
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)13277-X
10.1016/j.heliyon.2024.e37246
e37246
Research Article
Advancements in Bullen-type inequalities via fractional integral operators and their applications
Samraiz Muhammad muhammad.samraiz@uos.edu.pk
a
Hassan Zohaib hassanzohaib127@gmail.com
a
Naheed Saima saima.naheed@uos.edu.pk
a
Vivas-Cortez Miguel mjvivas@puce.edu.ec
b⁎
Ali Rifaqat rrafat@kku.edu.sa
c
Lamoudan Tarik lamoudan@kku.edu.sa
c
a Department of Mathematics, University of Sargodha, P.O. Box 40100, Sargodha, Pakistan
b Facultad de Ciencias Exactas y Naturales, Pontificia Universidad Católica del Ecuador, Av. 12 de octubre 1076, Apartado, Quito 17-01-2184, Ecuador
c Department of Mathematics, Applied College in Mohayil Asir, King Khalid University, Abha, Saudi Arabia
⁎ Corresponding author. mjvivas@puce.edu.ec
03 9 2024
15 9 2024
03 9 2024
10 17 e372468 5 2024
29 8 2024
29 8 2024
© 2024 The Author(s)
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
In this paper, we investigate Bullen-type inequalities applicable to functions that are twice-differentiable. To explore these advanced inequalities, we utilize generalized convexity and Riemann-type fractional integrals. A comparative analysis is provided to highlight the more refined inequalities from among the explored results. By exploring the limiting cases, a relation with existing literature is established. Several examples are also presented to illustrate the outcomes and their accuracy is validated through graphical analysis. Additionally, applications in generalized means are also discussed.

MSC

26A33
26A51
39B62
Keywords

Bullen-type inequalities
2D and 3D graphs
Error estimates
Hölder's inequality
Fractional integrals
==== Body
pmc1 Introduction

A subfield of mathematical analysis known as fractional calculus extends the idea of differentiation and integration to non-integer orders, enabling a deeper understanding of complex processes and enhancing our ability to model and analyze complex phenomena. The role of fractional calculus in producing more precise and realistic models for processes and systems displaying non-local or memory-dependent behavior is what makes it so important [1], [2], [3]. Its large-scale applications across diverse scientific domains like physics [4], engineering [5], economics [6], finance [7] and control systems [8] have contributed to its growing importance. It has powerful mathematical tools for describing systems and phenomena that show fractal behavior, exceptional diffusion and comprehensive interactions.

Convexity has been developed, updated and extended in a number of ways [9], [10], [11]. It is important because it makes complicated issues easier to understand and guarantees effective computer solutions [12]. Our daily lives are greatly impacted by convexity due to its many uses in optimization [13], geometry [14], industry [15] and arts [16]. Basic mathematical expressions known as inequalities define the relative magnitude or order of two numbers. Mathematicians have proposed improvements, alterations and expansions of traditional inequalities like that Bullen [17], [18], Hermite-Hadamard [19], [20], [21], Ostrowski [22], Hardy-type [23], Jensen's [24] and Simpson's type inequalities [25]. Inequalities serve as a convenient and effective tool for modeling constraints, describing connections, and solving problems across various fields [26].

In 1978, Bullen introduced his inequality for the first time [27]. It has been studied and generalized by mathematicians for different fractional operators with applications [28], [29]. This new inequality is a generalization and expansion of previously existing inequalities, especially those involving means and other mathematical functions. Bullen-type inequalities offer more precise and accurate bounds for a range of functions and mathematical expressions. There are multiple applications for these inequalities in areas such as mathematical analysis [30], probability theory and optimization [31].

Motivated by the interesting connections between fractional calculus, convex functions and inequalities, this paper aims to investigate Bullen type inequalities via h-convex functions with composition of generalized fractional integrals. The class of h-convex functions generalizes the concept of convexity and allows us to study the properties and behavior of functions with fractional convexity.

2 Preliminaries

We need to recall the following definitions and fundamental results to maintain the flow of our work. The beta function [32] is defined as follows: Definition 2.1 The definition of the beta function is given byB(ξ,ζ)=∫01ωξ−1(1−ω)ζ−1dω,

where Re(ξ)>0, Re(ζ)>0. The significant relationship between the gamma and beta functions is expressed as follows:B(ξ,ζ)=B(ζ,ξ)=Γ(ξ)Γ(ζ)Γ(ξ+ζ).

The following is the definition of the h-convex function as given in [33]. Definition 2.2 If T:J→ℜ is a non-negative function, then T is called h-convex ifT(ωξ+(1−ω)ζ)≤h(ω)T(ξ)+h(1−ω)T(ζ)

is satisfied for all ξ,ζ∈J and ω∈[0,1].

According to [34], the Riemann-Liouville fractional integrals are defined as follows. Definition 2.3 If T∈L[ξ,ζ], then the Riemann-Liouville fractional integrals Fξ+αT and Fζ−αT of order α>0 with ξ≥0 are defined byFξ+αT(ρ)=1Γ(α)∫ξρ(ρ−ϖ)α−1T(ϖ)dϖ,ρ>ξ

andFζ−αT(ρ)=1Γ(α)∫ρζ(ϖ−ρ)α−1T(ϖ)dϖ,ρ<ζ,

where Γ(α)=∫0∞e−uuα−1du. It is notable that Fξ+0T(x)=Fζ−0T(ρ)=T(ρ). Regarding the case of α=1, the fractional integral simplifies to the standard integral.

According to [35], the generalized Riemann-Liouville fractional integrals are defined as follows. Definition 2.4 If T∈L[ξ,ζ], then the integral of order α defined by(Fξ+,καT)(ρ)=1κΓκ(α)∫ξρ(ρ−ϖ)ακ−1T(ϖ)dϖ,(0≤ξ<ρ)

and(Fζ−,καT)(ρ)=1κΓκ(α)∫ρζ(ϖ−ρ)ακ−1f(ϖ)dϖ,(0≤ρ<ζ).

In [36], Mitrinovic et al. stated the Hölder's inequality as follows. Theorem 2.5 Letλ>1,1λ+1ϑ=1andT(ϖ),g(ϖ):[ξ,ζ]→ℜ. If|T|λ,|g|ϑ∈L[ξ,ζ], then the inequality∫ξζ|T(ϖ)g(ϖ)|dϖ≤(∫ξζ|T(ϖ)|λdϖ)1λ(∫ξζ|g(ϖ)|ϑdϖ)1ϑ

holds. The equality holds if and only ifA|T(ϖ)|λ=B|g(ϖ)|ϑnearly everywhere. Here, A and B are fixed.

In [37], Imdat Iscan stated the improved Hölder's integral inequality as follows. Theorem 2.6 Letλ>1,1λ+1ϑ=1andT(ϖ),g(ϖ):[ξ,ζ]→ℜ. If|T|λ,|g|ϑ∈L[ξ,ζ], then the inequality below holds.∫ξζ|T(ϖ)g(ϖ)|dϖ≤1ζ−ξ(∫ξζ(ζ−ϖ)|T(ϖ)|λdϖ)1λ(∫ξζ(ζ−ϖ)|g(ϖ)|ϑdϖ)1ϑ+1ζ−ξ(∫ξζ(ϖ−ξ)|T(ϖ)|λdϖ)1λ(∫ξζ(ϖ−ξ)|g(ϖ)|ϑdϖ)1ϑ.

The power mean integral inequality derived from the Hölder's inequality can be expressed as follows. Theorem 2.7 Letϑ≥1,1λ+1ϑ=1andf(ϖ),g(ϖ):[ξ,ζ]→ℜ. If|T|λ,|g|ϑ∈L[ξ,ζ], then∫ξζ|T(ϖ)g(ϖ)|dϖ≤(∫ξζ|T(ϖ)|dϖ)1−1ϑ(∫ξζ|T(ϖ)||g(ϖ)|ϑdϖ)1ϑ

holds.

The improved power mean integral inequality as stated in [38] is as follows. Theorem 2.8 Letϑ≥1,1λ+1ϑ=1andT(ϖ),g(ϖ):[ξ,ζ]→ℜ. If|T|,|g|ϑ∈L[ξ,ζ]are the integrable functions on[ξ,ζ], then(2.1) ∫ξζ|T(ϖ)g(ϖ)|dϖ≤1ζ−ξ(∫ξζ(ζ−ϖ)|T(ϖ)|dϖ)1−1ϑ(∫ξζ(ζ−ϖ)|T(ϖ)||g(ϖ)|ϑdϖ)1ϑ+1ζ−ξ(∫ξζ(ϖ−ξ)|T(ϖ)|dϖ)1−1ϑ(∫ξζ(ϖ−ξ)|T(ϖ)||g(ϖ)|ϑdϖ)1ϑ

holds.

Recalling the lemma given in [39] helps to establish the main results. Lemma 2.9 LetT:[ξ,ζ]→ℜandT∈C2(ξ,ζ)withT″ ∈ L[ξ,ζ]. Then for all ν∈[0,1], the following equality holds.T(Ç)−Ϝ[α+κζ−ξ(Fη+,καT(ζ)+Fη−,καT(ξ))−(νFη+,κα−1T(ζ)+(1−ν)Fη−,κα−1T(ξ))]=(ζ−ξ)2νακ+(1−ν)ακ(Z1+Z2),

where Ç=νξ+(1−ν)ζ,α>1 and Ϝ=Γκ(α+κ)κ[νακ+(1−ν)ακ](ζ−ξ)ακ−1. Also(2.2) Z1=∫0νωακ(ν−ω)T″(ωξ+(1−ω)ζ)dω,Z2=∫ν1(1−ω)ακ(ω−ν)T″(ωξ+(1−ω)ζ)dω.

3 Main results

The estimates of Bullen-type inequalities are essential to mathematics because they offer bounds, analytical tools and fundamental ideas for many different mathematical fields. In this section, we study the advanced estimates of Bullen-type inequalities. The first main result is given by the following theorem. Theorem 3.1 LetT:[ξ,ζ]→ℜbe a function with the propertyT∈C2(ξ,ζ). IfT″∈L[ξ,ζ]and|T″|ϑis h-convex, then the inequality(3.1) |T(Ç)−Ϝ[α+kζ−ξ(Fη+,καT(ζ)+Fη−,καT(ξ))−(νFη+,καT(ζ)+(1−ν)Fη−,κα−1T(ξ))]|≤(ζ−ξ)2νακ+(1−ν)ακA1λ[νακ+1λ+1ϑ+1(M(|T″(ξ)|ϑ+|T″(ζ)|ϑ))1ϑ+(1−ν)ακ+1λ+1ϑ+1(M(|T″(ξ)|ϑ+|T″(ζ)|ϑ))1ϑ]

is satisfied for allα>1,ϑ>1andA=2λ−1(2αλκ+λ+2)(αλκ+1)(αλκ+λ+1). The values of Ϝ and Ç are defined byLemma 2.9and M is the upper bound of h.

Proof By utilizing the modulus condition and Lemma 2.9, we obtain(3.2) |T(Ç)−Ϝ[α+κζ−ξ(Fη+,καT(ζ)+Fη−,καT(ξ))−(νFη+,καT(ζ)+(1−ν)Fη−,κα−1T(ξ))]|≤(ζ−ξ)2νακ+(1−ν)ακ(|Z1|+|Z2|),

where Z1 and Z2 are defined by (2.2). Now|Z1|≤∫0ν|ωακ(ν−ω)T″(ωξ+(1−ω)ζ)|dω=∫0ν|ωακ(ν−ω)||T″(ωξ+(1−ω)ζ)|dω.

By using Hölder's inequality, h-convexity of |T″(ζ)|ϑ and the fact|ν+y|λ≤2λ−1(|ν|λ+|y|λ),

for λ≥0 and ν,y∈ℜ, we can write|Z1|≤(∫0ν|ωακ|λ|ν−ω|λdω)1λ(∫0νM(|T″(ξ)|ϑ+|T″(ζ)|ϑ)dω)1ϑ=(∫0ν|ωαλκ||ν+(−ω)|λdω)1λ(∫0νM(|T″(ξ)|ϑ+|T″(ζ)|ϑ)dω)1ϑ≤(2λ−1∫0ν(νλωαλκ+ωαλκ+λ)dω)1λ(νM(|T″(ξ)|ϑ+|T″(ζ)|ϑ))1ϑ=(2λ−1ναλκ+λ+1(2αλκ+λ+2)(αλκ+1)(αλκ+λ+1))1λ(νM(|T″(ξ)|ϑ+|T″(ζ)|ϑ))1ϑ=νακ+1λ+1ϑ+1A1λ(M(|T″(ξ)|ϑ+|T″(ζ)|ϑ))1ϑ.

By utilizing the absolute value of Z2 and Hölder's inequality, we get|Z2|≤∫ν1|(1−ω)ακ(ω−ν)T″(ωξ+(1−ω)ζ)|dω.≤(∫ν1|(1−ω)ακ(ω−ν)|λdω)1λ(∫ν1|T″(ωξ+(1−ω)ζ)|ϑdω)1ϑ.

By replacing ω by 1−z|Z2|≤(∫1−ν0(|1−z−ν|λzαλκ)(−dz))1λ((1−ν)M(|T″(ξ)|ϑ+|T″(ζ)|ϑ))1ϑ=(∫1−ν0|(1−ν)+(−z)|λ(zαλκ)(−dz))1λ((1−ν)M(|T″(ξ)|ϑ+|T″(ζ)|ϑ))1ϑ≤(2λ−1(1−ν)αλκ+λ+1(2αλκ+λ+2)(αλκ+1)(αλκ+λ+1))1λ((1−ν)M(|T″(ξ)|ϑ+|T″(ζ)|ϑ))1ϑ|Z2|≤(1−ν)ακ+1λ+1A1λ[(1−ν)M(|T″(ξ)|ϑ+|T″(ζ)|ϑ)]1ϑ.

By summing |Z1| and |Z2| and taking (3.2) into account, we obtain (3.1). □

Remark 3.2 By changing h(ω)=ω in Theorem 3.1, we reach to [39, Theorem 6].

Remark 3.3 By changing h(ω)=ω and κ=1 in Theorem 3.1, we reach to [40, Theorem 6].

Example 3.4 Case (i): The validity of the double inequality (3.1) can be confirmed in the following three-dimensional illustration.Figure 1 The resulting conclusion of the inequality (3.1) is verified through a visual illustration with values of M=1,ν=12, κ = 1, α = 1, T(ξ)=eξ, λ = ϑ = 2, ξ ∈ [2,4], ζ ∈ [4,6] in Fig. 1.

Figure 1

Case (ii): In the provided 2D illustration, we confirm the rationality of the dual inequality (3.1).

Figure 2 The resulting conclusion of the inequality (3.1) is verified through a visual illustration with values of M=1,ν=12, κ = 1, M = 1, α = 1, T(ξ)=eξ, ξ = 0, ζ = 1 and λ ∈ [1.1,10] in Fig. 2.

Figure 2

Theorem 3.5 LetT:[ξ,ζ]→ℜbe a function with the propertyT∈C2(ξ,ζ). IfT″ ∈ L[ξ,ζ] and |T″|ϑ is h-convex, then the inequality(3.3) |T(Ç)−Ϝ[α+kζ−ξ(Fη+,καT(ζ)+Fη−,καT(ξ))−(νFη+,καT(ζ)+(1−ν)Fη−,κα−1T(ξ))]|≤(ζ−ξ)2νακ+(1−ν)ακ[(M(|T″(ξ)|ϑ+|T″(ζ)|ϑ))1ϑ(νακ+2λM1+(1−ν)ακ+2λM2)]×[B1λ(αλκ+1,λ+2)+B1λ(αλκ+2,λ+1)]

is satisfied for all α>1 and M1=[ν22]1ϑ,M2=[(1−ν)22]1ϑ. The values of Ϝ and Ç are defined by Lemma 2.9 and M is the upper bound of h.

Proof By utilizing the modulus condition and Lemma 2.9, we obtain(3.4) |T(Ç)−Ϝ[α+kζ−ξ(Fη+,καT(ζ)+Fη−,καf(ξ))−(νFη+,καT(ζ)+(1−ν)Fη−,κα−1T(ξ))]|≤(ζ−ξ)2νακ+(1−ν)ακ(|Z1|+|Z2|).

By utilizing the absolute value for Z1 and using improved Hölder's inequality|Z1|≤∫0ν|ωακ(ν−ω)T″(ωξ+(1−ω)ζ)|dω≤∫0ν|ωακ(ν−ω)||T″(ωξ+(1−ω)ζ)|dω≤1ν(∫0ν(ν−ω)|ωακ(ν−ω)|λdω)1λ(∫0ν(ν−ω)|T″(ωξ+(1−ω)ζ)|ϑdω)1ϑ+1ν(∫0νω|ωακ(ν−ω)|λdω)1λ(∫0νω|T″(ωξ+(1−ω)ζ)|ϑdω)1ϑ≤(M(|T″(ξ)|ϑ+|T″(ζ)|ϑ))1ϑν[(∫0νωαλκ(ν−ω)1+λdω)1λ(∫0ν(ν−ω)dω)1ϑ+(∫0νωαλκ+1(ν−ω)λdω)1λ(∫0νωdω)1ϑ].

Replacing ω by νz|Z1|≤[M(|T″(ξ)|ϑ+|T″(ζ)|ϑ)]1ϑν[(ναλκ+λ+2∫01zαλκ(1−z)1+λdz)1λ(ν22)1ϑ+(ναλκ+λ+2∫01zαλκ+1(1−z)λdz)1λ(ν22)1ϑ]=[M(|T″(ξ)|ϑ+|T″(ζ)|ϑ)]1ϑν[(ναλκ+λ+2B(αλκ+1,λ+2))1λ(ν22)1ϑ+(ναλκ+λ+2B(αλκ+2,λ+1))1λ(ν22)1ϑ].

Similarly|Z2|≤∫ν1|(1−ω)ακ(ω−ν)T″(ωξ+(1−ω)ζ)|dω.

Replacing ω by (1−ω) and using improved Hölder's inequality|Z2|≤∫0τ|ωακ(τ−ω)||T″((1−ω)ξ+ωζ)|dω.≤1τ(∫0τ(τ−ω)|ωακ(τ−ω)|λdω)1λ(∫0τ(τ−ω)|T″((1−ω)ξ+ωζ)|ϑdω)1ϑ+1τ(∫0τω|ωακ(τ−ω)|λdω)1λ(∫0τω|T″((1−ω)ξ+ωζ)|ϑdω)1ϑ≤1τ(∫0τ(τ−τz)1+λ(τz)αλκτdz)1λ(M(|T″(ξ)|ϑ+|T″(ζ)|ϑ)(τ22))1ϑ+1τ(∫01((τ−τz)λ(τz)αλκ+1τ)dz)1λ(M(|T″(ξ)|ϑ+|T″(ζ)|ϑ)(τ22))1ϑ

This can be written as|Z2|≤1(1−ν)[(1−ν)αλκ+λ+2B(αλκ+1,λ+2)]1λ[M(|T″(ξ)|ϑ+|T″(ζ)|ϑ)((1−ν)22)]1ϑ+1(1−ν)[(1−ν)αλκ+λ+2B(αλκ+2,λ+1)]1λ[M(|T″(ξ)|ϑ+|T″(ζ)|ϑ)((1−ν)22)]1ϑ|Z2|≤(1−ν)ακ+2λ[M(|T″(ξ)|ϑ+|T″(ζ)|ϑ)((1−ν)22)]1ϑ×[B1λ(αλκ+1,λ+2)+B1λ(αλκ+2,λ+1)].

The integrals are summed and we obtain the following result(3.5) |Z1|+|Z2|≤[(M(|T″(ξ)|ϑ+|T″(ζ)|ϑ))1ϑ(νακ+2λM1+(1−ν)ακ+2λM2)]×[B1λ(αλκ+1,λ+2)+B1λ(αλκ+2,λ+1)].

By substituting (3.5) into (3.4), we obtain (3.3). □

Remark 3.6 By setting h(ω)=ω in Theorem 3.5, we reach to [39, Theorem 7].

Remark 3.7 By setting h(ω)=ω and κ=1 in Theorem 3.5, we reach to [40, Theorem 7].

Example 3.8 Case (i): The validity of the double inequality (3.3) can be confirmed in the following three-dimensional illustration.Figure 3 The resulting conclusion of the inequality (3.3) is verified through a visual illustration with values of M=1,ν=12, κ = 1, α = 1, T(ξ)=ξ2, λ = ϑ = 2, ξ ∈ [2,4], ζ ∈ [4,6] in Fig. 3.

Figure 3

Case (ii): In the provided 2D illustration, we confirm the rationality of the dual inequality (3.3).

Figure 4 The resulting conclusion of the inequality (3.3) is verified through a visual illustration with values of M=1,ν=12, κ = 1, α = 1, T(ξ)=ξ2, ξ = 0, ζ = 1 and λ ∈ [1.1,10] in Fig. 4.

Figure 4

Theorem 3.9 LetT:[ξ,ζ]→ℜbe a function with the propertyT∈C2(ξ,ζ). IfT″ ∈ L[ξ,ζ] and |T″|ϑ is h-convex, then the inequality(3.6) |T(Ç)−Ϝ[α+kζ−ξ(Fη+,καT(ζ)+Fη−,καT(ξ))−(νFη+,καT(ζ)+(1−ν)Fη−,κα−1T(ξ))]|≤(ζ−ξ)2νακ+(1−ν)ακP1(P2+P3)

is satisfied for all α>1 and P1=1(ακ+1)(ακ+2),P2=νακ+2[M(|T″(ξ)|λ+|T″(ζ)|λ)]1λ,P3=(1−ν)ακ+2[M(|T″(ξ)|λ+|T″(ζ)|λ)]1λ. The values of Ϝ and Ç are defined by Lemma 2.9 and M is the upper bound of h.

Proof By utilizing the modulus condition and Lemma 2.9, we obtain(3.7) |T(Ç)−F[α+κζ−ξ(Fη+,καT(ζ)+Fη−,καT(ξ))−(νFη+,καT(ζ)+(1−ν)Fη−,κα−1T(ξ))]|≤(ζ−ξ)2νακ+(1−ν)ακ(|Z1|+|Z2|).

By utilizing the absolute value and power mean inequality for Z1, we get|Z1|≤∫0ν|ωακ(ν−ω)T″(ωξ+(1−ω)ζ)|dω≤∫0ν|ωακ(ν−ω)||T″(ωξ+(1−ω)ζ)|dω≤[∫0νωακ(ν−ω)dω]1−1λ[∫0νωακ(ν−ω)|T″(ωξ+(1−ω)ζ)|λdω]1λ≤[M(|T″(ξ)|λ+|T″(ζ)|λ)]1λ[(∫0νωακ(ν−ω)dω)1−1λ(∫0νωακ(ν−ω)dω)1λ]=[M(|T″(ξ)|λ+|T″(ζ)|)λ]1λ[(νακ+2(ακ+1)(ακ+2))1−1λ(νακ+2(ακ+1)(ακ+2))1λ].

This can also be written as|Z1|≤[M(|T″(ξ)|λ+|T″(ζ)|)λ]1λ[νακ+2(ακ+1)(ακ+2)].

Similarly|Z2|≤∫ν1|(1−ω)ακ(ω−ν)T″(ωξ+(1−ω)ζ)|dω≤[∫ν1(1−ω)ακ(ω−ν)dω]1−1λ[∫ν1(1−ω)ακ(ω−ν)|T″(ωξ+(1−ω)ζ)|λdω]1λ=[∫01−νzακ(1−z−ν)dz]1−1λ[∫01−νzακ(1−z−ν)|T″((1−z)ξ+zζ)|λdz]1λ≤[M(|T″(ξ)|λ+|T″(ζ)|)λ]1λ[(1−ν)ακ+2(ακ+1)(ακ+2)].

By summing |Z1| and |Z2|, we obtain the following result(3.8) |Z1|+|Z2|≤[νακ+2(ακ+1)(ακ+2)][M(|T″(ξ)|λ+|T″(ζ)|λ)1λ]+[(1−ν)ακ+2(ακ+1)(ακ+2)][M(|T″(ξ)|λ+|T″(ζ)|λ)1λ].

By using (3.8) in (3.7), we obtain (3.6). □

Remark 3.10 By substituting h(ω)=ω in Theorem 3.6, we reach to [39, Theorem 8].

Remark 3.11 By substituting h(ω)=ω and κ=1 in Theorem 3.6, we reach to [40, Theorem 8].

Example 3.12 Case (i): The validity of the double inequality (3.6) can be confirmed in the following three-dimensional illustration.Figure 5 The resulting conclusion of the inequality (3.6) is verified through a visual illustration with values of M=1,ν=12, κ = 1, α = 1, T(ξ)=3ξ5, λ = ϑ = 2, ξ ∈ [1.1,2], ζ ∈ [8,10] in Fig. 5.

Figure 5

Case (ii): In the provided 2D illustration, we confirm the rationality of the dual inequality (3.6).

Figure 6 The resulting conclusion of the inequality (3.6) is verified through a visual illustration with values of M=1,ν=12, κ = 1, α = 1, T(ξ)=3ξ5, ξ = 0, ζ = 1 and λ ∈ [1.1,10] in Fig. 6.

Figure 6

Theorem 3.13 Let T:[ξ,ζ]→ℜ be a function with the property T∈C2(ξ,ζ) . If T″∈L[ξ,ζ] and |T″|ϑ is h-convex on [ξ,ζ] , then the inequality (3.9) |T(Ç)−Ϝ[α+kζ−ξ(Fη+,καT(ζ)+Fη−,καT(ξ))−(νFη+,καT(ζ)+(1−ν)Fη−,κα−1T(ξ))]|≤(ζ−ξ)2νακ+(1−ν)ακ[B(ακ+1,3)(νακ+2P+(1−ν)ακ+2P)+B(ακ+2,2)(νακ+2P+(1−ν)ακ+2P)]

holds for all α>1 , ϑ≥1 , 1λ+1ϑ=1 and P=[M(|T″(ξ)|ϑ+|T″(ζ)|ϑ)]1ϑ . The values of Ϝ and Ç are defined by Lemma 2.9 and M is the upper bound of h.

Proof By utilizing the modulus condition and Lemma 2.9, we obtain(3.10) |T(Ç)−Ϝ[α+kζ−ξ(Fη+,καT(ζ)+Fη−,καT(ξ))−(νFη+,καT(ζ)+(1−ν)Fη−,κα−1T(ξ))]|≤(ζ−ξ)2νακ+(1−ν)ακ(|Z1|+|Z2|).

By utilizing the absolute value and using the improved power mean inequality (2.1) for |Z1|, we obtain|Z1|≤1ν[∫0ν(ν−ω)|ωακ(ν−ω)|dω]1−1ϑ[∫0ν(ν−ω)|ωακ(ν−ω)||T″(ωξ+(1−ω)ζ)|ϑdω]1ϑ+1ν[∫0νω|ωακ(ν−ω)|dω]1−1ϑ[∫0νω|ωακ(ν−ω)||T″(ωξ+(1−ω)ζ)|ϑdϑ]1ϑ=1ν[∫0ν(ν−ω)2|ωακ|dω]1−1ϑ[∫0νωακ(ν−ω)2|T″(ωξ+(1−ω)ζ)|ϑdω]1ϑ+1ν[∫0νωακ+1(ν−ω)dω]1−1ϑ[∫0νωακ+1(ν−ω)|T″(ωξ+(1−ω)ζ)|ϑdω]1ϑ.

By substituting ω=νz, we can have∫0νωακ(ν−ω)2dω=∫0ν(νz)ακ(ν−νz)2νdz=νακ+3∫0νzακ(1−z)2dz=νακ+3B(ακ+1,3).∫0νωακ+1(ν−ω)dω=∫0ν(νz)ακ+1(ν−νz)νdz=νακ+3∫0νzακ+1(1−z)dz=νακ+3B(ακ+2,2).

Now, using the definition of h-convexity, we obtain the following result|Z1|≤1ν[νακ+3B(ακ+1,3)]1−1ϑ×[M(|T″(ξ)|ϑ+|T″(ζ)|ϑ)]1ϑ[νακ+3B(ακ+1,3)]1ϑ+1ν[νακ+3B(ακ+2,2)]1−1ϑ×[M(|T″(ξ)|ϑ+|T″(ζ)|ϑ)]1ϑ[νακ+3B(ακ+2,2)]1ϑ.

This can be written as|Z1|≤νακ+2[M(|T″(ξ)|ϑ+|T″(ζ)|ϑ)]1ϑ[B(ακ+1,3)+B(ακ+2,2)].

If we replace ω by 1−ω in |Z2|, we get|Z2|≤∫ν1|(ω−ν)(1−ω)ακ||T″(ωξ+(1−ω)ζ)|dω=∫1−ν0|(1−ω−ν)ωακ||T″((1−ω)ξ+ωζ)|(−dω)=∫01−ν|(1−ν−ω)ωακ||T″((1−ω)ξ+ωζ)|dω.

Substituting τ=1−ν|Z2|≤∫0τ|(τ−ω)ωακ||T″((1−ω)ξ+ωζ)|dω≤1τ[∫0τ(τ−ω)|ωακ(τ−ω)|dω]1−1ϑ[∫0τ(τ−ω)|ωακ(τ−ω)||T″((1−ω)ξ+ωζ)|ϑdω]1ϑ+1τ[∫0τω|ωακ(τ−ω)|dω]1−1ϑ[∫0τω|ωακ(τ−ω)||T″((1−ω)ξ+ωζ)|ϑdω]1ϑ=1τ[∫0τ(τ−ω)2ωακdω]1−1ϑ[∫0τωακ(τ−ω)2|T″((1−ω)ξ+ωζ)|ϑdω]1ϑ+1τ[∫0τωακ+1(τ−ω)dω]1−1ϑ[∫0τωακ+1(τ−ω)|T″((1−ω)ξ+ωζ)|ϑdω]1ϑ.

This can be written as|Z2|≤(1−ν)ακ+2[M(|T″(ξ)|ϑ+|T″(ζ)|ϑ)]1ϑ[B(ακ+1,3)+B(ακ+2,2)].

By adding the values of Z1 and Z2, we obtain(3.11) |Z1|+|Z2|≤νακ+2[M(|T″(ξ)|ϑ+|f″(ζ)|ϑ)]1ϑ[B(ακ+1,3)+B(ακ+2,2)]+(1−ν)ακ+2[M(|T″(ξ)|ϑ+|T″(ζ)|ϑ)]1ϑ[B(ακ+1,3)+B(ακ+2,2)].

By using (3.11) in (3.10), we obtain (3.9). □

Remark 3.14 By substituting h(ω)=ω in Theorem 3.13, we reach to [39, Theorem 9].

Remark 3.15 By setting h(ω)=ω and κ=1 in Theorem 3.13, we reach to [40, Theorem 9].

Example 3.16 Case (i): The validity of the double inequality (3.9) can be confirmed in the following three-dimensional illustration.

Figure 7 The resulting conclusion of the inequality (3.9) is verified through a visual illustration with values of M=1,ν=12, κ = 1, α = 1, T(ξ)=5e3ξ, λ = ϑ = 2, ξ ∈ [2,4], ζ ∈ [6,8] in Fig. 7.

Figure 7

Case (ii): In the provided 2D illustration, we confirm the rationality of the dual inequality (3.9).

Figure 8 The resulting conclusion of the inequality (3.9) is verified through a visual illustration with values of M=1,ν=12, κ = 1, α = 1, T(ξ)=5e3ξ, ξ = 0, ζ = 1 and λ ∈ [1.1,10] in Fig. 8.

Figure 8

Theorem 3.17 LetT:[ξ,ζ]→ℜbe a function with the propertyT∈C2(ξ,ζ). IfT″ ∈ L[ξ,ζ] and |T″| is h-convex, then the inequality(3.12) |T(c)−Ϝ[(α+κ)ζ−ξ(Fη+,καT(ζ)+Fη−,καT(ξ))−(νFη+,καT(ζ)+(1−ν)Fη−,κα−1T(ξ))]|≤(ζ−ξ)2νακ+(1−ν)ακ[M(|T″(ξ)|+|T″(ζ)|)][νακ+2+(1−ν)ακ+2(ακ+1)(ακ+2)]

is satisfied for all α>1. The values of Ϝ and Ç are defined by Lemma 2.9 and M is the upper bound of h.

Proof By utilizing the modulus condition for Lemma 2.9|T(Ç)−Ϝ[α+κζ−ξ(Fη+,καT(ζ)+Fη−,καT(ξ))−(νFη+,καT(ζ)+(1−ν)Fη−,κα−1T(ξ))]|≤(ζ−ξ)2νακ+(1−ν)ακ[∫0ν|ωακ(ν−ω)T″(ωξ+(1−ω)ζ)|dω+∫ν1|(1−ω)ακ(ω−ν)T″(ωξ+(1−ω)ζ)|dω].

By using h-convexity|T(Ç)−Ϝ[α+κζ−ξ(Fη+,καT(ζ)+Fη−,καT(ξ))−(νFη+,καT(ζ)+(1−ν)Fη−,κα−1T(ξ))]|≤(ζ−ξ)2νακ+(1−ν)ακ[∫ν1ωακ(ν−ω)(h(ω)|T″(ξ)|+h(1−ω)|T″(ζ)|)dω+∫ν1(1−ω)ακ(ω−ν)(h(ω)|T″(ξ)|+h(1−ω)|T″(ζ)|)dω].

Since h(ω)≤M and h(1−ω)≤M, therefore(3.13) |T(Ç)−Ϝ[α+κζ−ξ(Fη+,καT(ζ)+Fη−,καT(ξ))−(νFη+,καT(ζ)+(1−ν)Fη−,κα−1T(ξ))]|≤(ζ−ξ)2νακ+(1−ν)ακ[M(|T″(ξ)|+|T″(ζ)|)]×[∫0νωακ(ν−ω)dω+∫ν1(1−ω)ακ(ω−ν)dω]

I1=∫0νωακ(ν−ω)dω,I2=∫ν1(ω−ν)(1−ω)ακdω.

Integrating by parts, we obtain the following resultI1=νακ+2(ακ+1)(ακ+2),

I2=ν(1−ν)ακ+1(ακ+1)+(1−ν)ακ+2(ακ+1)−ν(1−ν)ακ+1(ακ+1).

By using the values of I1 and I2 in (3.13), we have the following result.|T(Ç)−Ϝ[α+κζ−ξ(Fη+,καT(ζ)+Fη−,καT(ξ))−(νFη+,καT(ζ)+(1−ν)Fη−,κα−1T(ξ))]|≤(ζ−ξ)2νακ+(1−ν)ακ[M(|T″(ξ)|+|T″(ζ)|)]×[νακ+2(ακ+1)(ακ+2)+ν(1−ν)ακ+1(ακ+1)+(1−ν)ακ+2(ακ+1)−ν(1−ν)ακ+1(ακ+1)]=(ζ−ξ)2νακ+(1−ν)ακ[M(|T″(ξ)|+|T″(ζ)|)][νακ+2+(1−ν)ακ+2(ακ+1)(ακ+2)].

Therefore|T(Ç)−Ϝ[α+κζ−ξ(Fη+,καT(ζ)+Fη−,καT(ξ))−(νFη+,καT(ζ)+(1−ν)Fη−,κα−1T(ξ))]|≤(ζ−ξ)2νακ+(1−ν)ακ[M(|T″(ξ)|+|T″(ζ))][νακ+2+(1−ν)ακ+2(ακ+1)(ακ+2)].

 □

Remark 3.18 By setting h(ω)=ω in Theorem 3.12, we reach to [39, Theorem 5].

Remark 3.19 By choosing h(ω)=ω and κ=1 in Theorem 3.12, we reach to [40, Theorem 5].

Example 3.20 The validity of the double inequality (3.12) can be confirmed in the following three-dimensional illustration.

Figure 9 The resulting conclusion of the inequality (3.12) is verified through a visual illustration with values of M=1,ν=12, κ = 1, α = 1, T(ξ)=eξ, ξ ∈ [2,4], ζ ∈ [6,8] in Fig. 9.

Figure 9

A comparative analysis of the results obtained using Hölder's inequality and improved Hölder's inequality is given by the following examples. Example 3.21 If we set T(ζ)=eζ, ϖ>0, then |T″(ζ)|ϑ=eζϑ for ϑ>1 and ϖ>0 is a convex. For the situation in which α=1, ξ=0, ζ=1, ν=12 and ϑ=2. Let us identify the right portion of the inequalities (3.1) and (3.3).

(a) For relation (3.1)|T(Ç)−Ϝ[α+κζ−ξ(Fη+,καT(ζ)+Fη−,καT(ξ))−(νFη+,καT(ζ)+(1−ν)Fη−,κα−1T(ξ))]|≤(ζ−ξ)2νακ+(1−ν)ακA1λ[νακ+1λ+1ϑ+1(M(|T″(ξ)|ϑ+|T″(ζ)|ϑ))1ϑ+(1−ν)ακ+1λ+1ϑ+1(M(|T″(ξ)|ϑ+|T″(ζ)|ϑ))1ϑ]=(1615)12(12)2(1+(2.7183)2)12≈0.74784.

(b) For relation (3.3)|T(Ç)−Ϝ[α+κζ−ξ(Fη+,καT(ζ)+Fη−,καT(ξ))−(νFη+,καT(ζ)+(1−ν)Fη−,κα−1T(ξ))]|≤(ζ−ξ)2νακ+(1−ν)ακ[B1λ(αλκ+1,λ+2)(M(|T″(ξ)|ϑ+|T″(ζ)|ϑ))1ϑ×(νακ+2λM1+(1−ν)ακ+2λM2)+B1λ(αλκ+2,λ+1)(M(|T″(ξ)|ϑ+|T″(ζ)|ϑ))1ϑ×(νακ+2λM1+(1−ν)ακ+2λM2)]=2(0.12911)(2.8964)(0.1768)≈0.13223.

Discussion. Due to the fact that 0.74784>0.13223, the improved Hölder's inequality provides more accurate estimates compared to the classical Hölder's inequality. The visual representations of the Example 3.21 is represented in Fig. 10.Figure 10 This graph is sketched corresponding to the choice of parameters ξ = 1, ζ = 2 and λ ∈ [1.1,2].

Figure 10

Example 3.22 If we set T(ζ)=eζ, ϖ>0, then |T″(ζ)|ϑ=eζϑ for ϑ>1 and ϖ>0 is a convex. For the case in which α=1, ξ=0, ζ=1, ν=12 and ϑ=2, let us identify the right portion of the inequalities (3.6) and (3.9).

(a) For relation (3.6)|T(Ç)−Ϝ[α+κζ−ξ(Fη+,καT(ζ)+Fη−,καT(ξ))−(νFη+,καT(ζ)+(1−ν)Fη−,κα−1T(ξ))]|≤(ζ−ξ)2νακ+(1−ν)ακP1(P2+P3)=(0.1667)(0.36205+0.36205)≈0.12071.

(a) For relation (3.9)|T(Ç)−Ϝ[α+κζ−ξ(Fη+,καT(ζ)+Fη−,καT(ξ))−(νFη+,καT(ζ)+(1−ν)Fη−,κα−1T(ξ))]|≤(ζ−ξ)2νακ+(1−ν)ακ[B1λ(ακ+1,3)(νακ+2.P+(1−ν)ακ+2.P)+B1λ(ακ+2,2)(νακ+2.P+(1−ν)ακ+2.P)]=2(0.28868)(0.25)(0.36205)≈0.04626.

Discussion. Due to the fact that 0.12071>0.04626, the extended power mean inequality provides more accurate estimates compared to the classical power mean inequality. The visual representation of Example 3.22 is mentioned in Figs. 11.Figure 11 This graph is sketched corresponding to the choice of the parameters M = 1, ξ = 1, ζ = 2 and λ ∈ [1.01,1.05].

Figure 11

4 Applications

The use of means helps us learn how to solve issues, examine and understand data and arrive at well informed and mathematical conclusions. The purpose of this section is to provide real world examples of significant results related to these methods.

(1) A = A(ξ,ζ)=ξ+ζ2, ξ,ζ∈ℜ+

(2)The logarithmic mean=L(ξ,ζ)=ζ−ξlnζ−lnξ, ξ≠ζ,ξ,ζ∈ℜ+.

Proposition 4.1 Let ξ,ζ∈ℜ+ , with ξ<ζ , then we have the following bounds of mean differences (4.1) |12(eA(ξ,ζ)+A(eξ,eζ))−L(eξ,eζ)|≤(ξ−ζ)2(1615)(18)12(e2ζ+e2ξ2)12.

Proof If we set ν=12, κ=1, M=1, α=1 and T(t)=et in Theorem 3.1, we obtain the following result.|12(eξ+ζ2+eζ+eξ2)−eζ−eξζ−ξ|≤(ζ−ξ)2(1615)(18)12(e2ζ+e2ξ2)12.

The validity of the relationship expressed by (4.1) has been demonstrated. □

Proposition 4.2 (4.2) |12(eA(ξ,ζ)+A(eξ,eζ))−L(eξ,eζ)|≤(0.0456)(ζ−ξ)2(e2ξ+e2ζ)12.

Proof If we set ν=12, κ=1, M=1, α=1 and T(t)=et in Theorem 3.5, we obtain|12(eξ+ζ2+eζ+eξ2)−eζ−eξζ−ξ|≤(0.0456)(ζ−ξ)2(e2ξ+e2ζ)12.

The validity of the relationship expressed by (4.2) has been demonstrated. □

Proposition 4.3 (4.3) |12(eA(ξ,ζ)+A(eξ,eζ))−L(eξ,eζ)|≤(0.041675)(ζ−ξ)2(e2ξ+e2ζ)12.

Proof If we set ν=12, κ=1, M=1, α=1 and T(t)=et in Theorem 3.9, we obtain|12(eξ+ζ2+eζ+eξ2)−eζ−eξζ−ξ|≤(0.041675)(ζ−ξ)2(e2ξ+e2ζ)12.

The validity of the relationship expressed by (4.3) has been demonstrated. □

Proposition 4.4 (4.4) |12(eA(ξ,ζ)+A(eξ,eζ))−L(eξ,eζ)|≤(0.14434)(ζ−ξ)2(e2ξ+e2ζ)12.

Proof If we set ν=12, κ=1, M=1, α=1 and T(t)=et in Theorem 3.13, we obtain|12(eξ+ζ2+eζ+eξ2)−eζ−eξζ−ξ|≤(0.14434)(ζ−ξ)2(e2ξ+e2ζ)12.

The validity of the relationship expressed by (4.4) has been demonstrated. □

Proposition 4.5 (4.5) |12(eA(ξ,ζ)+A(eξ,eζ))−L(eξ,eζ)|≤(ζ−ξ)296[eζ+eξ].

Proof If we set ν=12, κ=1, M=1, α=1 and T(t)=et in Theorem 3.17, we can have|12(eξ+ζ2+eζ+eξ2)−eζ−eξζ−ξ|≤(ζ−ξ)296[eζ+eξ].

The validity of the relationship expressed by (4.5) has been demonstrated. □

5 Conclusions

Refining inequalities is a top goal for civilizations everywhere. Recognizing and actively attempting to remove structural obstacles would help us to promote more opportunity, fairness and inclusion for everyone. Inequalities appear in many areas of day-to-day existence, impacting outcomes, opportunities, and the availability of resources. Bullen-type inequalities help people and organizations to communicate more effectively and convincingly by making data easier to understand and deliver. These estimates are also used to analyze inequalities related to income distribution, wealth accumulation and financial risk. They can help in understanding disparities in wealth and income and inform policy decisions aimed at promoting economic equality and stability. In this study, we established Bullen-type inequalities applicable to functions that are twice differentiable. By applying h-convexity, we introduce new estimates for fractional Bullen-type inequalities applicable to twice-differentiable functions. The improvement in estimations is demonstrated through various specific cases and graphical representations. Additionally, Riemann-type fractional integral operators are utilized to establish the main findings of the presented study. It may be beneficial to explore similar outcomes for additional classes of convex functions, particularly coordinate convex functions.

Funding

No funding.

CRediT authorship contribution statement

Muhammad Samraiz: Writing – original draft, Project administration, Methodology, Investigation. Zohaib Hassan: Writing – original draft, Formal analysis, Conceptualization. Saima Naheed: Writing – review & editing, Validation, Supervision, Software. Miguel Vivas-Cortez: Writing – review & editing, Funding acquisition, Data curation, Conceptualization. Rifaqat Ali: Writing – original draft, Formal analysis, Data curation. Tarik Lamoudan: Writing – original draft, Formal analysis, Data curation, Conceptualization.

Declaration of Competing Interest

The authors declare that they have no competing interests.

Data availability

No data were used for the research described in the article. Moreover, no data are associated with our study.

Acknowledgements

The authors extend their appreciation to the Deanship of Research and Graduate Studies at 10.13039/501100007446 King Khalid University for funding this work through Large Group Project under grant number RGP2/218/45 .
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