Next Article in Journal
A Two-Tier Partition Algorithm for the Optimization of the Large-Scale Simulation of Information Diffusion in Social Networks
Next Article in Special Issue
An Eigenvalues Approach for a Two-Dimensional Porous Medium Based Upon Weak, Normal and Strong Thermal Conductivities
Previous Article in Journal
Design and Optimization of Plasmon Resonance Sensor Based on Micro–Nano Symmetrical Localized Surface
Previous Article in Special Issue
Determination of Forces and Moments Per Unit Length in Symmetric Exponential FG Plates with a Quasi-Triangular Hole
Open AccessArticle

New Weighted Opial-Type Inequalities on Time Scales for Convex Functions

1
Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt
2
Department of Mathematics, Cankaya University, Ankara 06530, Turkey
3
Institute of Space Science, 07650 Magurele-Bucharest, Romania
4
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(5), 842; https://doi.org/10.3390/sym12050842
Received: 21 March 2020 / Revised: 21 April 2020 / Accepted: 1 May 2020 / Published: 21 May 2020
(This article belongs to the Special Issue Composite Structures with Symmetry)

Abstract

Our work is based on the multiple inequalities illustrated in 1967 by E. K. Godunova and V. I. Levin, in 1990 by Hwang and Yang and in 1993 by B. G. Pachpatte. With the help of the dynamic Jensen and Hölder inequality, we generalize a number of those inequalities to a general time scale. In addition to these generalizations, some integral and discrete inequalities will be obtained as special cases of our results.
Keywords: opial-type inequality; dynamic inequality; convexity; time scale opial-type inequality; dynamic inequality; convexity; time scale

1. Introduction

The following inequality [1] is well-known in the literature as Opial’s inequality.
Theorem 1.
If δ is an absolutely continuous function on [ 0 , h ] R with δ ( 0 ) = δ ( h ) = 0 , then
0 h | δ ( t ) δ ( t ) | d t h 4 0 h | δ ( t ) | 2 d t .
In 1967 E. K. Godunova and V. I. Levin [2] proved the following two theorems which are a generalization of Opial’s inequality (1).
Theorem 2.
Let δ be real-valued absolutely continuous function on [ a , b ] R with δ ( a ) = 0 . Let f be real-valued convex and increasing function on [ 0 , ) R with f ( 0 ) = 0 . Then, the following inequality holds
a b f ( | δ ( t ) | ) | δ ( t ) | d t f a b | δ ( t ) | d t .
Theorem 3.
Let δ be real-valued absolutely continuous function on [ a , b ] R with δ ( a ) = δ ( b ) = 0 . Assume f and g are real-valued convex and increasing functions on [ 0 , ) R with f ( 0 ) = 0 . Further let p 0 on [ a , b ] R and a b p ( t ) d t = 1 . Then, the following inequality holds
a b f ( | δ ( t ) | ) | δ ( t ) | d t 2 f g 1 a b p ( t ) g | δ ( t ) | 2 p ( t ) d t .
In 1990, Hwang and Yang [3] established the following result:
Theorem 4.
Assuming f 0 and g 0 are continuous functions on [ 0 , ) R with f ( 0 ) = 0 such that f 0 and g 0 exist and non-decreasing continuous functions on [ 0 , ) R . Suppose x and y are absolutely continuous functions on [ a , τ ] R , and x ( a ) = y ( a ) = 0 . Then, for all m 1 , we get
a τ f ( | δ ( t ) | m ) g ( | γ ( t ) | m ) | γ ( t ) | m + g ( | γ ( t ) | m ) f ( | δ ( t ) | m ) | δ ( t ) | m d t 1 λ f λ a τ | δ ( t ) | m d t g λ a τ | γ ( t ) | m d t ,
where λ = ( τ a ) m 1 .
S. Hilger [4], suggested time scales theory to unify discrete and continuous analysis. Continuous calculus, discrete calculus, and quantum calculus can be said as the three most common examples of calculus on time scales i.e., for continuous calculus T = R , for discrete calculus T = h Z and for quantum calculus T = q Z ¯ = { q z : z Z } { 0 } where q > 1 . The book due to Bohner and Peterson [5] on the subject of time scales briefs and organizes much of time scales calculus. For some Opial-type integral, dynamic inequalities and other types of inequalities on time scales, see the papers [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. More results on inequalities see, [33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53].
The following essential relations on some time scales such as R , Z , h Z and q Z ¯ will be used in the following section. Note that:
(i)
If T = R , then
σ ( t ) = t ,   μ ( t ) = 0 ,   ψ Δ ( t ) = ψ ( t ) ,   α β ψ ( t ) Δ t = α β ψ ( t ) d t .
(ii)
If T = Z , then
σ ( t ) = t + 1 ,   μ ( t ) = 1 ,   ψ Δ ( t ) = ψ ( t + 1 ) ψ ( t ) ,   α β ψ ( t ) Δ t = t = α β 1 ψ ( t ) .
(iii)
If T = h Z , then
σ ( t ) = t + h ,   μ ( t ) = h ,   ψ Δ ( t ) = ψ ( t + h ) ψ ( t h ) ,   α β ψ ( t ) Δ t = t = α h β h 1 ψ ( t h ) h .
(iv)
If T = q Z ¯ , then
σ ( t ) = q t ,   μ ( t ) = ( q 1 ) t ,   ψ Δ ( t ) = ψ ( q t ) ψ ( t ) ( q 1 ) t ,   α β ψ ( t ) Δ t = ( q 1 ) t = ( log q α ) ( log q β ) 1 ψ ( q t ) q t .
Next is Hölder’s and Jensen’s inequality:
Lemma 1
(Hölder’s inequality [5]). Let a , b T . For f, g C r d ( T , R ) , we have
a b f ( t ) g ( t ) Δ t a b f p ( t ) Δ t 1 p a b g q ( t ) Δ t 1 q ,
where p > 1 and 1 p + 1 q = 1 .
Lemma 2
(Jensen’s inequality [16]). Let a, b T and c, d R . Assume that g C r d [ a , b ] T , [ c , d ] and r C r d [ a , b ] T , R are nonnegative with a b r ( t ) Δ t > 0 . If Φ ˘ C r d ( c , d ) , R is a convex function, then
Φ ˘ a b r ( t ) g ( t ) Δ t a b r ( t ) Δ t a b r ( t ) Φ ˘ ( g ( t ) ) Δ t a b r ( t ) Δ t .
Lemma 3
(Chain rule [5]). Assume g : R R , is continuous, g : T R is delta-differentiable on T κ , and f : R R is continuously differentiable. Then there exists c in the real interval [ t , σ ( t ) ] with
( f g ) Δ ( t ) = f ( g ( c ) ) g Δ ( t ) .
In the proofs of our results, the following inequality will be used:
a k + b k ( a + b ) k 2 k 1 a k + b k ,   if a 0 , b 0 and k 1 .
In this article, we prove some dynamic Opial-type inequalities involving convex functions on time scales. Our results generalize some of the mentioned results of Pachpatte [54,55], E. K. Godunova and V. I. Levin [2] and Hwang and Yang [3], in time scales. Furthermore, our results extend some existing dynamic Opial-type inequalities in the literature, and give some integral and discrete inequalities as special cases.

2. Main Results

Theorem 5.
Assuming T is a time scale with Γ, α T , g 0 and f 0 are continuous functions on [ 0 , ) R with f ( 0 ) = 0 such that g 0 and f 0 exist and non-decreasing continuous functions on [ 0 , ) R . Suppose x and y are rd-continuous functions on [ α , Γ ] T , and x ( α ) = y ( α ) = 0 . Then, for all m 1 , we get
α Γ f | x ( θ ) | m g | y ( θ ) | m y Δ ( θ ) m + g | y ( θ ) | m f | x ( θ ) | m x Δ ( θ ) m Δ θ 1 λ 1 f λ 1 α Γ x Δ ( θ ) m Δ θ g λ 1 α Γ y Δ ( θ ) m Δ θ ,
where λ 1 = ( Γ α ) m 1 .
Proof. 
For t [ α , Γ ] T . Define φ ( t ) : = α t x Δ ( θ ) m Δ θ and ψ ( t ) : = α t y Δ ( θ ) m Δ θ so that φ Δ ( t ) = x Δ ( t ) m and ψ Δ ( t ) = y Δ ( t ) m . Thus,
| x ( t ) | m = α t x Δ ( θ ) Δ θ m α t x Δ ( θ ) Δ θ m , | y ( t ) | m = α t y Δ ( θ ) Δ θ m α t y Δ ( θ ) Δ θ m .
Next, applying the dynamic Hölder’s inequality (1) on (11) with indices m and m m 1 , we get
| x ( t ) | m ( t α ) m 1 α t x Δ ( θ ) m Δ θ ( t α ) m 1 α t x Δ ( θ ) m Δ θ λ 1 φ ( t ) .
and similarly
| y ( t ) | m λ 1 ψ ( t ) .
Since f 0 , f 0 , g 0 , and g 0 are non-decreasing continuous on [ 0 , ) R . Since σ ( t ) t , we get g ( λ 1 ψ ) g σ ( λ 1 ψ ) , and we find that
α Γ f | x ( θ ) | m g | y ( θ ) | m y Δ ( θ ) m + g | y ( θ ) | m f | x ( θ ) | m x Δ ( θ ) m Δ θ α Γ f λ 1 φ ( θ ) g λ 1 ψ ( θ ) ψ Δ ( θ ) + g λ 1 ψ ( θ ) f λ 1 φ ( θ ) φ Δ ( θ ) Δ θ α Γ f λ 1 φ ( θ ) g λ 1 ψ ( θ ) ψ Δ ( θ ) + g σ λ 1 ψ ( θ ) f λ 1 φ ( θ ) φ Δ ( θ ) Δ θ .
From Lemma 3 for c [ t , σ ( t ) ] R , we get
1 λ 1 f Δ ( λ 1 φ ) ( t ) = f ( λ 1 φ ( c ) ) φ Δ ( t ) f ( λ 1 φ ( t ) ) φ Δ ( t ) ,
and similarly
1 λ 1 g Δ ( λ 1 ψ ( t ) ) g ( λ 1 ψ ( t ) ) ψ Δ ( t ) .
Then, by substituting (15) and (16) into (14), we get
α Γ f | x ( θ ) | m g | y ( θ ) | m y Δ ( θ ) m + g | y ( θ ) | m f | x ( θ ) | m x Δ ( θ ) m Δ θ α Γ f λ 1 φ ( θ ) g λ 1 ψ ( θ ) ψ Δ ( θ ) + g σ λ 1 ψ ( θ ) f λ 1 φ ( θ ) φ Δ ( θ ) Δ θ = 1 λ 1 α Γ f ( λ 1 φ ) g ( λ 1 ψ ) Δ ( θ ) Δ θ = 1 λ 1 f λ 1 α Γ x Δ ( θ ) m Δ θ g λ 1 α Γ y Δ ( θ ) m Δ θ ,
which is the desired inequality (10). This proves our claim. □
Remark 1.
When T = R in Theorem 5, then, by the relation (4), we obtain Hwang and Yang inequality (3).
Corollary 1.
If we take T = h Z in Theorem 5, then, by the relation (6), inequality (10) becomes
h n = α h Γ h 1 f | x ( n h ) | m g | y ( n h ) | m Δ y ( n h ) m + g | y ( n h ) | m f | x ( n h ) | m Δ x ( n h ) m 1 λ 1 f h λ 1 n = α h Γ h 1 Δ x ( n h ) m g h λ 1 n = α h Γ h 1 Δ y ( n h ) m .
Remark 2.
In Corollary 2, if we take h = 1 , then inequality (10) becomes
n = α Γ 1 f | x ( n ) | m g | y ( n ) | m Δ y ( n ) m + g | y ( n ) | m f | x ( n ) | m Δ x ( n ) m 1 λ 1 f λ 1 n = α Γ 1 Δ x ( n ) m g λ 1 n = α Γ 1 Δ y ( n ) m .
Corollary 2.
If we take T = q Z ¯ in Theorem 5, then, by the relation (7), inequality (10) becomes
( q 1 ) n = ( log q α ) ( log q Γ ) 1 f | x ( q n ) | m g | y ( q n ) | m Δ y ( q n ) m + g | y ( q n ) | m f | x ( q n ) | m Δ x ( q n ) m q n 1 λ 1 f ( q 1 ) λ 1 n = ( log q α ) ( log q Γ ) 1 Δ x ( q n ) m q n g ( q 1 ) λ 1 n = ( log q α ) ( log q Γ ) 1 Δ y ( q n ) m q n .
Remark 3.
If g = m = 1 , and x = y in inequality (10), then, we obtain
α Γ f | x ( θ ) | x Δ ( θ ) Δ θ f α Γ x Δ ( θ ) Δ θ
with weak conditions for f.
Remark 4.
If T = R , then inequality (17) gives E. K. Godunova and V. I. Levin’s inequality in [2].
Corollary 3.
Assuming f ( t ) = t + m m , g = 1 , x = y and 0 in Theorem 5. Then, from (10), and Lemma 1 with indices + m and + m m , we obtain
α Γ | x ( θ ) | x Δ ( θ ) m Δ θ m + m 1 λ 1 λ 1 α Γ x Δ ( θ ) m Δ θ + m m m + m λ 1 m ( Γ α ) m α Γ x Δ ( θ ) + m Δ θ = m + m ( Γ α ) α Γ x Δ ( θ ) + m Δ θ .
Corollary 4.
When m = 1 and = 1 in (18), we obtain the following inequality
α Γ | x ( θ ) | x Δ ( θ ) Δ θ ( Γ α ) 2 α Γ x Δ ( θ ) 2 Δ θ .
Corollary 5.
When T = R , m = 1 and α = 0 in (18), we obtain the inequality of Hua [56]
0 Γ | x ( θ ) | x ( θ ) d θ Γ + 1 0 Γ x ( θ ) + 1 d θ .
Moreover, in (20), equality holds if and only if x ( t ) = c t .
Corollary 6.
If T = R , then (18), gives the inequality of Yang [57].
α Γ | x ( θ ) | x ( θ ) m d θ m + m ( Γ α ) α Γ x ( θ ) + m d θ .
Corollary 7.
Assuming p 0 is rd-continuous on [ α , Γ ] T with α Γ Δ θ p ( θ ) < , and let q > 0 is non-increasing bounded on [ α , Γ ] T . By using f ( t ) = t 2 , g ( t ) = 1 , m = 1 and x ( t ) = y ( t ) = α t q ( θ ) z Δ ( θ ) Δ θ and x Δ ( t ) = q ( t ) z Δ ( t ) . Then, we get from Theorem 5, the following inequality
2 α Γ α θ q ( ξ ) x Δ ( ξ ) Δ ξ q ( θ ) z Δ ( θ ) Δ θ α Γ q ( θ ) z Δ ( θ ) Δ θ 2 .
However, since
α t q ( θ ) z Δ ( θ ) Δ θ q ( t ) α t z Δ ( θ ) Δ θ q ( t ) z ( t ) ,
it follows that
2 α Γ q ( θ ) | x ( θ ) | x Δ ( θ ) Δ θ α Γ 1 p ( θ ) p ( θ ) q ( θ ) x Δ ( θ ) Δ θ 2 .
By applying the Cauchy–Schwarz inequality, we get that
α Γ q ( θ ) | x ( θ ) | x Δ ( θ ) Δ θ 1 2 α Γ Δ θ p ( θ ) α Γ p ( θ ) q ( θ ) x Δ ( θ ) 2 Δ θ .
Remark 5.
When T = R , the inequality (22), reduces to Yang inequality [57]
α Γ q ( θ ) | x ( θ ) | x ( θ ) d θ 1 2 α Γ d θ p ( θ ) α Γ p ( θ ) q ( θ ) x ( θ ) 2 d θ .
Corollary 8.
Take f ( t ) = t 2 , g ( t ) = 1 , m = 1 and x = y in Theorem 5, the inequality (10) becomes
α Γ | x ( θ ) | | x Δ ( θ ) | Δ θ 1 2 ( α Γ | x Δ ( θ ) | Δ θ ) 2 = 1 2 α Γ p 1 ν ^ ( θ ) p 1 ν ^ ( θ ) x Δ ( θ ) Δ θ 2 .
From Lemma 1 with indices ν ^ , μ ^ such that 1 ν ^ + 1 μ ^ = 1 , we obtain
α Γ | x ( θ ) | | x Δ ( θ ) | Δ θ 1 2 α Γ p 1 μ ^ ( θ ) Δ θ 2 μ ^ ( α Γ p ( θ ) | x Δ ( θ ) | ν ^ Δ θ ) 2 ν ^ .
Remark 6.
If T = R , then (23), gives the inequality of Maroni [58].
α Γ | x ( θ ) | | x Δ ( θ ) | d θ 1 2 α Γ p 1 μ ^ ( θ ) d θ 2 μ ^ α Γ p ( θ ) | x ( θ ) | ν d θ 2 ν .
Theorem 6.
Under the hypotheses of Theorem 5. Let p > 0 and p > 0 be defined on [ α , Γ ] T with α Γ p ( θ ) Δ θ = 1 and α Γ p ( θ ) Δ θ = 1 . Further, let η > 0 be convex and increasing on [ 0 , ) R . Then, for all m 1 , the following inequality holds
α Γ f | x ( θ ) | m g | y ( θ ) | m y Δ ( θ ) m + g | y ( θ ) | m f | x ( θ ) | m x Δ ( θ ) m Δ θ 1 λ 1 f 2 λ 1 η 1 α Γ p ( θ ) η | x Δ ( θ ) | m 2 p ( θ ) Δ θ × g 2 λ 1 η 1 α Γ p ( θ ) η | y Δ ( θ ) | m 2 p ( θ ) Δ θ ,
where λ 1 defined as in Theorem 5.
Proof. 
Dynamic Jensen inequality (2) provides
η 1 2 α Γ | x Δ ( θ ) | m Δ θ α Γ p ( θ ) η | x Δ ( θ ) | m 2 p ( θ ) Δ θ .
Further, since η is increasing, we have
α Γ | x Δ ( θ ) | m Δ θ 2 η 1 α Γ p ( θ ) η | x Δ ( θ ) | m 2 p ( θ ) Δ θ .
Similarly, we have
α Γ | y Δ ( θ ) | m Δ θ 2 η 1 α Γ p ( θ ) η | y Δ ( θ ) | m 2 p ( θ ) Δ θ .
From (10), (26) and (27) immediately gives
α Γ f | x ( θ ) | m g | y ( θ ) | m y Δ ( θ ) m + g | y ( θ ) | m f | x ( θ ) | m x Δ ( θ ) m Δ θ 1 λ 1 f 2 λ 1 η 1 α Γ p ( θ ) η | x Δ ( θ ) | m 2 p ( θ ) Δ θ × g 2 λ 1 η 1 α Γ p ( θ ) η | y Δ ( θ ) | m 2 p ( θ ) Δ θ .
This gives our claim. □
Remark 7.
When T = R in Theorem 6, then, by the relation (4), we get the inequality of Hwang and Yang [3].
Corollary 9.
If we take T = h Z in Theorem 6, then, by the relation (6), inequality (25) becomes
h n = α h Γ h 1 f | x ( n h ) | m g | y ( n h ) | m Δ y ( n h ) m + g | y ( n h ) | m f | x ( n h ) | m Δ x ( n h ) m 1 λ 1 f 2 h λ 1 η 1 n = α h Γ h 1 p ( n h ) η | Δ x ( n h ) | m 2 p ( n h ) × g 2 h λ 1 η 1 n = α h Γ h 1 p ( n h ) η | Δ y ( n h ) | m 2 p ( n h ) .
Remark 8.
In Corollary 10, if we take h = 1 , then inequality (25) becomes
n = α Γ 1 f | x ( n ) | m g | y ( n ) | m Δ y ( n ) m + g | y ( n ) | m f | x ( n ) | m Δ x ( n ) m 1 λ 1 f 2 λ 1 η 1 n = α Γ 1 p ( n ) η | Δ x ( n ) | m 2 p ( n ) × g 2 λ 1 η 1 n = α Γ 1 p ( n ) η | Δ y ( n ) | m 2 p ( n ) .
Corollary 10.
If we take T = q Z ¯ in Theorem 6, then, by the relation (7), inequality (25) becomes
( q 1 ) n = ( log q α ) ( log q Γ ) 1 f | x ( q n ) | m g | y ( q n ) | m Δ y ( q n ) m + g | y ( q n ) | m f | x ( q n ) | m Δ x ( q n ) m q n 1 λ 1 f 2 ( q 1 ) λ 1 η 1 n = ( log q α ) ( log q Γ ) 1 q n p ( q n ) η | Δ x ( q n ) | m 2 p ( q n ) × g 2 ( q 1 ) λ 1 η 1 n = ( log q α ) ( log q Γ ) 1 q n p ( q n ) η | Δ y ( q n ) | m 2 p ( q n ) .
Remark 9.
When T = R , if we take m = 1 and p ( t ) = p ( t ) , in the inequality (25), we get Pachpatte inequality [54]
α Γ f ( | x ( θ ) | ) g ( | y ( θ ) | ) | y ( θ ) | + g ( | y ( θ ) | ) f ( | x ( θ ) | ) | x ( θ ) | d θ f η 1 α Γ p ( θ ) η | x ( θ ) | p ( θ ) d θ × g η 1 α Γ p ( θ ) η | y ( θ ) | p ( θ ) d θ .
Remark 10.
Taking f ( θ ) = g ( θ ) and x ( θ ) = y ( θ ) in inequality (28), we get the Pachpatte inequality [54]
α Γ f ( | x ( θ ) | ) f ( | x ( θ ) | ) | x ( θ ) | d θ 1 2 f η 1 α Γ p ( θ ) η | x ( θ ) | p ( θ ) d θ 2 .
Corollary 11.
Suppose T is a time scale, α, β, ( α + β ) 2 T , and f, g are defined as in Theorem 5. Furthermore, Assume x and y are rd-continuous functions on [ α , β ] T such that x ( α ) = y ( α ) = 0 and x ( β ) = y ( β ) = 0 . Then, we get
α β f ( | x ( θ ) | m ) g ( | y ( θ ) | m ) | y Δ ( θ ) | m + g ( | y ( θ ) | m ) f ( | x ( θ ) | m ) | x Δ ( θ ) | m Δ θ 2 m 1 λ 2 g 2 1 m λ 2 α β | y Δ ( θ ) | m Δ θ × f 2 1 m λ 2 α α + β 2 | x Δ ( θ ) | m Δ θ + f 2 1 m λ 2 α + β 2 β | x Δ ( θ ) | m Δ θ ,
where λ 2 = ( β α ) m 1 .
Proof. 
Let Γ [ α , β ] T , the functions x and y satisfy the conditions of Theorem 5 on [ α , Γ ] T . Thus, inequality (10) holds. Next, in the interval [ Γ , β ] T , the functions x and y are rd-continuous, and x ( β ) = y ( β ) = 0 . Thus, by defining φ ( t ) = t β | x Δ ( θ ) | m Δ θ , ϕ ( t ) = t β | y Δ ( θ ) | m Δ θ t [ Γ , β ] , and following an argument similar to Theorem 5, we obtain
Γ β f ( | x ( θ ) | m ) g ( | y ( θ ) | m ) | y Δ ( θ ) | m + g ( | y ( θ ) | m ) f ( | x ( θ ) | m ) | x Δ ( θ ) | m Δ θ 1 λ 3 f λ 3 Γ β | x Δ ( θ ) | m Δ θ g λ 3 Γ β | y Δ ( θ ) | m Δ θ ,
where λ 3 = ( β Γ ) m 1 . A combination of the inequalities (10) and (30), we get
α β f ( | x ( θ ) | m ) g ( | y ( θ ) | m ) | y Δ ( θ ) | m + g ( | y ( θ ) | m ) f ( | x ( θ ) | m ) | x Δ ( θ ) | m Δ θ = α Γ f ( | x ( θ ) | m ) g ( | y ( θ ) | m ) | y Δ ( θ ) | m + g ( | y ( θ ) | m ) f ( | x ( θ ) | m ) | x Δ ( θ ) | m Δ θ + Γ β f ( | x ( θ ) | m ) g ( | y ( t ) | m ) | y Δ ( θ ) | m + g ( | y ( θ ) | m ) f ( | x ( θ ) | m ) | x Δ ( θ ) | m Δ θ 1 λ 1 f λ 1 α Γ | x Δ ( θ ) | m Δ θ g λ 1 α Γ y Δ ( θ ) | m Δ θ + 1 λ 3 f λ 3 Γ β | x Δ ( θ ) | m Δ θ g λ 3 Γ β | y Δ ( θ ) | m Δ θ ,
for Γ = α + β 2 , we find that λ 1 = λ 3 = ( β α 2 ) m 1 , where λ 2 = ( β α ) m 1 , then λ 1 = λ 3 = λ 2 2 m 1 . Then, by substituting in (31), we get
α β f ( | x ( θ ) | m ) g ( | y ( θ ) | m ) | y Δ ( θ ) | m + g ( | y ( θ ) | m ) f ( | x ( θ ) | m ) | x Δ ( θ ) | m Δ θ 2 m 1 λ 2 f 2 1 m λ 2 α α + β 2 | x Δ ( θ ) | m Δ θ g 2 1 m λ 2 α α + β 2 | y Δ ( θ ) | m Δ θ + 2 m 1 λ 2 f 2 1 m λ 2 α + β 2 β | x Δ ( θ ) | m Δ θ g 2 1 m λ 2 α + β 2 β | y Δ ( θ ) | m Δ θ .
Since g is non-decreasing function, we have
α β f ( | x ( θ ) | m ) g ( | y ( θ ) | m ) | y Δ ( θ ) | m + g ( | y ( θ ) | m ) f ( | x ( t ) | m ) | x Δ ( θ ) | m Δ θ 2 m 1 λ 2 g 2 1 m λ 2 α β | y Δ ( θ ) | m Δ θ [ f 2 1 m λ 2 α α + β 2 | x Δ ( θ ) | m Δ θ + f 2 1 m λ 2 α + β 2 β | x Δ ( θ ) | m Δ θ ] .
This gives our claim (29). □
Corollary 12.
If we take T = h Z in Corollary 11, then, by the relation (6), inequality (29) becomes
h n = α h β h 1 f ( | x ( n h ) | m ) g ( | y ( n h ) | m ) | Δ y ( n h ) | m + g ( | y ( n h ) | m ) f ( | x ( n h ) | m ) | Δ x ( n h ) | m 2 m 1 λ 2 g 2 1 m λ 2 h n = α h β h 1 | Δ y ( n h ) | m × f 2 1 m λ 2 h n = α h α + β 2 h 1 | Δ x ( n h ) | m + f 2 1 m λ 2 h n = α + β 2 h β h 1 | Δ x ( n h ) | m .
Remark 11.
In Corollary 12, if we take h = 1 , then inequality (29) becomes
n = α β 1 f ( | x ( n ) | m ) g ( | y ( n ) | m ) | Δ y ( n ) | m + g ( | y ( n ) | m ) f ( | x ( n ) | m ) | Δ x ( n ) | m 2 m 1 λ 2 g 2 1 m λ 2 n = α β 1 | Δ y ( n ) | m × f 2 1 m λ 2 n = α α + β 2 1 | Δ x ( n ) | m + f 2 1 m λ 2 n = α + β 2 β 1 | Δ x ( n ) | m .
Corollary 13.
If we take T = q Z ¯ in Corollary 11, then, by the relation (7), inequality (29) becomes
( q 1 ) n = ( log q α ) ( log q β ) 1 f ( | x ( q n ) | m ) g ( | y ( q n ) | m ) | Δ y ( q n ) | m + g ( | y ( q n ) | m ) f ( | x ( q n ) | m ) | Δ x ( q n ) | m q n 2 m 1 λ 2 g 2 1 m λ 2 ( q 1 ) n = ( log q α ) ( log q β ) 1 | Δ y ( q n ) | m q n × [ f 2 1 m λ 2 ( q 1 ) n = ( log q α ) ( log q α + β 2 ) 1 | Δ x ( q n ) | m q n + f 2 1 m λ 2 ( q 1 ) n = ( log q α + β 2 ) ( log q β ) 1 | Δ x ( q n ) | m q n ] .
Corollary 14.
In Corollary 11. For m = 1 , we can get the following inequality
α β f ( | x ( θ ) | ) g ( | y ( θ ) | ) | y Δ ( θ ) | + g ( | y ( t ) | ) f ( | x ( θ ) | ) | x Δ θ | Δ θ 2 f 1 2 α β | x Δ ( θ ) | Δ θ g 1 2 α β | y Δ ( θ ) | Δ θ .
Proof. 
By substituting in (32) by m = 1 , we get
α β f ( | x ( θ ) | ) g ( | y ( θ ) | ) | y Δ ( θ ) | + g ( | y ( θ ) | ) f ( | x ( θ ) | ) | x Δ ( θ ) | Δ θ f α α + β 2 | x Δ ( θ ) | Δ θ g α α + β 2 | y Δ ( θ ) | Δ θ + f α + β 2 β | x Δ ( θ ) | Δ θ g α + β 2 β | y Δ ( θ ) | Δ θ .
If we choose Γ such that
α Γ x Δ ( θ ) Δ θ = Γ β x Δ ( θ ) Δ θ = 1 2 α β x Δ ( θ ) Δ θ .
Then, by substituting from (35) in (34), we can get
α β f ( | x ( θ ) | ) g ( | y ( θ ) | ) | y Δ ( θ ) | + g ( | y ( θ ) | ) f ( | x ( θ ) | ) | x Δ ( θ ) | Δ θ 2 f 1 2 α β | x Δ ( θ ) | Δ θ g 1 2 α β | y Δ ( θ ) | Δ θ .
This gives our claim. □
Corollary 15.
Let f ( t ) = t + m m , 0 , g ( t ) = 1 and x ( t ) = y ( t ) in (29), to obtain
α β + m m | x ( θ ) | | x Δ ( θ ) | m Δ θ β α 2 ( m 1 ) m [ α α + β 2 | x Δ ( θ ) | m Δ θ + m m + α + β 2 β | x Δ ( θ ) | m Δ θ + m m ] .
By using the inequality (9) in inequality (36), we get
α β | x ( θ ) | | x Δ ( θ ) | m Δ θ m + m β α 2 ( m 1 ) m × α α + β 2 | x Δ ( θ ) | m Δ θ + m m + α + β 2 β | x Δ ( θ ) | m Δ θ + m m m + m β α 2 ( m 1 ) m α α + β 2 | x Δ ( θ ) | m Δ θ + α + β 2 β | x Δ ( θ ) | m Δ θ + m m = m + m β α 2 ( m 1 ) m α β | x Δ ( θ ) | m Δ θ + m m .
From Lemma 1 with indices + m and + m m on (37), we have
α β | x ( θ ) | | x Δ ( θ ) | m Δ θ m + m β α 2 α β | x Δ ( θ ) | + m Δ θ .
Remark 12.
Taking T = R , then (38) gives Yang inequality [59]
α β | x ( θ ) | | x ( θ ) | m d t m + m β α 2 α β | x ( θ ) | + m d θ .
Theorem 7.
Under the hypotheses of Theorem 5. Assuming that s 0 , r 0 and s Δ 0 , r Δ 0 , t [ α , Γ ] T and r ( α ) = 0 , s ( α ) = 0 and χ 0 and π 0 are convex and increasing functions on ( 0 , ) R . Then, we get
α Γ [ s Δ ( θ ) π | y Δ ( θ ) | s Δ ( θ ) f r ( θ ) χ | x ( θ ) | r ( θ ) g s ( θ ) π | y ( θ ) | s ( θ ) + r Δ ( θ ) χ | x Δ ( θ ) | r Δ ( θ ) f ( r ( θ ) χ | x ( θ ) | r ( θ ) ) g s ( θ ) π | y ( θ ) | s ( θ ) ] Δ θ f α Γ r Δ ( θ ) χ | x Δ ( θ ) | r Δ ( θ ) Δ θ g α Γ s Δ ( θ ) π | y Δ ( θ ) | s Δ ( θ ) Δ θ .
Proof. 
For t [ α , Γ ] T , we define φ ( t ) : = α t x Δ ( θ ) Δ θ and ψ ( t ) : = α t y Δ ( θ ) Δ θ so that φ Δ ( t ) = x Δ ( t ) and ψ Δ ( t ) = y Δ ( t ) .
Thus,
| x ( t ) | = α t x Δ ( θ ) Δ θ α t x Δ ( θ ) Δ θ = φ ( t ) , | y ( t ) | = α t y Δ ( θ ) Δ θ α t y Δ ( θ ) Δ θ = ψ ( t ) .
Then, we obtain
| x ( t ) | | r ( t ) | α t r Δ ( θ ) φ Δ ( θ ) Δ θ r Δ ( θ ) α t r Δ ( θ ) Δ θ ,
| y ( t ) | | s ( t ) | α t s Δ ( θ ) ψ Δ ( θ ) Δ θ s Δ ( θ ) α t s Δ ( θ ) Δ θ .
Thus, from Lemma 2, we get
χ | x ( t ) | r ( t ) 1 r ( t ) α t r Δ ( θ ) χ φ Δ ( θ ) r Δ ( θ ) Δ θ .
π | y ( t ) | s ( t ) 1 s ( t ) α t s Δ ( θ ) π ψ Δ ( θ ) s Δ ( θ ) Δ θ .
Using the above inequalities, we get
α Γ [ s Δ ( θ ) π | y Δ ( θ ) | s Δ ( θ ) f r ( θ ) χ | x ( θ ) | r ( θ ) g s ( θ ) π | y ( θ ) | s ( θ ) + r Δ ( θ ) χ | x Δ ( θ ) | r Δ ( θ ) f r ( θ ) χ | x ( θ ) | r ( θ ) g s ( θ ) π | y ( θ ) | s ( θ ) ] Δ θ α Γ [ s Δ ( θ ) π ψ Δ ( θ ) s Δ ( θ ) f α t r Δ ( θ ) χ φ Δ ( θ ) r Δ ( θ ) Δ θ g α t s Δ ( θ ) π ψ Δ ( θ ) s Δ ( θ ) Δ θ + r Δ ( θ ) χ φ Δ ( θ ) r Δ ( θ ) f α t r Δ ( θ ) χ φ Δ ( θ ) r Δ ( θ ) Δ θ g α t s Δ ( θ ) π ψ Δ ( θ ) s Δ ( θ ) Δ θ ] Δ θ .
Since f, r, χ , and φ are increasing and t σ ( t ) , we get
f α t r Δ ( θ ) χ φ Δ ( θ ) r Δ ( θ ) Δ θ f σ α t r Δ ( θ ) χ φ Δ ( θ ) r Δ . ( θ ) Δ θ .
Then by substituting in (40), we get
α Γ [ s Δ ( θ ) π | y Δ ( θ ) | s Δ ( θ ) f r ( θ ) χ | x ( θ ) | r ( θ ) g s ( θ ) π | y ( θ ) | s ( θ ) + r Δ ( θ ) χ | x Δ ( θ ) | r Δ ( θ ) f r ( θ ) χ | x ( θ ) | r ( θ ) g s ( θ ) π | y ( θ ) | s ( θ ) ] Δ θ   α Γ [ s Δ ( θ ) π ψ Δ ( θ ) s Δ ( θ ) f σ α t r Δ ( θ ) χ φ Δ ( θ ) r Δ ( θ ) Δ θ g α t s Δ ( θ ) π ψ Δ ( θ ) s Δ ( θ ) Δ θ + r Δ ( θ ) χ φ Δ ( θ ) r Δ ( θ ) f α t r Δ ( θ ) χ φ Δ ( θ ) r Δ ( θ ) Δ θ g α t s Δ ( θ ) π ψ Δ ( θ ) s Δ ( θ ) Δ θ ] Δ θ .
From Lemma 3 for c [ t , σ ( t ) ] , we get
f Δ α t r Δ ( θ ) χ φ Δ ( θ ) r Δ ( θ ) Δ θ = r Δ ( t ) χ φ Δ ( t ) r Δ ( t ) f α c r Δ ( θ ) χ φ Δ ( θ ) r Δ ( θ ) Δ θ r Δ ( t ) χ φ Δ ( t ) r Δ ( t ) f α t r Δ ( θ ) χ φ Δ ( θ ) r Δ ( θ ) Δ θ .
g Δ α t s Δ ( θ ) π ψ Δ ( θ ) s Δ ( θ ) Δ θ = s Δ ( t ) χ ψ Δ ( t ) s Δ ( t ) g α c s Δ ( θ ) χ ψ Δ ( θ ) s Δ ( θ ) Δ θ s Δ ( t ) χ φ Δ ( t ) s Δ ( t ) g α t s Δ ( θ ) π ψ Δ ( θ ) s Δ ( θ ) Δ θ .
Then, by substituting from (42) and (42) in (41), we get
α Γ [ s Δ ( θ ) π | y Δ ( θ ) | s Δ ( θ ) f r ( θ ) χ | x ( θ ) | r ( θ ) g s ( θ ) π | y ( θ ) | s ( θ ) + r Δ ( θ ) χ | x Δ ( θ ) | r Δ ( θ ) f r ( θ ) χ | x ( θ ) | r ( θ ) g s ( θ ) π | y ( θ ) | s ( θ ) ] Δ θ α Γ [ g Δ α t s Δ ( θ ) π ψ Δ ( θ ) s Δ ( θ ) Δ θ f σ α t r Δ ( θ ) χ φ Δ ( θ ) r Δ ( θ ) Δ θ + f Δ α t r Δ ( θ ) χ φ Δ ( θ ) r Δ ( θ ) Δ θ g α t s Δ ( θ ) π ψ Δ ( θ ) s Δ ( θ ) Δ θ ] Δ θ = α Γ [ f α t r Δ ( θ ) χ φ Δ ( θ ) r Δ ( θ ) Δ θ g α t s Δ ( θ ) π ψ Δ ( θ ) s Δ ( θ ) Δ θ ] Δ = f α Γ r Δ ( θ ) χ | x Δ ( θ ) | r Δ ( θ ) Δ θ g α Γ s Δ ( θ ) π | y Δ ( θ ) | s Δ ( θ ) Δ θ .
This gives our claim. □
Remark 13.
Taking T = R , then, by the relation (6), inequality (39) gives Pachpatte inequality in [55]
α Γ [ s ( θ ) π | y ( θ ) | s ( θ ) f r ( θ ) χ | x ( θ ) | r ( θ ) g s ( θ ) π | y ( θ ) | s ( θ ) + r ( θ ) χ | x ( θ ) | r ( θ ) f ( r ( θ ) χ | x ( θ ) | r ( θ ) ) g s ( θ ) π | y ( θ ) | s ( θ ) ] d θ f α Γ r ( θ ) χ | x ( θ ) | r ( θ ) d θ g α Γ s ( θ ) π | y ( θ ) | s ( θ ) d θ .
Corollary 16.
If we take T = h Z in Theorem 7, then, by the relation (6), inequality (39) becomes
h n = α h Γ h 1 [ Δ s ( n h ) π | Δ y ( n h ) | Δ s ( n h ) f r ( n h ) χ | x ( n h ) | r ( n h ) g s ( n h ) π | y ( n h ) | s ( n h ) + Δ r ( n h ) χ | Δ x ( n h ) | Δ r ( n h ) f ( r ( n h ) χ | x ( n h ) | r ( n h ) ) g s ( n h ) π | y ( n h ) | s ( n h ) ] f h n = α h Γ h 1 Δ r ( n h ) χ | Δ x ( n h ) | Δ r ( n h ) g h n = α h Γ h 1 Δ s ( n h ) π | Δ y ( n h ) | Δ s ( n h ) .
Remark 14.
In Corollary 16, if we take h = 1 , then inequality (39) becomes
n = α Γ 1 [ Δ s ( n ) π | Δ y ( n ) | Δ s ( n ) f r ( n ) χ | x ( n ) | r ( n ) g s ( n ) π | y ( n ) | s ( n ) + Δ r ( n ) χ | Δ x ( n ) | Δ r ( n ) f ( r ( n ) χ | x ( n ) | r ( n ) ) g s ( n ) π | y ( n ) | s ( n ) ] f n = α Γ 1 Δ r ( n ) χ | Δ x ( n ) | Δ r ( n ) g n = α Γ 1 Δ s ( n ) π | Δ y ( n ) | Δ s ( n ) .
Corollary 17.
If we take T = q Z ¯ in Theorem 7, then, by the relation (7), inequality (39) becomes
( q 1 ) n = ( log q α ) ( log q Γ ) 1 [ Δ s ( q n ) π | Δ y ( q n ) | Δ s ( q n ) f r ( q n ) χ | x ( q n ) | r ( q n ) g s ( q n ) π | y ( q n ) | s ( q n ) + Δ r ( q n ) χ | Δ x ( q n ) | Δ r ( q n ) f ( r ( q n ) χ | x ( q n ) | r ( q n ) ) g s ( q n ) π | y ( q n ) | s ( q n ) ] q n f ( q 1 ) n = ( log q α ) ( log q Γ ) 1 q n Δ r ( q n ) χ | Δ x ( q n ) | Δ r ( q n ) × g ( q 1 ) n = ( log q α ) ( log q Γ ) 1 q n Δ s ( q n ) π | Δ y ( q n ) | Δ s ( q n ) .
Remark 15.
For s ( t ) = r ( t ) , χ ( t ) = π ( t ) , y ( t ) = x ( t ) and g ( t ) = f ( t ) , the inequality (39) becomes
α Γ r Δ ( θ ) χ | x Δ ( θ ) | r Δ ( θ ) f r ( θ ) χ | x ( θ ) | r ( θ ) f r ( θ ) χ | x ( θ ) | r ( θ ) Δ θ 1 2 [ f α Γ r Δ ( θ ) χ | x Δ ( θ ) | r Δ ( θ ) Δ θ ] 2 .
Corollary 18.
With the assumptions of Theorem 7. Suppose ω 0 , t [ α , Γ ] T and α Γ ω ( θ ) Δ θ = 1 . Assuming Ψ 0 is increasing and convex on [ 0 , ) R . Then, we get
α Γ [ s Δ ( θ ) π | y Δ ( θ ) | s Δ ( θ ) f r ( θ ) χ | x ( θ ) | r ( θ ) g s ( θ ) π | y ( θ ) | s ( θ ) + r Δ ( θ ) χ | x Δ ( θ ) | r Δ ( θ ) f r ( θ ) χ | x ( θ ) | r ( θ ) g s ( θ ) π | y ( θ ) | s ( θ ) ] Δ θ f Ψ 1 α Γ ω ( θ ) Ψ r Δ ( θ ) χ | x Δ ( θ ) | r Δ ( θ ) ω ( θ ) Δ θ g Ψ 1 α Γ ω ( θ ) Ψ s Δ ( θ ) π | y Δ ( θ ) | s Δ ( θ ) ω ( θ ) Δ θ .
Proof. 
Since
α Γ r Δ ( θ ) χ | x Δ ( θ ) | r Δ ( θ ) α Γ r Δ ( θ ) χ | x Δ ( θ ) | r Δ ( θ ) ω ( θ ) ω ( θ ) Δ θ α Γ ω ( θ ) Δ θ
and
α Γ s Δ ( θ ) π | y Δ ( θ ) | s Δ ( θ ) α Γ s Δ ( θ ) π | y Δ ( θ ) | s Δ ( θ ) ω ( θ ) ω ( θ ) Δ θ α Γ ω ( θ ) Δ θ .
From Lemma 2, we have
Ψ α Γ r Δ ( θ ) χ | x Δ ( θ ) | r Δ ( θ ) α Γ ω ( θ ) Ψ r Δ ( θ ) π | x Δ ( θ ) | r Δ ( θ ) ω ( θ ) Δ θ
and
Ψ α Γ s Δ ( θ ) π | y Δ ( θ ) | s Δ ( θ ) α Γ ω ( θ ) Ψ s Δ ( θ ) π | y Δ ( θ ) | s Δ ( θ ) ω ( θ ) Δ θ
and hence
α Γ r Δ ( θ ) χ | x Δ ( θ ) | r Δ ( θ ) Ψ 1 α Γ ω ( θ ) Ψ r Δ ( θ ) π | x Δ ( θ ) | r Δ ( θ ) ω ( θ ) Δ θ
and
α Γ s Δ ( θ ) π | y Δ ( θ ) | s Δ ( θ ) Ψ 1 α Γ ω ( θ ) Ψ s Δ ( θ ) π | y Δ ( θ ) | s Δ ( θ ) ω ( θ ) Δ θ
Using (46) and (47) in (39), we get (45). This completes the proof. □

3. Conclusions

In this paper, with the help of the dynamic Jensen inequality, dynamic Hölder inequality and a simple consequence of Keller’s chain rule on time scales, we generalized a number of Opial-type inequalities to a general time scale. Besides that, in order to illustrate the theorems for each type of inequality applied to various time scales such as R , h Z , q Z ¯ and Z as a sub case of h Z . For future studies, researchers may obtain some different generalizations for dynamic Opial inequality and its companion inequalities by using the results presented in this paper.

Author Contributions

A.A.E.-D. contributed in conceptualization, methodology, resources, validation and original draft preparation. D.B. contributed in investigation, formal analysis, review, editing and funding acquisition. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Opial, Z. Sur une inégalité. Ann. Polon. Math. 1960, 8, 29–32. [Google Scholar] [CrossRef]
  2. Godunova, E.K.; Levin, V.I. An inequality of Maroni. Mat. Zametki 1967, 2, 221–224. [Google Scholar] [CrossRef]
  3. Hwang, T.S.; Yang, G.S. On integral inequalities related to Opial’s inequality. Tamkang J. Math. 1990, 21, 177–183. [Google Scholar]
  4. Hilger, S. Ein maßkettenkalkül mit anwendung auf zentrumsmannigfaltigkeiten. Ph.D. Thesis, Universität Würzburg, Würzburg, Germany, 1988. [Google Scholar]
  5. Bohner, M.; Peterson, A. Dynamic Equations on Time Scales: An Introduction with Applications; Birkhäuser Boston, Inc.: Boston, MA, USA, 2001. [Google Scholar] [CrossRef]
  6. Karpuz, B.; Kaymakçalan, B.; Özkan, U.M. Some multi-dimenstonal Opial-type inequalities on time scales. J. Math. Ineq. 2010, 4, 207–216. [Google Scholar] [CrossRef]
  7. Zhao, C.J.; Cheung, W.S. On Opial-type integral inequalities and applications. Math. Inequal. Appl. 2014, 17, 223–232. [Google Scholar] [CrossRef]
  8. Cheung, W.S. Opial-type inequalities with m functions in n variables. Mathematika 1992, 39, 319–326. [Google Scholar] [CrossRef]
  9. Cheung, W.S. Some generalized Opial-type inequalities. J. Math. Anal. Appl. 1991, 162, 317–321. [Google Scholar] [CrossRef]
  10. Zhao, C.J.; Cheung, W.S. On Opial’s type inequalities for an integral operator with homogeneous kernel. J. Inequalities Appl. 2012, 2012, 123. [Google Scholar] [CrossRef]
  11. Abdeldaim, A.; El-Deeb, A.; Ahmed, R.G. On Retarded Nonlinear Integral Inequalities Of Gronwall and Applications. J. Math. Inequal. 2019, 13, 1023–1038. [Google Scholar] [CrossRef]
  12. El-Deeb, A.; El-Sennary, H.; Agarwal, P. Some opial-type inequalities with higher order delta derivatives on time scales. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales Serie A Matemáticas 2020, 114, 29. [Google Scholar] [CrossRef]
  13. El-Deeb, A.A.; Makharesh, S.D.; Baleanu, D. Dynamic Hilbert-Type Inequalities with Fenchel-Legendre Transform. Symmetry 2020, 12, 582. [Google Scholar] [CrossRef]
  14. El-Deeb, A.A.; Khan, Z.A. Certain new dynamic nonlinear inequalities in two independent variables and applications. Bound. Value Probl. 2020, 2020, 31. [Google Scholar] [CrossRef]
  15. Abdeldaim, A.; El-Deeb, A.A.; Agarwal, P.; El-Sennary, H.A. On some dynamic inequalities of Steffensen type on time scales. Math. Methods Appl. Sci. 2018, 41, 4737–4753. [Google Scholar] [CrossRef]
  16. Agarwal, R.; O’Regan, D.; Saker, S. Dynamic Inequalities on Time Scales; Springer: Cham, Switzerland, 2014; Volume 2014. [Google Scholar]
  17. Akin-Bohner, E.; Bohner, M.; Akin, F. Pachpatte inequalities on time scales. JIPAM. J. Inequal. Pure Appl. Math. 2005, 6, 23. [Google Scholar]
  18. Bohner, M.; Matthews, T. The Grüss inequality on time scales. Commun. Math. Anal. 2007, 3, 1–8. [Google Scholar]
  19. Bohner, M.; Matthews, T. Ostrowski inequalities on time scales. JIPAM J. Inequal. Pure Appl. Math. 2008, 9, 8. [Google Scholar]
  20. Dinu, C. Hermite-Hadamard inequality on time scales. J. Inequal. Appl. 2008, 24. [Google Scholar] [CrossRef]
  21. El-Deeb, A.A. On some generalizations of nonlinear dynamic inequalities on time scales and their applications. Appl. Anal. Discret. Math. 2019, 440–462. [Google Scholar] [CrossRef]
  22. El-Deeb, A.A.; Cheung, W.S. A variety of dynamic inequalities on time scales with retardation. J. Nonlinear Sci. Appl. 2018, 11, 1185–1206. [Google Scholar] [CrossRef]
  23. El-Deeb, A.A.; El-Sennary, H.A.; Khan, Z.A. Some Steffensen-type dynamic inequalities on time scales. Adv. Differ. Equ. 2019, 246. [Google Scholar] [CrossRef]
  24. El-Deeb, A.A.; Elsennary, H.A.; Cheung, W.S. Some reverse Hölder inequalities with Specht’s ratio on time scales. J. Nonlinear Sci. Appl. 2018, 11, 444–455. [Google Scholar] [CrossRef]
  25. El-Deeb, A.A.; Elsennary, H.A.; Nwaeze, E.R. Generalized weighted Ostrowski, trapezoid and Grüss type inequalities on time scales. Fasc. Math. 2018, 123–144. [Google Scholar] [CrossRef]
  26. El-Deeb, A.A.; Xu, H.; Abdeldaim, A.; Wang, G. Some dynamic inequalities on time scales and their applications. Adv. Differ. Equ. 2019, 19. [Google Scholar] [CrossRef]
  27. El-Deeb, A.A. Some Gronwall-Bellman type inequalities on time scales for Volterra-Fredholm dynamic integral equations. J. Egypt. Math. Soc 2018, 26, 1–17. [Google Scholar] [CrossRef]
  28. Hilscher, R. A time scales version of a Wirtinger-type inequality and applications. J. Comput. Appl. Math. 2002, 141, 219–226. [Google Scholar] [CrossRef]
  29. Li, W.N. Some delay integral inequalities on time scales. Comput. Math. Appl. 2010, 59, 1929–1936. [Google Scholar] [CrossRef]
  30. Řehák, P. Hardy inequality on time scales and its application to half-linear dynamic equations. J. Inequal. Appl. 2005, 495–507. [Google Scholar] [CrossRef]
  31. Saker, S.H.; El-Deeb, A.A.; Rezk, H.M.; Agarwal, R.P. On Hilbert’s inequality on time scales. Appl. Anal. Discrete Math. 2017, 11, 399–423. [Google Scholar] [CrossRef]
  32. Tian, Y.; El-Deeb, A.A.; Meng, F. Some nonlinear delay Volterra-Fredholm type dynamic integral inequalities on time scales. Discret. Dyn. Nat. Soc. 2018, 8. [Google Scholar] [CrossRef]
  33. Saker, S.H. Opial’s type inequalities on time scales and some applications. Ann. Polon. Math. 2012, 104, 243–260. [Google Scholar] [CrossRef]
  34. Saker, S.H. Some Opial-type inequalities on time scales. Abstr. Appl. Anal. 2011. [Google Scholar] [CrossRef]
  35. Saker, S.H. New inequalities of Opial’s type on time scales and some of their applications. Discrete Dyn. Nat. Soc. 2012, 23. [Google Scholar] [CrossRef]
  36. Bohner, M.J.; Mahmoud, R.R.; Saker, S.H. Discrete, continuous, delta, nabla, and diamond-alpha Opial inequalities. Math. Inequal. Appl. 2015, 18, 923–940. [Google Scholar] [CrossRef]
  37. Saker, S.H. Some new inequalities of Opial’s type on time scales. Abstr. Appl. Anal. 2012, 14. [Google Scholar] [CrossRef]
  38. Karpuz, B.; Özkan, U.M. Some generalizations for Opial’s inequality involving several functions and their derivatives of arbitrary order on arbitrary time scales. Math. Inequal. Appl. 2011, 14, 79–92. [Google Scholar] [CrossRef]
  39. Li, J.D. Opial-type integral inequalities involving several higher order derivatives. J. Math. Anal. Appl. 1992, 167, 98–110. [Google Scholar]
  40. Lee, C.S. On Some Generalization of Inequalities of Opial, Yang and Shum. Canad. Math. Bull. 1980, 23. [Google Scholar] [CrossRef]
  41. Abdeldaim, A.; El-Deeb, A.A. Some new retarded nonlinear integral inequalities with iterated integrals and their applications in retarded differential equations and integral equations. J. Fract. Calc. Appl. 2014, 5, 9. [Google Scholar] [CrossRef]
  42. Abdeldaim, A.; El-Deeb, A.A. On generalized of certain retarded nonlinear integral inequalities and its applications in retarded integro-differential equations. Appl. Math. Comput. 2015, 256, 375–380. [Google Scholar] [CrossRef]
  43. Abdeldaim, A.; El-Deeb, A.A. On some generalizations of certain retarded nonlinear integral inequalities with iterated integrals and an application in retarded differential equation. J. Egypt. Math. Soc. 2015, 23, 470–475. [Google Scholar] [CrossRef]
  44. Abdeldaim, A.; El-Deeb, A.A. On some new nonlinear retarded integral inequalities with iterated integrals and their applications in integro-differential equations. Br. J. Math. Comput. Sci. 2015, 5, 479–491. [Google Scholar] [CrossRef]
  45. Agarwal, R.P.; Lakshmikantham, V. Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations; Volume 6 of Series in Real Analysis; World Scientific Publishing: Singapore, 1993. [Google Scholar]
  46. El-Deeb, A.A. On Integral Inequalities and Their Applications Lambert; LAP Lambert Academic Publishing: Saarbrücken, Germany, 2017. [Google Scholar]
  47. El-Deeb, A.A. A Variety of Nonlinear Retarded Integral Inequalities of Gronwall Type and Their Applications. In Advances in Mathematical Inequalities and Applications; Springer: Berlin, Germany, 2018; pp. 143–164. [Google Scholar]
  48. El-Deeb, A.A.; Ahmed, R.G. On some explicit bounds on certain retarded nonlinear integral inequalities with applications. Adv. Inequalities Appl. 2016, 2016, 19. [Google Scholar]
  49. El-Deeb, A.A.; Ahmed, R.G. On some generalizations of certain nonlinear retarded integral inequalities for Volterra-Fredholm integral equations and their applications in delay differential equations. J. Egypt. Math. Soc. 2017, 25, 279–285. [Google Scholar] [CrossRef]
  50. El-Owaidy, H.; Abdeldaim, A.; El-Deeb, A.A. On some new retarded nonlinear integral inequalities and their applications. Math. Sci. Lett. 2014, 3, 157. [Google Scholar] [CrossRef]
  51. El-Owaidy, H.M.; Ragab, A.A.; Eldeeb, A.A.; Abuelela, W.M.K. On some new nonlinear integral inequalities of Gronwall-Bellman type. Kyungpook Math. J. 2014, 54, 555–575. [Google Scholar] [CrossRef]
  52. El-Deeb, A.A.; Kh, F.M.; Ismail, G.A.F.; Khan, Z.A. Weighted dynamic inequalities of Opial-type on time scales. Adv. Differ. Eq. 2019, 2019, 393. [Google Scholar] [CrossRef]
  53. Kh, F.M.; El-Deeb, A.A.; Abdeldaim, A.; Khan, Z.A. On some generalizations of dynamic Opial-type inequalities on time scales. Adv. Differ. Eq. 2019, 2019, 1–14. [Google Scholar] [CrossRef]
  54. Pachpatte, B.G. On integral inequalities similar to Opial’s inequality. Demonstr. Math. 1989, 22, 21–27. [Google Scholar] [CrossRef]
  55. Pachpatte, B.G. A note on generalized Opial type inequalities. Tamkang J. Math. 1993, 24, 229–235. [Google Scholar]
  56. Hua, L.G. On an inequality of Opial. Sci. Sin. 1965, 14, 789–790. [Google Scholar]
  57. Yang, G.S. On a certain result of Z. Opial. Proc. Jpn. Acad. 1966, 42, 78–83. [Google Scholar] [CrossRef]
  58. Maroni, P. Sur l’inégalité d’Opial-Beesack. C. R. Acad. Sci. Paris Sér. A-B 1967, 264, A62–A64. [Google Scholar]
  59. Yang, G.S. A note on some integro-differential inequalities. Soochow J. Math. 1983, 9, 231–236. [Google Scholar]
Back to TopTop