View of Multivariate Opial-type Inequalities on Time Scales

https://doi.org/10.34198/ejms.12123.1326

Multivariate Opial-type Inequalities on Time Scales

Yisa Oluwatoyin Anthonio¹, Kamilu Rauf^2,*, Abdullai Ayinla Abdurasid³ and Oluwaseun Raphael Aderele⁴

1Department of Mathematics and Statistics, Lagos State University of Science and Technology, Ikorodu, Lagos, Nigeria; e-mail: anthonioii@yahoo.com

2 Department of Mathematics, University of Ilorin, Ilorin, Nigeria;

e-mail: krauf@unilorin.edu.ng

3Department of Mathematics and Statistics, Lagos State University of Science and Technology, Ikorodu, Lagos, Nigeria; e-mail: abdulrasid.abdullai@yahoo.com

4Department of Mathematics and Statistics, Lagos State University of Science and Technology, Ikorodu, Lagos, Nigeria; e-mail: seunaderaph@yahoo.com

Abstract

Opial inequality was developed to provide bounds for integral of functions and their derivatives. It has become an indispensable tool in the theory of mathematical analysis due to its usefulness. A refined Jensen inequality for multivariate functions is employed to establish new Opial-type inequalities for convex functions of several variables on time scale.

1 Introduction

Opial [11] discussed problems involving functions and their derivatives. His work motivated many researchers to obtain a general version of the results and several methods have been used to extend the inequality in the following directions:

Received: September 12, 2022; Revised & Accepted: October 26, 2022; Published: January 27, 2023 2020 Mathematics Subject Classification: 15A39, 97H30.

Keywords and phrases: Opial-type inequalities, convex functions, refined Jensen inequality, time scale.

Godunova and Levin [6] extended and provided sharper inequality compare to [11]. Rozanova [14] adopted convex function to improve [6] by using absolutely continuous function. In [12], Qi refined the results in [11] and obtained a more generalized Opial-type inequalities. In [13], Rauf and Anthonio generalized [5]

through convexity and reiterated that the absolute value on both sides of the inequalities therein are not necessary. Bohner and Kaymakcalan [4] presented a version of Opial inequality for time scales and pointed out some of its applications to so-called dynamic equations. Such dynamic equations contain both differential and difference equations as special cases. Various extensions of their inequality were provided as well. Saker [15] proved some new Opial dynamic inequalities involving higher order derivatives on time scales. The results were proved by using Holder’s inequality, a simple consequence of Keller’s chain rule and Taylor monomials on time scales. Some continuous and discrete inequalities were used to derive special cases in their results. Some weighted generalization of Opial type inequalities in two independent variables for two functions was established in [9].

They also obtain weighted Opial-type inequalities by usingp-norms.

An interval, in the time-scale context, is always understood as the intersection of a real interval with a given time-scale. We shall write the delta derivativesf^∆for a function f defined onTand it turns out that:

(i) f^∆=f⁰ is the usual derivative ifT=R; and

(ii) f^∆= ∆f is the usual forward difference operator if T=Z.

1.1 Some Basic Definitions

The derivative of a functionf :T−→R denoted by f^∆ is as follows: Let t∈T, if a numberα∈Rsuch that for all >0 there exists a neighborhoodU oftwith

|f(σ(t))−f(s)−α(σ(t)−s)| ≤|σ(t)−s)|. (1.1)

The fundamentals of time scales calculus, Opial inequality and its applications can be sourced from [1], [2], [3], [4], [6], [7], [8], [9], [10], [11] [13] & [16].

Throughout the work, refined Jensen inequality of the form (1.2) is employed Z t

f(x(t))dλ(s) ς

≤ Z t

dλ(s)

^ς−ζZ t

ϕ(f(x(t)))^1ζdλ(s) ζ

. (1.2)

2 Some Results on Opial-Type Inequalities

In this section, we shall discuss Opial-type inequalities on time scale. We begin with the following theorem.

Theorem 2.1. Let T be a time scale with x, y, t ∈ T. Suppose ς and ζ are real numbers, x, y, t ∈C_rd([0, t]_T,R) where φ(y) is positive rd-continuous function on [0, t]_T such that φ(y)^ζ1 ≤ φ(y). Let φ(x) be a convex and increasing function on [0,∞) with φ(0) = 0 and t(x) be absolutely continuous on [0, t], x(0) = 0 with Lebesque-Stieltjes integrable function with respect to g(x). Then, it follows that

Z t 0

∆g(x)φ⁰(t(x))t⁰(x)≤t

1−ζ ζ

Z t 0

∆g(x)t⁰(x) 1+ζ

. (2.1)

Proof. By Jensen inequality, we let ϕ(t) =t^ζ in (1.2) and have

ϕ⁻¹ Z t

0

∆g(y)φ(y)

≤t

1−ζ ζ

Z t 0

∆g(y)ϕ⁻¹(φ(y))

φ⁰(y)≤t

ζ−1 ζ ϕ

Z t 0

∆g(y)φ(y)^ζ1

that is

φ⁰(y)≤t

ζ−1 ζ ϕ

Z t 0

∆g(y)φ(y)

. (2.2)

Assuming

y(x) = Z t

0

∆g(y)t⁰(x) which can be written as

t⁰(x) =y⁰(x)^ζ. (2.3)

Combining (2.2) and (2.3) and also integrate with respect to g(x) yields Z t

0

∆g(x)φ⁰(t(x))t⁰(x)≤t

1−ζ ζ

Z t 0

∆g(x)φ⁰(y(t))y⁰(t)^ζ=t

ζ−1

ζ (y(t))^1+ζ

=t

1−ζ ζ

Z t 0

∆g(x)t⁰(x) ζ+1

(2.4) which is Godunova and Levin result whenT=R.

Remark 2.2. IfRt

0∆g(y)φ(y)^1ζ ≤Rt

0∆g(y)φ(y), then φ⁰(y)≤t

1−ζ ζ ϕ

Z t 0

∆g(y)φ(y)

(2.5)

(2.5) is a direct consequence of (2.2). Further simplification yields Z t

0

∆g(x)φ⁰(t(x))t⁰(x)≤t

1−ζ ζ

Z t 0

∆g(x)φ⁰(y(t))y⁰(t) =t

1−ζ ζ (y(t))^ζ

=t

1−ζ ζ

Z t 0

∆g(x)t⁰(x) ζ

. (2.6)

However, if we chooset so that y(t) =

Z t a

∆g(x)t⁰(x) = Z Ω

t

∆g(x)t⁰(x) = 1 2

Z Ω 0

∆g(x)t⁰(x) ∀ t∈[0,Ω].

(2.7) Combination of (2.6) and (2.7) gives

Z _Ω

0

∆g(x)φ⁰(t(x))t⁰(x)≤t

ζ−1 ζ

1 2

Z _Ω

0

∆g(x)t(x) ^ζ

. (2.8)

Theorem 2.3. Let T be a time scale with x, y, t ∈ T. If ζ is a real number, x, y, t ∈ C_rd([0, t]_T,R) where φ(y) is positive rd-continuous function on [0, t]_T. Supposeφ(t)andΦ(t)are convex and increasing functions on[0,∞)withφ(0) = 0 and t(x) is absolutely continuous on [0, t]satisfying t(0) =t(x) = 0. Then,

Z Ω 0

∆g(x)φ⁰(t(x))t⁰(x)≤2





"

Z Ω 0

∆g(x)µ(x)

#^ζ−1_ζ Z Ω

0

∆g(x) t⁰(x)

2µ(x) ^ζ

µ(x)

!¹_ζ



ζ

.

Proof. From (1.2) we have







 _RΩ

0 ∆g(x)t⁰(x) 2µ(x)

µ(x) RΩ

0 ∆g(x)µ(x)









ζ

≤ RΩ

0 ∆g(x)_t0(x) 2µ(x)

ζ

µ(x) RΩ

0 ∆g(x)µ(x) , (2.9)

RΩ

0 t⁰(x)∆g(x) 2µ(x)

! µ(x)

!ζ

≤ Z Ω

0

∆g(x)µ(x)

^ζ−1Z Ω 0

∆g(x)

t⁰(x) 2µ(x)

ζ

µ(x), (2.10) 1

2 Z Ω

0

∆g(x)t⁰(x)≤ Z Ω

0

∆g(x)µ(x)

ζ−1

ζ Z Ω

0

∆g(x)

t⁰(x) 2µ(x)

ζ

µ(x)

!¹_ζ

. (2.11) By combining (2.8) and (2.11) yields

Z Ω 0

∆g(x)φ⁰(t(x))t⁰(x)≤2





"

Z Ω 0

∆g(x)µ(x)

#^ζ−1_ζ Z Ω

0

∆g(x) t⁰(x)

2µ(x) ζ

µ(x)

!¹_ζ



ζ

. (2.12)

Theorem 2.4. Let T be a time scale with x, y, t ∈ T. Suppose ς and ζ are real numbers, x, y, t ∈ C_rd([0, x]_T,R) where y and g(y) are positive rd-continuous functions on [0,Ω]_T such that R

[0,t]∆g(s)φ(s) < ∞. If φ(t) is a convex and increasing function on [0,∞) with φ(0) = 0, t₁(x) and t₂(x) are absolutely continuous on [0, x] with t(0) = 0. Suppose φ1(t) and φ2(t) are continuous differentiable functions defined on [0, t] and also are positive convex on [0,∞) Then, the inequality below follows:

Z t 0

∆g(x) φ₁(t₁(x))φ⁰₂(t₂(x))t⁰₂(x) +φ₂(t₂(x))φ⁰₁(t₁(x))t⁰₁(x)

≤



 Z t

0

∆g(x)µ(x)

ζ−1

ζ Z t

0

∆g(x)

t⁰₁(x) µ(x)

ζ

µ(x)

!¹_ζ



ζ

×



 Z t

0

∆g(x)µ(x)

^ζ−1_ζ Z t 0

∆g(x)

t⁰₂(x) µ(x)

ζ

µ(x)

!¹

ζ





ζ

.

(2.13)

Proof. The proof is similar to the latter Theorem.

By applying (1.2), it gives

φ1(t1(x))φ⁰₂(t2(x))≤t

ζ−1 ζ

Z t 0

∆g(t)φ(y1)^{ζ 1}_ζ

(2.14) and

φ₂(t₂(x))φ⁰₁(t₁(x))≤t

ζ−1 ζ

Z t 0

∆g(t)φ(y₂)^{ζ 1}_ζ

. (2.15)

Taking addition of (2.14) and (2.15) and integrate over [0, t] with respect tog(x) yields

Z t 0

∆g(x) φ₁(t₁(x))φ⁰₂(t₂(x))t⁰₂(x) +φ₂(t₂(x))φ⁰₁(t₁(x))t⁰₁(x)

≤



 Z t

0

∆g(x)µ(x)

ζ−1

ζ Z t

0

∆g(x)

t⁰₁(x) µ(x)

ζ

µ(x)

!¹_ζ



ζ

×



 Z t

0

∆g(x)µ(x)

^ζ−1_ζ Z t 0

∆g(x)

t⁰₂(x) µ(x)

ζ

µ(x)

!¹_ζ



ζ

.

(2.16)

Ift1(x) =t1(x) =t(x), φ2((x)) =φ1(x) andζ = 1, (2.16) reduces to the following interesting inequality

Z t 0

∆g(x) φ1(t1(x))φ⁰₂(t2(x))t⁰₂(x) +φ2(t2(x))φ⁰₁(t1(x))t⁰₁(x)

≤ Z t

0

∆g(x)

t⁰₁(x) µ(x)

µ(x)

2

. (2.17)

Furthermore, let φ(y) and Φ(y) be convex functions on [0,∞), φ(0) = Φ(0) = 0, φ⁰(y)≤φ(y) and by applying (2.8) we have,

φ(t(x))Φ(t⁰(x))≤t

ζ−1 ζ

Z t 0

φ(y)^ζ∆g(t) ¹_ζ

Φ(t⁰(x)). (2.18) Integrate (2.18) with respect tog(x) yields

Z Ω 0

∆g(x)φ(t(x))Φ(t⁰(x))≤t

ζ−1 ζ

Z Ω 0

∆g(x) Z t

0

∆g(x)φ(t(x))^{ζ 1}_ζ

Φ(t⁰(x)).

(2.19)

Assuming

Φ Z t

0

∆g(x)t⁰(x)

=t Φ Rt

0∆g(x)t⁰(x) Rt

0 ∆g(x)

!!

which is Z t

0

∆g(x)t⁰(x)

≤ Z t

0

∆g(x)t⁰(x)^ζ

=tΦ⁻¹ Φ Rt

0∆g(x)t⁰(x)^ζ Rt

0 ∆g(x)

!!

≤tΦ⁻¹ 1

t Z t

0

∆g(x)Φ(t⁰(x))^ζ

.

(2.20)

In view of (2.18) and (2.20) we have Z Ω

0

∆g(x)φ(t(x))Φ(t⁰(x))≤t

ζ−1 ζ +1Z Ω

0

Φ⁻¹ 1

t Z t

0

∆g(x)Φ(t⁰(x))^ζ _ζ¹!!

×Φ(t⁰(x))∆g(x) if a= 1

t Z t

0

Φ(t⁰(x))∆g(x) and b = Φ(t⁰(x)) in latter inequality yields Z Ω

0

∆g(x)φ(t(x))Φ(t⁰(x))≤t

ζ−1 ζ +1Z a

0

∆g(t)φ tΦ⁻¹(t)

= Z _Ω

0

Z _b

0

∆g(t)∆g(x)φ tΦ⁻¹(t)

(2.21) which is Qi’s result whenT=R.

3 Refined Sub-variate and Multi-variate Opial-type Inequalities on Time Scales

We present our main results on Jensen inequality for functions of several variables on time scale as follows:

Theorem 3.1. Let T be a time scale with x, y, t ∈T. Let ζ be real numbers, let x, y, t ∈ C_rd([0, t]_T,R) where y and φ(y) are positive rd-continuous functions on [0, t]_T. Lett(x) be absolutely continuous function which is non-decreasing on[0, x]

and let t_i, µ_i≥0∀i= 1,2 and ϕ(x) be convex function, µ(x) be Lebesque-Stieltjes integrable function with respect to g(x). Then the following inequality holds:

Z x 0

∆g(s)

t⁰₁(x) +t⁰₂(x) (µ⁰₁(x) +µ⁰₂(x))

ζ

(µ⁰₁(x) +µ⁰₂(x))ϕ⁰

t₁(x) +t₂(x)

(µ1(x) +µ2(x))(µ1(x) +µ2(x))

≤ϕ Z x

0

∆g(s)(µ⁰₁(x) +µ⁰₂(x))

y₁⁰(s) +y⁰₂(s) µ⁰₁(x) +µ⁰₂(x)

^ζ! .

Proof. From (1.2), we have



 Rx

0 ∆g(s)_µ^t010^(s)+t02(s)

1(s)+µ⁰₂(s)(µ⁰₁(s) +µ⁰₂(s)) Rx

0 ∆g(s)µ⁰₁(s) +µ⁰₂(s)





ζ

≤ 1

µ1(x) +µ2(x) Z x

0

∆g(s)(µ⁰₁(s) +µ⁰₂(s))

y₁⁰(s) +y⁰₂(s) µ⁰₁(s) +µ⁰₂(s)

ζ

.

(3.1)

(3.1) can be written as Rx

0 ∆g(s)^(t

0

1(s)+t⁰₂(s))

(µ⁰₁(s)+µ⁰₂(s))(µ⁰₁(s) +µ⁰₂(s)) ζ

Rx

0 ∆g(s)(µ⁰₁(s) +µ⁰₂(s))ζ

≤ 1

µ1(x) +µ2(x) Z x

0

∆g(s)(µ⁰₁(s) +µ⁰₂(s))

y₁⁰(s) +y⁰₂(s) µ⁰₁(s) +µ⁰₂(s)

ζ

= Rx

0 ∆g(s)(µ⁰₁(s) +µ⁰₂(s))ζ

µ₁(x) +µ₂(x)

Z _x

0

∆g(s)(µ⁰₁(s) +µ⁰₂(s))

y₁⁰(s) +y₂⁰(s) µ⁰₁(s) +µ⁰₂(s)

ζ

= (µ₁(s) +µ₂(s))^ζ µ₁(x) +µ₂(x)

Z x 0

∆g(s)(µ⁰₁(s) +µ⁰₂(s))

y⁰₁(s) +y₂⁰(s) µ⁰₁(s) +µ⁰₂(s)

ζ

≤ (µ₁(s) +µ₂(s)) µ1(x) +µ2(x)

Z x 0

∆g(s)(µ⁰₁(s) +µ⁰₂(s))

y₁⁰(s) +y⁰₂(s) µ⁰₁(s) +µ⁰₂(s)

ζ

(3.2) which can be expressed in the following form with the fact that y(t) = Rx

0 µ(x)dg(x) Z x

0

∆g(s) t⁰₁(s) +t⁰₂(s)

(µ⁰₁(s) +µ⁰₂(s))(µ⁰₁(s) +µ⁰₂(s))≤ Z x

0

∆g(s)(µ⁰₁(s)+µ⁰₂(s))ϕ

y₁⁰(s) +y₂⁰(s) (µ⁰₁(s) +µ⁰₂(s))

(3.3)

further estimation of the inequality yields

Z x 0

∆g(s)

t⁰₁(x) +t⁰₂(x) (µ⁰₁(x) +µ⁰₂(x))

ζ

(µ⁰₁(x) +µ⁰₂(x))ϕ⁰

t₁(x) +t₂(x)

(µ1(x) +µ2(x))(µ₁(x) +µ₂(x))

≤ Z x

0

∆g(s) µ⁰₁(x) +µ⁰₂(x)

y₁⁰(x) +y₂⁰(x) µ⁰₁(x) +µ⁰₂(x)

ζ

×ϕ⁰ Z x

0

∆g(s) µ⁰₁(s) +µ⁰₂(s)

y⁰₁(s) +y₂⁰(s) µ⁰₁(s) +µ⁰₂(s)

ζ!

= Z x

0

∆g(s)ϕ Z x

0

∆g(s) µ⁰₁(s) +µ⁰₂(s)

y⁰₁(s) +y₂⁰(s) µ⁰₁(s) +µ⁰₂(s)

ζ!0

=ϕ Z x

0

∆g(x) µ⁰₁(x) +µ⁰₂(x)

y₁⁰(x) +y⁰₂(x) µ⁰₁(x) +µ⁰₂(x)

ζ!

(3.4) that is

Z x 0

∆g(x)

t⁰₁(x) +t⁰₂(x) (µ⁰₁(x) +µ⁰₂(x))

ζ

(µ⁰₁(x) +µ⁰₂(x))ϕ⁰

t1(x) +t2(x)

(µ1(x) +µ2(x))(µ1(x) +µ2(x))

≤ϕ Z x

0

∆g(x) µ⁰₁(x) +µ⁰₂(x)

y₁⁰(s) +y⁰₂(s) µ⁰₁(x) +µ⁰₂(x)

ζ!

(3.5) which is Rozanova’s result. In general, the above can be extended to n^th term as

Z x 0

∆g(x)

t⁰₁(x) +...+t⁰_n(x) (µ⁰₁(x) +...+µ⁰_n(x))

ζ

(µ⁰₁(x) +· · ·+µ⁰₂(x))

×ϕ⁰

t₁(x) +· · ·+t_n(x)

(µ1(x) +· · ·+µn(x))(µ₁(x) +· · ·+µ_n(x))

≤ϕ Z x

0

∆g(x)µ⁰₁(x) +· · ·+µ⁰_n(x)

µ⁰₁(x) +· · ·+µ⁰_n(x) µ⁰₁(x) +· · ·+µ⁰_n(x)

ζ!

(3.6)

which implies Z _x

0

∆g(x)

n

Y

i=1

Pn

i=1t⁰_n(x) Pn

i=1µ⁰_n(x)

ζ n

X

i=1

µ⁰_n(x)

!

×ϕ⁰

Pn

i=1tn(x) (Pn

i=1µ_n(x))(

n

X

i=1

µn(x))

!

≤ϕ Z x

0

∆g(x)

n

X

i=1

µ⁰_n(x)

! Pn

i=1y_n⁰(x) P_n

i=1µ⁰_n(x) ζ!

.

(3.7)

Theorem 3.2. Let T be a time scale withv, µ, x, y, t∈T. Let ζ be real numbers, let x, y, t ∈ C_rd([0, t]_T,R) where y and φ(y) are positive rd-continuous functions on [0, t]_T. Let ϕi(t) = t^ζi are non-negative, convex and increasing functions on (0,∞). If t_i(x) are absolutely continuous defined on [0,Ω], satisfying t_i(0) = 0

∀i∈N. Then,

Z Ω 0

∆g(x)µ⁰₂(s)

t⁰₂(s) µ⁰₂(s)

ζ2

v₁(s) µ₁(s)

t₁(s) µ₁(s)

ζ1!

v₂⁰(s) µ₂(s)

t₂(s) µ₂(s)

ζ2!

+ Z Ω

0

∆g(x)µ⁰₁(s)

t⁰₁(s) µ⁰₁(s)

ζ1

v₂(s) µ₂(s)

t₂(s) µ₂(s)

ζ2!

v⁰₁(s) µ₁(s)

t₁(s) µ₁(s)

ζ1!

+ Z Ω

0

∆g(x)µ⁰₀(s)

t⁰₀(s) µ⁰₀(s)

ζ0

v₃(s) µ₃(s)

t₃(s) µ3(s)

ζ2!

v⁰₀(s) µ₀(s)

t₀(s) µ0(s)

ζ1!

+· · ·+ Z Ω

0

∆g(x)µ⁰_3−n(s)

t⁰_3−n(s) µ⁰_3−n(s)

ζ3−n

v_n(s) µ_n(s)

t_n(s) µn(s)

ζn!

×v⁰_3−n(s) µ3−n(s)

t₀(s) µ3−n(s)

ζ3−n!

≤v₁ Z Ω

0

∆g(s)µ⁰₁(s)

t⁰₁(t) µ⁰₁(s)

ζ1!

×v₂ Z Ω

0

µ⁰₂(s)∆g(x)

t⁰₂(s) µ⁰₂(s)

ζ2!

×v3

Z Ω 0

∆g(s)µ⁰₃(s)

t⁰₃(t) µ⁰₃(s)

ζ3!

× · · · ×vn

Z Ω 0

vµ⁰_n

t⁰_n(s) µ⁰_n(s)

ζn! .

(3.8)

Proof. The proof is similar to the proof of Theorem 3.1 by combining (1.2) and (3.1), it follows that



 Rx

0 ∆g(s)^t

0 1(s) µ⁰₁(s)µ⁰₁(s) Rx

0 ∆g(s)µ⁰₁(s)





ζ

≤ 1 µ₁(x)

Z x 0

∆g(s)µ⁰₁(s)

y⁰₁(s) µ⁰₁(s)

ζ

. (3.9) By simplification, (3.9) yields

Z Ω 0

∆g(x)µ⁰₂(s) t⁰₂(s)

µ⁰₂(s) ζ₂

v1(s) µ1(s) t1(s)

µ1(s) ζ₁!

v⁰₂(s) µ2(s) t2(s)

µ2(s) ζ₂!

+ Z Ω

0

∆g(x)µ⁰₁(s) t⁰₁(s)

µ⁰₁(s) ζ₁

v2(s) µ2(s) t2(s)

µ2(s) ζ₂!

v₁⁰(s) µ1(s) t1(s)

µ1(s) ζ₁!

+ Z Ω

0

∆g(x)µ⁰₀(s) t⁰₀(s)

µ⁰₀(s) ζ0

v3(s) µ3(s) t3(s)

µ3(s) ζ₂!

v₀⁰(s) µ0(s) t0(s)

µ0(s) ζ₁!

+· · ·+ Z_Ω

0

∆g(x)µ⁰_3−n(s) t⁰_3−n(s) µ⁰_3−n(s)

!ζ3−n

vn(s) µn(s) tn(s)

µn(s) ζ_n!

×v3−n⁰ (s) µ3−n(s) t0(s)

µ3−n(s) ζ_3−n!

≤ ZΩ

0

∆g(x)µ⁰₂(s) y⁰₂(s)

µ⁰₂(s) ζ2

v1(s) Z t

0

∆g(s)µ⁰₁(s) y⁰₁(s)

µ⁰₁(s) ζ1!

v₂⁰(s) Zt

0

∆g(s)µ2(s) y⁰₂(s)

µ⁰₂(s) ζ2!

+ Z Ω

0

∆g(s)µ⁰₁(s) y₁⁰(s)

µ⁰₁(s) ζ1

v2(s) Z t

0

∆g(s)µ⁰₂(s) y⁰₂(s)

µ⁰₂(s) ζ2!

v⁰₁(s) Zt

0

∆g(s)µ1(s) y₁⁰(s)

µ⁰₁(s) ζ1!

+ Z Ω

0

∆g(x)µ⁰₀(s) y⁰₀(s)

µ⁰₀(s) ζ₀

v3(s) Zt

0

dg(s)µ⁰₃(s) y⁰₃(s)

µ⁰₃(s) ζ₃!

v⁰₀(s) Z t

0

dg(s)µ0(s) y₀⁰(s)

µ⁰₀(s) ζ₀!

+· · ·+ ZΩ

0

dg(x)µ⁰3−n(s) y_3−n⁰ (s) µ⁰_3−n(s)

!ζ3−n

vn(s) Z t

0

dg(s)µ⁰_n(s) y_n⁰(s)

µ⁰_n(s) ζ_n!

×v_3−n⁰ (s)



 Z t

0

dg(s)µ3−n(s) y⁰_3−n(s) µ⁰_3−n(s)

!ζ_3−n



= ZΩ

0

∆g(x)

"

v1 µ⁰₁(s) Z x

0

∆g(s) y⁰₁(s)

µ⁰₁(s) ζ1!

×v2 µ⁰₂(s) Z x

0

∆g(s) y₂⁰(s)

µ⁰₂(s) ζ2!

×v3 µ⁰₃(s) Z x

0

∆g(s) y⁰₃(s)

µ⁰₃(s) ζ3!

× · · ·.×vn µ⁰_n(s) Z x

0

∆g(s) y_n⁰(s)

µ⁰_n(s) ζ_n!#0

=v1

Z Ω 0

∆g(x)µ⁰₁(s) t⁰₁(t)

µ⁰₁(s) ζ1!

×v2

Z Ω 0

∆g(x)µ⁰₂(s) t⁰₂(s)

µ⁰₂(s) ζ2!

×v3

Z Ω 0

∆g(x)µ⁰₃(s) t⁰₃(t)

µ⁰₁ ζ₃!

× · · ·.×vn

Z Ω 0

∆g(x)µ⁰_n t⁰_n(s)

µ⁰_n(s) ζn!

.

(3.10)

In conclusion, if R = T, µ ≥2, µ₁(x) = µ₂(x) = µ(x), ϕ₁(x) = ϕ₂(x) = ϕ(x), t1(x) =t2(x) =t(x) and v1(x) =v2(x) =v(x) = 1, then (3.10) yields

Z Ω 0

∆g(x)µ⁰₂(s)

t⁰₂(s) µ⁰₂(s)

ζ2

v⁰₂(s) µ₂(s)

t₂(s) µ₂(s)

ζ2! µ₁(s)

t₁(s) µ₁(s)

ζ1!!

+ Z Ω

0

∆g(x)µ⁰₁(s)

t⁰₁(s) µ⁰₁(s)

ζ1

v⁰₁(s) µ₁(s)

t₁(s) µ₁(s)

ζ1! µ₂(s)

t₂(s) µ₂(s)

ζ2!

≤v₁ Z Ω

0

∆g(x)µ⁰₁(s)

t⁰₁(t) µ⁰₁(s)

ζ1! v₂

Z Ω 0

∆g(x)µ⁰₂(s)

t⁰₂(s) µ⁰₂(s)

ζ2! .

(3.11) Also if R=T and v1(x) =v2(x) =v(x) in (3.11), we obtain

Z Ω 0

∆g(x)µ⁰(s) t⁰(s)

µ⁰(s) ζ

v⁰(s) µ(s) t(s)

µ(s) ζ!

µ(s) t(s)

µ(s) ζ!

≤ 1 2

v

Z Ω 0

∆g(x)µ⁰₁(s) t⁰(s)

µ⁰(s) 2

.

(3.12)

Remark 3.3. Inequality (1.2) was employed to obtain the generalized Opial-type inequalities for convex functions of several variables on time scales.

Conflicts of Interest

The authors declare that they have no competing interests.

Acknowledgments

We thank the referees for their insightful and constructive recommendations and comments, which enhanced the quality of the paper.

References

[1] R. Agarwal, D. O’Regan and S. Saker, Dynamic Inequalities on Time Scales, Springer International Publishing Switzerland, 2014.

[2] M. Anwar, R. Bibi, M. Bohner and J. Pe˘cari´c, Jensen functionals on time scales for several variables, Int. J. Anal.2014, Art. ID 126797, 14 pp.

https://doi.org/10.1155/2014/126797

[3] P. R. Beesack, Elementary proofs of some Opial-type integral inequalities, J. Anal.

Math. 36 (1979), 1-14.https://doi.org/10.1007/bf02798763

[4] M. Bohner and B. Kaymakcalan, Opial inequalities on time scales, Ann. Polon.

Math. 77(1) (2001), 11-20.https://doi.org/10.4064/ap77-1-2

[5] J. Calvert, Some generalizations of Opial’s inequality, Proc. Amer. Math. Soc. 18 (1967), 72-75.https://doi.org/10.1090/s0002-9939-1967-0204594-1

[6] E. K. Godunova and V. I. Levin, O niekotoryh integralnyh nerabenstvah, soderzascih proizvodnye (Russian),Izv. vuzov, Matem.91 (1969), 20-24.

[7] S. Hilger,Ein Makettenkalkl mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D.

Thesis, Universtat Wurzburg, 1988.

[8] S. Hilger, Differential and difference calculus—unified!, Proceedings of the Second World Congress of Nonlinear Analysts, Part 5 (Athens, 1996),Nonlinear Anal.30(5) (1997), 2683-2694.https://doi.org/10.1016/s0362-546x(96)00204-0

[9] H¨useyin Budak, Mehmet Zeki Sarikaya and Artion Kashuri, On weighted generalization of Opial type inequalities in two variables, Korean J. Math. 28(4) (2020), 717-737.

[10] Josip Pe˘cari´c and Ilko Brneti´c, Note on generalization of Godunova Levin-Opial Inequality, Demonstratio Math.30(3) (1997), 545-549.

https://doi.org/10.1515/dema-1997-0310

[11] Z. Opial, Sur une in´egalit´e,Ann. Polon. Math. 8 (1960), 29-32.

https://doi.org/10.4064/ap-8-1-29-32

[12] Z. Qi, Further generalization of Opial’s inequality, Acta Math. Sinica (N.S.) 1(3) (1985), 196-200.

[13] K. Rauf and Y. O. Anthonio, Results on an integral inequality of the Opial-type, Glob. J. Pure Appl. Sci. 23 (2017), 151-156.

https://doi.org/10.4314/gjpas.v23i1.15

[14] G. I. Rozanova, Integral inequalities with derivatives and with arbitrary convex functions (Russian), Moskov. Gos. Ped. Inst. Vcen Zap. 460 (1972), 58-65.

[15] S. H. Saker, Some Opial-type inequalities on time scales. Abstr. Appl. Anal. 2011 (2011), Article ID 265316, 19 pp.https://doi.org/10.1155/2011/265316 [16] Chang-Jian Zhao and Wing-Sum Cheung, On some Opial-type inequalities, J.

Inequal. Appl. 2011 (2011), 7.https://doi.org/10.1186/1029-242x-2011-7

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted, use, distribution and reproduction in any medium, or format for any purpose, even commercially provided the work is properly cited.