuhammad Jibril Shahab Sahir1

E-mail: jibrielshahab@gmail.comORCID ID: http://orcid.org/0000-0002-7854-82661 Department of Mathematics, University of Sargodha, Sub-

Campus Bhakkar, Bhakkar, Pakistan.

Citation informationCite this article as: Fractional dynamic inequalities harmonized on time scales, Muhammad Jibril Shahab Sahir, Cogent Mathematics & Statistics (2018), 5: 1438030.

ReferencesAczél, J., & Beckenbach, E. F. (1978). On Hölder’s inequality,

General inequalities 2. Proceedings of the Second International Conference Oberwolfach, pp. 145–150.

Agarwal, R. P., Bohner, M., & Peterson, A. (2001). Inequalities on time scales: A survey. Mathematical Inequalities & Applications, 4, 535–557.

Agarwal, R. P., O’Regan, D., & Saker, S. H. (2014). Dynamic inequalities on time scales. Cham: Springer International Publishing.

Anastassiou, G. A. (2010a). Principles of delta fractional calculus on time scales and inequalities. Mathematical & Computer Modelling, 52(3–4), 556–566.

Anastassiou, G. A. (2010b). Foundations of nabla fractional calculus on time scales and inequalities. Computers & Mathematics with Applications, 59(12), 3750–3762.

Anastassiou, G. A. (2012). Integral operator inequalities on time scales. International Journal of Difference Equations, 7(2), 111–137.

Anderson, D. R., Bullock, J., Erbe, L., Peterson, A., & Tran, H. (2003). Nabla dynamic equations on time scales. Pan-American Mathematical Journal, 13(1), 1–48.

Bohner, M., & Peterson, A. (2001). Dynamic equations on time scales. Boston, MA: Birkhäuser Boston Inc.

Bohner, M., & Peterson, A. (2003). Advances in dynamic equations on time scales. Boston, MA: Birkhäuser Boston.

Dahmani, Z. (2012). About some integral inequalities using Riemann-Liouville integrals. General Mathematics, 20(4), 63–69.

Hilger, S. (1990). Analysis on measure chains - A unified approach to continuous and discrete calculus. Results in Mathematics, 18, 18–56.

Hölder, O. (1889). Über einen Mittelwerthsatz. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 2, 38–47.

Sahir, M. J. S. (2017). Dynamic inequalities for convex functions harmonized on time scales. Journal of Applied Mathematics & Physics, 5, 2360–2370.

(3.12)n∏

i=1

(J�a(|w(t)||fi(t)|

pi )) 1

pi ≤ 1.