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Article

Local Fractional Perspective on Weddle’s Inequality in Fractal Space

1
Department of Mathematics of Humanities College, Zhejiang Guangsha Vocational and Technical University of Construction, Jinhua 321004, China
2
Department of Mathematics, Government College University Faisalabad, Faisalabad 38000, Pakistan
3
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia
*
Author to whom correspondence should be addressed.
Fractal Fract. 2025, 9(10), 662; https://doi.org/10.3390/fractalfract9100662 (registering DOI)
Submission received: 3 September 2025 / Revised: 1 October 2025 / Accepted: 2 October 2025 / Published: 14 October 2025
(This article belongs to the Special Issue Advances in Fractional Integral Inequalities: Theory and Applications)

Abstract

The Yang local fractional setting provides the generalized framework to explore the non-differentiable mappings considering the local properties. Due to the dominance of these concepts, mathematicians have investigated multiple problems, including mathematical modelling, optimization, and inequalities. Incorporating these useful concepts, this study aims to derive Weddle-type integral inequalities within the context of fractal space. To achieve the intended results, we establish a new local fractional identity. By using this identity, the convexity property, the bounded property of mappings, the L-Lipschitzian property of mappings, and other famous inequalities, we develop numerous upper bounds. Additionally, we provide 2D and 3D graphical representations and numerous applications, which show the significance of our main findings. To the best of our knowledge, this is the first study concerning error inequalities of Weddle’s quadrature formulation within the fractal space.
MSC:
26A33; 26A51; 26D07; 26D10; 26D15; 26D20

1. Introduction

Convex mappings have potential applications in many domains, which include optimization, coding hypothesis, designing, and inequality theory. But their role is unprecedented for the derivation of mathematical inequalities. Working on this benchmark class of mappings, it is not difficult to conclude the fundamental results of analysis. In the fields of applied science, mathematical inequalities are considered to be the essential structure for connecting qualitative and quantitative analysis. First, we recall the definition of convex mapping.
Definition 1
([1]). Any mapping T : I R R is termed as convex, if
T ( ( 1 ε ) ν 1 + ε κ 1 ) ( 1 ε ) T ( ν 1 ) + ε T ( κ 1 ) ,
for every value of ε [ 0 , 1 ] and ν 1 , κ 1 I .
Many inequalities have been proposed in the literature using convex mappings and their different classes. For more details, see [2,3,4,5]. The following double inequality is commonly called Hadamard’s inequality in the literature:
Theorem 1
([1]). Let T : I R be a convex mapping. Then
T ν 1 + κ 1 2 1 κ 1 ν 1 ν 1 κ 1 T ( δ ) d δ T ( ν 1 ) + T ( κ 1 ) 2 ,
for all ν 1 , κ 1 I , and ν 1 < κ 1 .
It was introduced by Hermite in 1881 (see [6]) and Hadamard in 1893 (see [7]). The inequality (2) also holds in reverse direction if we take T is a concave mapping. In scientific research, the most prominent Simpson’s type of inequality is described as follows:
Theorem 2
([8]). Consider a four time continuously differentiable mapping T : [ ν 1 , κ 1 ] R . Then,
1 6 T ( ν 1 ) + 4 T ν 1 + κ 1 2 + T ( κ 1 ) 1 κ 1 ν 1 ν 1 κ 1 T ( δ ) d δ ( κ 1 ν 1 ) 4 2880 T 4 ,
where T 4 = s u p δ [ ν 1 , κ 1 ] | T 4 ( δ ) | < .
The Newton’s inequality, which is based on the four-point closed formula, is described as follows:
Theorem 3
([9]). Let T : [ ν 1 , κ 1 ] R be four times continuously differentiable mapping. Then,
1 8 T ( ν 1 ) + 3 T 2 ν 1 + κ 1 3 + 3 T ν 1 + 2 κ 1 3 + T ( κ 1 ) 1 κ 1 ν 1 ν 1 κ 1 T ( δ ) d δ ( κ 1 ν 1 ) 4 6480 T 4 ,
where T 4 = s u p δ [ ν 1 , κ 1 ] | T 4 ( δ ) | < .
The Maclaurin method does not correlate with any boundary points in the quadrature method. It is utilized to alleviate the drawbacks of Simpson’s approach. It is formulated as follows:
Theorem 4
([9]). Let T : [ ν 1 , κ 1 ] R be a four times continuously differentiable mapping on [ ν 1 , κ 1 ] and T 4 = s u p δ [ ν 1 , κ 1 ] | T 4 ( δ ) | < . Then,
1 8 3 T 5 ν 1 + κ 1 6 + 2 T ν 1 + κ 1 2 + 3 T ν 1 + 5 κ 1 6 1 κ 1 ν 1 ν 1 κ 1 T ( δ ) d δ 7 ( κ 1 ν 1 ) 4 51840 T 4 .
Now, we present Boole’s formula type inequality, which is widely recognized in the literature. This formula executes a polynomial of degree four to estimate the integral over five points.
Theorem 5
([9]). Suppose that T : [ ν 1 , κ 1 ] R be a six times continuously differentiable mapping on ( ν 1 , κ 1 ) . Then,
1 90 7 ( T ( ν 1 + T ( κ 1 ) ) ) + 32 T 3 ν 1 + κ 1 4 + T ν 1 + 3 κ 1 4 + 12 T ν 1 + κ 1 2 1 κ 1 ν 1 ν 1 κ 1 T ( δ ) d δ ( κ 1 ν 1 ) 6 1935360 T 6 ,
where T 6 = s u p δ [ ν 1 , κ 1 ] | T 6 ( δ ) | < .
The exploration of non-differentiable mappings along with local attributes is a fascinating domain of mathematical sciences. Various strategies have been discussed in literature, including quantum calculus and setups within the fractal domain. Working on that problem, Yang defined λ level sets of real numbers and the operations over them. Also, he published two monographs, in which fractal calculus and local fractional functional analysis are discussed in detail. Over the years, mathematicians dealing with fractal geometries have been taking the benefit of it. Fractals are those sets whose Hausdorff dimension is greater than topological dimensions, and Hausdorff dimension measures the complexity.
Now, we recall the basics of Yang calculus.

Yang Local Fractional Calculus

In this section, we recall the basic concepts of local fractional calculus which are essential to our main findings given in [10].
  • Z λ : = { ± 0 λ , ± 1 λ , ± 2 λ , } ,
  • Q λ : = { v λ = r 1 r 2 λ : r 1 , r 2 Z , r 2 0 } ,
  • Q λ : = { v λ r 1 r 2 λ : r 1 , r 2 Z , r 2 0 } ,
  • R λ : = Q λ Q λ .
Also, the multiplication and addition are defined as
c λ d λ = c λ d λ : = ( c d ) λ and c λ + d λ : = ( c + d ) λ
and both c λ d λ , c λ + d λ R λ . Clearly, ( R λ , + , ) is a field.
The local fractional continuity is defined as follows.
Definition 2
([10]). The local fractional derivative of T ( δ ) of order λ at δ = δ 0 is described as
T λ ( δ ) = δ 0 D δ λ T ( δ ) = d λ T ( δ ) ( d δ ) λ δ = δ 0 = lim δ δ 0 λ ( T ( δ ) T ( δ 0 ) ) ( δ δ 0 ) λ .
where λ ( T ( δ ) T ( δ 0 ) ) = Γ ( 1 + λ ) ( T ( δ ) T ( δ 0 ) ) . Let s N and T λ ( δ ) = D δ λ T ( δ ) . If
T ( s + 1 ) λ ( δ ) = D δ λ T ( δ ) · D δ λ T ( δ ) D δ λ T ( δ ) ( s + 1 ) times exists for any δ [ ν 1 , κ 1 ] ,
then we say T D ( s + 1 ) λ .
We now present the local anti-derivative operator of T ( δ ) C λ ( ν 1 , κ 1 ) .
Definition 3
([10]). Let = { δ 0 , δ 1 , δ 2 , , δ i , δ i + 1 , , δ σ } , where σ N , be a division of [ ν 1 , κ 1 ] such that δ 0 < δ 1 < δ 2 < < δ σ . Then the local fractional integral of T on [ ν 1 , κ 1 ] is defined by
I κ 1 λ ν 1 T ( δ ) = 1 Γ ( 1 + λ ) ν 1 κ 1 Ψ ( δ ) ( d δ ) λ = 1 Γ ( 1 + λ ) lim δ i 0 i = 1 σ T ( δ i ) ( δ ) λ ,
where δ i = δ i + 1 δ i for i = 1 , 2 , 3 σ . The space of all local integrable mappings is represented by I δ λ [ ν 1 , κ 1 ] .
From the above, one can easily conclude that I κ 1 λ ν 1 T ( δ ) = 0 if ν 1 = κ 1 and I κ 1 λ ν 1 T ( δ ) = I ν 1 λ κ 1 T ( δ ) when ν 1 < κ 1 . Now, we present some significant findings about integration and local derivatives that will help us in our future work.
Lemma 1.
Consider 0 < λ 1 . If T ( δ ) = r λ ( δ ) C λ [ ν 1 , κ 1 ] . Then
I κ 1 λ ν 1 g ( δ ) = r ( κ 1 ) r ( ν 1 ) .
Furthermore, we have
d λ δ s λ ( d δ ) λ = Γ ( 1 + s λ ) Γ ( 1 + ( s 1 ) λ ) δ ( s 1 ) λ
and
1 Γ ( 1 + λ ) ν 1 κ 1 δ s λ ( d δ ) λ = Γ ( 1 + s λ ) Γ ( 1 + ( s + 1 ) λ ) κ 1 ( s + 1 ) λ ν 1 ( s + 1 ) λ .
The fractal calculus has been utilized to study new mathematical inequalities. Motivated by ongoing research in this area, Mo et al. in [11] generalized the concept of convexity in fractal space as follows.
Definition 4.
A mapping T : [ ν 1 , κ 1 ] R λ is known-as generalized convex in fractal space if
T ( ε ν 1 + ( 1 ε ) κ 1 ) ε λ T ( ν 1 ) + ( 1 ε ) λ T ( κ 1 )
for all ε [ 0 , 1 ] and 0 < λ 1 .
The trapezoidal inequality in fractal space is given next. For more details, see [11].
Theorem 6.
Let T : [ ν 1 , κ 1 ] R λ be a generalized convex mapping, where 0 < λ 1 . Then,
T ν 1 + κ 1 2 Γ ( 1 + λ ) ( κ 1 ν 1 ) λ I κ 1 λ ν 1 T ( δ ) T ( ν 1 ) + T ( κ 1 ) 2 λ .
Sarikaya and Budak [12] utilized the generalized convexity within the framework of Yang local calculus to investigate the weighted Hadamard’s type inequalities. In [13,14], the authors considered the local fractional operators along with generalized convexity to study the various error inequalities. In [15], Erden and Sarikaya explored the mean value theorem and some Pompeiu-like inequalities leveraging the concept of Yang calculus. Almutairi and Kilicman [16] investigated the trapezium-type inequalities through generalized Breckner convexity.
Lou et al. [17] discussed the fractal weighted Hadamard’s inequalities through generalized convexity and also delivered the utility of results. In [18], the authors focused on the applications of Hermite–Hadamard-like inequalities in signal processing. Butt et al. [19] proved the Jensen’s type in inequalities in fractal space and developed some new counterparts of Hadamard–Mercer-like inequalities. Luo et al. [20] worked on improved Hölder’s inequalities and their applications to construct some Simpson-like inequalities. Alsharari et al. [21] bridged the fractional operator and local Yang calculus to approximate the error bounds of Simpson’s rule. Meftah et al. [22] looked at error analysis of Milne rules considering generalized convexity within fractal calculus. Liu et al. [23] presented upper bounds of fractional Maclaurin’s inequality and presented the applications to nonlinear analysis. In [24], the authors proved some new advancements in Ostrowski’s inequalities relying on Mercer inequality for fractal Breckner convexity. For more details see [25,26,27,28].
Motivation and Structure of Study: Over the years authors have tried several approaches to assess the bounds of local fractional operators, the average value of fractal mappings, and the error analysis of Newton-Cotes schemes within the context of Yang fractal calculus. It is a common fact that approximations through higher-order polynomials show minimum error. The Generalized Weddle’s procedure is superior due to better accuracy and minimum error as compared to Simpson’s and Newton’s procedures. Here are the few questions that elaborate on the need for the study.
  • How can the error terms of Weddle’s rule be established for local fractional differentiable mappings? Mathematically,
    1 ( 20 ) λ T ( ν 1 ) + 5 λ T 5 ν 1 + κ 1 6 + T 2 ν 1 + κ 1 3 + 6 λ T ν 1 + κ 1 2 + T ν 1 + 2 κ 1 3 + 5 λ T ν 1 + 5 κ 1 6 + T ( κ 1 ) Γ ( 1 + λ ) ( κ 1 ν 1 ) λ I κ 1 ν 1 T ( δ ) ?
  • How different generalized classes of mappings can be used to evaluate the bounds of the above inequality?
  • What are the applications of the proposed bounds?
  • How can the bounds of several local fractional integrals be found incorporated with the proposed inequalities?
To address these problems, we have organized this study into multiple steps. First, we discuss the research problem background and fundamental concepts to carry out the further proceedings. Our approach includes the generation of error bounds through identity. In the following context, we will construct a fractal identity, which will later on be helpful to estimate our main findings. At the end of each result, we provide its 2D and 3D graphical validations. Lastly, some applicable analysis will be discussed. To the extent of our knowledge, this approach is new and novel for discussing error bounds of fractal Newton–Cotes schemes.

2. Fractal Estimates of Weddle’s Inequality

The space of fractal integrable mappings is denoted by L [ ν 1 , κ 1 ] .
Lemma 2.
If T : [ ν 1 , κ 1 ] R is a local differentiable mapping and T λ L [ ν 1 , κ 1 ] , then
W ( ν 1 , κ 1 ) Γ ( 1 + λ ) ( κ 1 ν 1 ) λ I κ 1 ν 1 T ( δ ) = κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 ( ε ) T λ ( ( 1 ε ) ν 1 + ε κ 1 ) T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ ,
where
( ε ) = ε 1 20 λ , ε 0 , 1 6 , ε 6 20 λ , ε 1 6 , 1 3 , ε 7 20 λ , ε 1 3 , 1 2 , ε 13 20 λ , ε 1 2 , 2 3 , ε 14 20 λ , ε 2 3 , 5 6 , ε 19 20 λ , ε 5 6 , 1 ,
and
W ( ν 1 , κ 1 ) = 1 ( 20 ) λ T ( ν 1 ) + 5 λ T 5 ν 1 + κ 1 6 + T 2 ν 1 + κ 1 3 + 6 λ T ν 1 + κ 1 2 + T ν 1 + 2 κ 1 3 + 5 λ T ν 1 + 5 κ 1 6 + T ( κ 1 ) .
Proof. 
From the definition of ( ε ) , we have
1 Γ ( 1 + λ ) 0 1 ( ε ) T λ ( ( 1 ε ) ν 1 + ε κ 1 ) T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ = 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 0 1 6 1 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 6 1 3 6 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 3 1 2 7 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 2 2 3 13 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 2 3 5 6 14 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 5 6 1 19 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ = i = 0 12 I i .
We can use integration by parts to solve these integrals
I 1 = 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ = 1 ( κ 1 ν 1 ) λ 7 60 λ T 5 ν 1 + κ 1 6 + 1 20 λ T ( ν 1 ) Γ ( 1 + λ ) ( κ 1 ν 1 ) 2 λ ν 1 I 5 ν 1 + κ 1 6 T ( δ ) , I 2 = 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ = 1 ( κ 1 ν 1 ) λ 1 30 λ T 2 ν 1 + κ 1 3 + 2 15 λ T 5 ν 1 + κ 1 6 Γ ( 1 + λ ) ( κ 1 ν 1 ) 2 λ I 2 ν 1 + κ 1 3 5 ν 1 + κ 1 6 T ( δ ) , I 3 = 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ = 1 ( κ 1 ν 1 ) λ 3 20 λ T ν 1 + κ 1 2 + 1 60 λ T 2 ν 1 + κ 1 3 Γ ( 1 + λ ) ( κ 1 ν 1 ) 2 λ I ν 1 + κ 1 2 2 ν 1 + κ 1 3 T ( δ ) , I 4 = 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ = 1 ( κ 1 ν 1 ) λ 1 60 λ T ν 1 + 2 κ 1 3 + 3 20 λ T ν 1 + κ 1 2 Γ ( 1 + λ ) ( κ 1 ν 1 ) 2 λ I ν 1 + 2 κ 1 3 ν 1 + κ 1 2 T ( δ ) , I 5 = 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ = 1 ( κ 1 ν 1 ) λ 2 15 λ T ν 1 + 5 κ 1 6 + 1 30 λ T ν 1 + 2 κ 1 3 Γ ( 1 + λ ) ( κ 1 ν 1 ) 2 λ I ν 1 + 5 κ 1 6 ν 1 + 2 κ 1 3 T ( δ ) , I 6 = 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ = 1 ( κ 1 ν 1 ) λ 1 20 λ T ( κ 1 ) + 7 60 λ T ν 1 + 5 κ 1 6 Γ ( 1 + λ ) ( κ 1 ν 1 ) 2 λ I κ 1 ν 1 + 5 κ 1 6 T ( δ ) , I 7 = 1 Γ ( 1 + λ ) 0 1 6 1 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ = 1 ( κ 1 ν 1 ) λ 7 60 λ T ν 1 + 5 κ 1 6 + 1 20 λ T ( κ 1 ) Γ ( 1 + λ ) ( κ 1 ν 1 ) 2 λ I κ 1 5 ν 1 + κ 1 6 T ( δ ) , I 8 = 1 Γ ( 1 + λ ) 1 6 1 3 6 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ = 1 ( κ 1 ν 1 ) λ 2 15 λ T ν 1 + 5 κ 1 6 + 1 30 λ T ν 1 + 2 κ 1 3 Γ ( 1 + λ ) ( κ 1 ν 1 ) 2 λ I ν 1 + 5 κ 1 6 ν 1 + 2 κ 1 3 T ( δ ) , I 9 = 1 Γ ( 1 + λ ) 1 3 1 2 7 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ = 1 ( κ 1 ν 1 ) λ 1 60 λ T ν 1 + 2 κ 1 3 + 3 20 λ T ν 1 + κ 1 2 Γ ( 1 + λ ) ( κ 1 ν 1 ) 2 λ I ν 1 + 2 κ 1 3 ν 1 + κ 1 2 T ( δ ) , I 10 = 1 Γ ( 1 + λ ) 1 2 2 3 13 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ = 1 ( κ 1 ν 1 ) λ 3 20 λ T ν 1 + κ 1 2 + 1 60 λ T 2 ν 1 + κ 1 3 Γ ( 1 + λ ) ( κ 1 ν 1 ) 2 λ I ν 1 + κ 1 2 2 ν 1 + κ 1 3 T ( δ ) , I 11 = 1 Γ ( 1 + λ ) 2 3 5 6 14 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ = 1 ( κ 1 ν 1 ) λ 1 30 λ T 2 ν 1 + κ 1 3 + 2 15 λ T 5 ν 1 + κ 1 6 Γ ( 1 + λ ) ( κ 1 ν 1 ) 2 λ I 2 ν 1 + κ 1 3 5 ν 1 + κ 1 6 T ( δ ) , I 12 = 1 Γ ( 1 + λ ) 5 6 1 19 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ = 1 ( κ 1 ν 1 ) λ 1 20 λ T ( ν 1 ) + 7 60 λ T 5 ν 1 + κ 1 6 Γ ( 1 + λ ) ( κ 1 ν 1 ) 2 λ ν 1 I 5 ν 1 + κ 1 6 T ( δ ) .
Now, by taking i = 0 12 I i and after simple computations we get (5). It ends the proof. □
Now we establish the first bound for Weddle’s rule.
Theorem 7.
Suppose that all constraints of lemma 2 are fulfilled. If | T λ | is a generalized convex mapping, then
W ( ν 1 , κ 1 ) Γ ( 1 + λ ) ( κ 1 ν 1 ) λ I κ 1 ν 1 T ( δ ) ( κ 1 ν 1 ) λ Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 4 75 λ + Γ ( 1 + 2 λ ) Γ ( 1 + 3 λ ) 1 150 λ | T λ ( ν 1 ) | + Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 14 225 λ Γ ( 1 + 2 λ ) Γ ( 1 + 3 λ ) 1 150 λ | T λ ( κ 1 ) | .
Proof. 
Through Lemma 2 and generalized convexity of | T λ | , we have
W ( ν 1 , κ 1 ) Γ ( 1 + λ ) ( κ 1 ν 1 ) λ I κ 1 a T ( δ ) κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 ( ε ) T λ ( ( 1 ε ) ν 1 + ε κ 1 ) T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 0 1 6 1 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 6 1 3 6 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 3 1 2 7 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 2 2 3 13 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 2 3 5 6 14 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 5 6 1 19 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ [ ( 1 ε ) λ | T λ ( ν 1 ) | + ε λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ [ ( 1 ε ) λ | T λ ( ν 1 ) | + ε λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ [ ( 1 ε ) λ | T λ ( ν 1 ) | + ε λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ [ ( 1 ε ) λ | T λ ( ν 1 ) | + ε λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ [ ( 1 ε ) λ | T λ ( ν 1 ) | + ε λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ [ ( 1 ε ) λ | T λ ( ν 1 ) | + ε λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 0 1 6 1 20 ε λ [ ε λ | T λ ( ν 1 ) | + ( 1 ε ) λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 1 6 1 3 6 20 ε λ [ ε λ | T λ ( ν 1 ) | + ( 1 ε ) λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 1 3 1 2 7 20 ε λ [ ε λ | T λ ( ν 1 ) | + ( 1 ε ) λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 1 2 2 3 13 20 ε λ [ ε λ | T λ ( ν 1 ) | + ( 1 ε ) λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 2 3 5 6 14 20 ε λ [ ε λ | T λ ( ν 1 ) | + ( 1 ε ) λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 5 6 1 19 20 ε λ [ ε λ | T λ ( ν 1 ) | + ( 1 ε ) λ | T λ ( κ 1 ) | ] ( d ε ) λ = κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 20 1 20 ε λ [ ( 1 ε ) λ | T λ ( ν 1 ) | + ε λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 1 20 1 6 ε 1 20 λ [ ( 1 ε ) λ | T λ ( ν 1 ) | + ε λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 1 6 6 20 6 20 ε λ [ ( 1 ε ) λ | T λ ( ν 1 ) | + ε λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 6 20 1 3 ε 6 20 λ [ ( 1 ε ) λ | T λ ( ν 1 ) | + ε λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 1 3 7 20 7 20 ε λ [ ( 1 ε ) λ | T λ ( ν 1 ) | + ε λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 7 20 1 2 ε 7 20 λ [ ( 1 ε ) λ | T λ ( ν 1 ) | + ε λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 1 2 13 20 13 20 ε λ [ ( 1 ε ) λ | T λ ( ν 1 ) | + ε λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 13 20 2 3 ε 13 20 λ [ ( 1 ε ) λ | T λ ( ν 1 ) | + ε λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 2 3 14 20 14 20 ε λ [ ( 1 ε ) λ | T λ ( ν 1 ) | + ε λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 14 20 5 6 ε 14 20 λ [ ( 1 ε ) λ | T λ ( ν 1 ) | + ε λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 5 6 19 20 19 20 ε λ [ ( 1 ε ) λ | T λ ( ν 1 ) | + ε λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 19 20 1 ε 19 20 λ [ ( 1 ε ) λ | T λ ( ν 1 ) | + ε λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 0 1 20 1 20 ε λ [ ε λ | T λ ( ν 1 ) | + ( 1 ε ) λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 1 20 1 6 ε 1 20 λ [ ε λ | T λ ( ν 1 ) | + ( 1 ε ) λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 1 6 6 20 6 20 ε λ [ ε λ | T λ ( ν 1 ) | + ( 1 ε ) λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 6 20 1 3 ε 6 20 λ [ ε λ | T λ ( ν 1 ) | + ( 1 ε ) λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 1 3 7 20 7 20 ε λ [ ε λ | T λ ( ν 1 ) | + ( 1 ε ) λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 7 20 1 2 ε 7 20 λ [ ε λ | T λ ( ν 1 ) | + ( 1 ε ) λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 1 2 13 20 13 20 ε λ [ ε λ | T λ ( ν 1 ) | + ( 1 ε ) λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 13 20 2 3 ε 13 20 λ [ ε λ | T λ ( ν 1 ) | + ( 1 ε ) λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 2 3 14 20 14 20 ε λ [ ε λ | T λ ( ν 1 ) | + ( 1 ε ) λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 14 20 5 6 ε 14 20 λ [ ε λ | T λ ( ν 1 ) | + ( 1 ε ) λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 5 6 19 20 19 20 ε λ [ ε λ | T λ ( ν 1 ) | + ( 1 ε ) λ | T λ ( κ 1 ) | ] ( d ε ) λ + 1 Γ ( 1 + λ ) 19 20 1 ε 19 20 λ [ ε λ | T λ ( ν 1 ) | + ( 1 ε ) λ | T λ ( κ 1 ) | ] ( d ε ) λ .
Finally, we get our intended inequality. □
Example 1.
If we take T ( δ ) = δ σ λ , ν 1 = 0 and κ 1 = 3 in Theorem 7. Then, 2D and 3D validations of Theorem 7 are
Figure 1a,b illustrate the comparison between sides of Theorem 7.
Now we give other bounds for Weddle’s rule by utilizing the well known Hölder’s inequality.
Theorem 8.
Suppose that all constraints of Lemma 2 are fulfilled. If | T λ | r 2 is a generalized convex mapping, then
W ( ν 1 , κ 1 ) Γ ( 1 + λ ) ( κ 1 ν 1 ) λ I κ 1 a T ( δ ) ( κ 1 ν 1 ) λ Γ ( 1 + λ r 1 ) Γ ( 1 + ( r 1 + 1 ) λ ) 7 60 ( r 1 + 1 ) λ + 1 20 ( r 1 + 1 ) λ 1 r 1 × Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 11 36 λ | T λ ( ν 1 ) | r 2 + 1 36 λ | T λ ( κ 1 ) | r 2 1 r 2 + Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 1 36 λ | T λ ( ν 1 ) | r 2 + 11 36 λ | T λ ( κ 1 ) | r 2 1 r 2 + Γ ( 1 + λ r 1 ) Γ ( 1 + ( r 1 + 1 ) λ ) 1 30 ( r 1 + 1 ) λ + 2 15 ( r 1 + 1 ) λ 1 r 1 × Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 1 4 λ | T λ ( ν 1 ) | r 2 + 1 12 λ | T λ ( κ 1 ) | r 2 1 r 2 + Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 1 12 λ | T λ ( ν 1 ) | r 2 + 1 4 λ | T λ ( κ 1 ) | r 2 1 r 2 + Γ ( 1 + λ r 1 ) Γ ( 1 + ( r 1 + 1 ) λ ) 3 20 ( r 1 + 1 ) λ + 1 60 ( r 1 + 1 ) λ 1 r 1 × Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 7 36 λ | T λ ( ν 1 ) | r 2 + 5 36 λ | T λ ( κ 1 ) | r 2 1 r 2 + Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 5 36 λ | T λ ( ν 1 ) | r 2 + 7 36 λ | T λ ( κ 1 ) | r 2 1 r 2 ,
where 1 r 1 + 1 r 2 = 1 .
Proof. 
Through Lemma 2, Hölder’s inequality and generalized convexity of | T λ | , we have
W ( ν 1 , κ 1 ) Γ ( 1 + λ ) ( κ 1 ν 1 ) λ I κ 1 a T ( δ ) κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 ( ε ) T λ ( ( 1 ε ) ν 1 + ε κ 1 ) T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 0 1 6 1 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 6 1 3 6 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 3 1 2 7 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 2 2 3 13 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 2 3 5 6 14 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 5 6 1 19 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ r 1 ( d ε ) λ 1 r 1 1 Γ ( 1 + λ ) 0 1 6 | T λ ( ( 1 ε ) ν 1 + ε κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ r 1 ( d ε ) λ 1 r 1 1 Γ ( 1 + λ ) 1 6 1 3 | T λ ( ( 1 ε ) ν 1 + ε κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ r 1 ( d ε ) λ 1 r 1 1 Γ ( 1 + λ ) 1 3 1 2 | T λ ( ( 1 ε ) ν 1 + ε κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ r 1 ( d ε ) λ 1 r 1 1 Γ ( 1 + λ ) 1 3 1 2 | T λ ( ( 1 ε ) ν 1 + ε κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ r 1 ( d ε ) λ 1 r 1 1 Γ ( 1 + λ ) 2 3 5 6 | T λ ( ( 1 ε ) ν 1 + ε κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ r 1 ( d ε ) λ 1 r 1 1 Γ ( 1 + λ ) 5 6 1 | T λ ( ( 1 ε ) ν 1 + ε κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ r 1 ( d ε ) λ 1 r 1 1 Γ ( 1 + λ ) 0 1 6 | T λ ( ε ν 1 + ( 1 ε ) κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ r 1 ( d ε ) λ 1 r 1 1 Γ ( 1 + λ ) 1 6 1 3 | T λ ( ε ν 1 + ( 1 ε ) κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ r 1 ( d ε ) λ 1 r 1 1 Γ ( 1 + λ ) 1 3 1 2 | T λ ( ε ν 1 + ( 1 ε ) κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ r 1 ( d ε ) λ 1 r 1 1 Γ ( 1 + λ ) 1 3 1 2 | T λ ( ε ν 1 + ( 1 ε ) κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ r 1 ( d ε ) λ 1 r 1 1 Γ ( 1 + λ ) 2 3 5 6 | T λ ( ε ν 1 + ( 1 ε ) κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ r 1 ( d ε ) λ 1 r 1 1 Γ ( 1 + λ ) 5 6 1 | T λ ( ε ν 1 + ( 1 ε ) κ 1 ) | r 2 ( d ε ) λ 1 r 2 κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ r 1 ( d ε ) λ 1 r 1 × 1 Γ ( 1 + λ ) 0 1 6 [ ( 1 ε ) λ | T λ ( ν 1 ) | r 2 + ε λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ r 1 ( d ε ) λ 1 r 1 × 1 Γ ( 1 + λ ) 1 6 1 3 [ ( 1 ε ) λ | T λ ( ν 1 ) | r 2 + ε λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ r 1 ( d ε ) λ 1 r 1 × 1 Γ ( 1 + λ ) 1 3 1 2 [ ( 1 ε ) λ | T λ ( ν 1 ) | r 2 + ε λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ r 1 ( d ε ) λ 1 r 1 × 1 Γ ( 1 + λ ) 1 3 1 2 [ ( 1 ε ) λ | T λ ( ν 1 ) | r 2 + ε λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ r 1 ( d ε ) λ 1 r 1 × 1 Γ ( 1 + λ ) 2 3 5 6 [ ( 1 ε ) λ | T λ ( ν 1 ) | r 2 + ε λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ r 1 ( d ε ) λ 1 r 1 × 1 Γ ( 1 + λ ) 5 6 1 [ ( 1 ε ) λ | T λ ( ν 1 ) | r 2 + ε λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ r 1 ( d ε ) λ 1 r 1 × 1 Γ ( 1 + λ ) 0 1 6 [ ε λ | T λ ( ν 1 ) | r 2 + ( 1 ε ) λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ r 1 ( d ε ) λ 1 r 1 × 1 Γ ( 1 + λ ) 1 6 1 3 [ ε λ | T λ ( ν 1 ) | r 2 + ( 1 ε ) λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ r 1 ( d ε ) λ 1 r 1 × 1 Γ ( 1 + λ ) 1 3 1 2 [ ε λ | T λ ( ν 1 ) | r 2 + ( 1 ε ) λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ r 1 ( d ε ) λ 1 r 1 × 1 Γ ( 1 + λ ) 1 3 1 2 [ ε λ | T λ ( ν 1 ) | r 2 + ( 1 ε ) λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ r 1 ( d ε ) λ 1 r 1 × 1 Γ ( 1 + λ ) 2 3 5 6 [ ε λ | T λ ( ν 1 ) | r 2 + ( 1 ε ) λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ r 1 ( d ε ) λ 1 r 1 × 1 Γ ( 1 + λ ) 5 6 1 [ ε λ | T λ ( ν 1 ) | r 2 + ( 1 ε ) λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 κ 1 ν 1 2 λ Γ ( 1 + λ r 1 ) Γ ( 1 + ( r 1 + 1 ) λ ) 7 60 ( r 1 + 1 ) λ + 1 20 ( r 1 + 1 ) λ 1 r 1 × Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 11 36 λ | T λ ( ν 1 ) | r 2 + 1 36 λ | T λ ( κ 1 ) | r 2 1 r 2 + Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 1 36 λ | T λ ( ν 1 ) | r 2 + 11 36 λ | T λ ( κ 1 ) | r 2 1 r 2 + Γ ( 1 + λ r 1 ) Γ ( 1 + ( r 1 + 1 ) λ ) 1 30 ( r 1 + 1 ) λ + 2 15 ( r 1 + 1 ) λ 1 r 1 × Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 1 4 λ | T λ ( ν 1 ) | r 2 + 1 12 λ | T λ ( κ 1 ) | r 2 1 r 2 + Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 1 12 λ | T λ ( ν 1 ) | r 2 + 1 4 λ | T λ ( κ 1 ) | r 2 1 r 2 + Γ ( 1 + λ r 1 ) Γ ( 1 + ( r 1 + 1 ) λ ) 3 20 ( r 1 + 1 ) λ + 1 60 ( r 1 + 1 ) λ 1 r 1 × Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 7 36 λ | T λ ( ν 1 ) | r 2 + 5 36 λ | T λ ( κ 1 ) | r 2 1 r 2 + Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 5 36 λ | T λ ( ν 1 ) | r 2 + 7 36 λ | T λ ( κ 1 ) | r 2 1 r 2 + Γ ( 1 + λ r 1 ) Γ ( 1 + ( r 1 + 1 ) λ ) 1 60 ( r 1 + 1 ) λ + 3 20 ( r 1 + 1 ) λ 1 r 1 × Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 5 36 λ | T λ ( ν 1 ) | r 2 + 7 36 λ | T λ ( κ 1 ) | r 2 1 r 2 + Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 7 36 λ | T λ ( ν 1 ) | r 2 + 5 36 λ | T λ ( κ 1 ) | r 2 1 r 2 + Γ ( 1 + λ r 1 ) Γ ( 1 + ( r 1 + 1 ) λ ) 2 15 ( r 1 + 1 ) λ + 1 30 ( r 1 + 1 ) λ 1 r 1 × Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 1 12 λ | T λ ( ν 1 ) | r 2 + 1 4 λ | T λ ( κ 1 ) | r 2 1 r 2 q u a d + Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 1 4 λ | T λ ( ν 1 ) | r 2 + 1 12 λ | T λ ( κ 1 ) | r 2 1 r 2 + Γ ( 1 + λ r 1 ) Γ ( 1 + ( r 1 + 1 ) λ ) 1 20 ( r 1 + 1 ) λ + 7 60 ( r 1 + 1 ) λ 1 r 1 × Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 1 36 λ | T λ ( ν 1 ) | r 2 + 11 36 λ | T λ ( κ 1 ) | r 2 1 r 2 + Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 11 36 λ | T λ ( ν 1 ) | r 2 + 1 36 λ | T λ ( κ 1 ) | r 2 1 r 2 .
This ends the proof. □
Example 2.
If we take T ( δ ) = δ σ λ , ν 1 = 0 and κ 1 = 3 in Theorem 8. Then, 2D and 3D validations of Theorem 8 are
Figure 2a,b illustrate the comparison between sides of Theorem 8.
Next, we construct a new estimate of Weddle’s inequality.
Theorem 9.
Suppose that all constraints of lemma 2 are fulfilled. If | T λ | r 2 is a generalized convex mapping, then
W ( ν 1 , κ 1 ) Γ ( 1 + λ ) ( κ 1 ν 1 ) λ I κ 1 a T ( δ ) κ 1 ν 1 2 λ Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 29 1800 λ 1 1 r 2 × A 1 | T λ ( ν 1 ) | r 2 + A 2 | T λ ( κ 1 ) | r 2 1 r 2 + A 2 | T λ ( ν 1 ) | r 2 + A 1 | T λ ( κ 1 ) | r 2 1 r 2 + Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 17 900 λ 1 1 r 2 × A 3 | T λ ( ν 1 ) | r 2 + A 4 | T λ ( κ 1 ) | r 2 1 r 2 + A 4 | T λ ( ν 1 ) | r 2 + A 3 | T λ ( κ 1 ) | r 2 1 r 2 + Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 41 1800 λ 1 1 r 2 × A 5 | T λ ( ν 1 ) | r 2 + A 6 | T λ ( κ 1 ) | r 2 1 r 2 + A 6 | T λ ( ν 1 ) | r 2 + A 5 | T λ ( κ 1 ) | r 2 1 r 2 + Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 41 1800 λ 1 1 r 2 × A 7 | T λ ( ν 1 ) | r 2 + A 8 | T λ ( κ 1 ) | r 2 1 r 2 + A 8 | T λ ( ν 1 ) | r 2 + A 7 | T λ ( κ 1 ) | r 2 1 r 2 + Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 17 900 λ 1 1 r 2 × A 9 | T λ ( ν 1 ) | r 2 + A 10 | T λ ( κ 1 ) | r 2 1 r 2 + A 10 | T λ ( ν 1 ) | r 2 + A 9 | T λ ( κ 1 ) | r 2 1 r 2 + Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 29 1800 λ 1 1 r 2 × A 11 | T λ ( ν 1 ) | r 2 + A 12 | T λ ( κ 1 ) | r 2 1 r 2 + A 12 | T λ ( ν 1 ) | r 2 + A 11 | T λ ( κ 1 ) | r 2 1 r 2 ,
r 2 1 , where
  • A 1 = Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 69 4000 λ Γ ( 1 + 2 λ ) Γ ( 1 + 3 λ ) 473 108000 λ ,
  • A 2 = Γ ( 1 + 2 λ ) Γ ( 1 + 3 λ ) 473 108000 λ Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 41 36000 λ ,
  • A 3 = Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 59 9000 λ + Γ ( 1 + 2 λ ) Γ ( 1 + 3 λ ) 37 3000 λ ,
  • A 4 = Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 37 3000 λ Γ ( 1 + 2 λ ) Γ ( 1 + 3 λ ) 37 3000 λ ,
  • A 5 = Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 761 12000 λ Γ ( 1 + 2 λ ) Γ ( 1 + 3 λ ) 8239 108000 λ ,
  • A 6 = Γ ( 1 + 2 λ ) Γ ( 1 + 3 λ ) 8239 108000 λ Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 1463 36000 λ ,
  • A 7 = Γ ( 1 + 2 λ ) Γ ( 1 + 3 λ ) 13819 108000 λ Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 901 12000 λ ,
  • A 8 = Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 3523 36000 λ Γ ( 1 + 2 λ ) Γ ( 1 + 3 λ ) 13819 108000 λ ,
  • A 9 = Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 1171 9000 λ Γ ( 1 + 2 λ ) Γ ( 1 + 3 λ ) 189 1000 λ ,
  • A 10 = Γ ( 1 + 2 λ ) Γ ( 1 + 3 λ ) 189 1000 λ Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 1001 9000 λ ,
  • A 11 = Γ ( 1 + 2 λ ) Γ ( 1 + 3 λ ) 14693 108000 λ Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 1067 12000 λ ,
  • A 12 = Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 3781 36000 λ Γ ( 1 + 2 λ ) Γ ( 1 + 3 λ ) 14693 108000 λ .
Proof. 
Through Lemma 2, power mean inequality and generalized convexity of | T λ | r 2 , we have
W ( ν 1 , κ 1 ) Γ ( 1 + λ ) ( κ 1 ν 1 ) λ I κ 1 a T ( δ ) κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 ( ε ) T λ ( ( 1 ε ) ν 1 + ε κ 1 ) T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 0 1 6 1 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 6 1 3 6 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 3 1 2 7 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 2 2 3 13 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 2 3 5 6 14 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 5 6 1 19 20 ε λ T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ | T λ ( ( 1 ε ) ν 1 + ε κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ | T λ ( ( 1 ε ) ν 1 + ε κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ | T λ ( ( 1 ε ) ν 1 + ε κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ | T λ ( ( 1 ε ) ν 1 + ε κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ | T λ ( ( 1 ε ) ν 1 + ε κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ | T λ ( ( 1 ε ) ν 1 + ε κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ | T λ ( ε ν 1 + ( 1 ε ) κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ | T λ ( ε ν 1 + ( 1 ε ) κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ | T λ ( ε ν 1 + ( 1 ε ) κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ | T λ ( ε ν 1 + ( 1 ε ) κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ | T λ ( ε ν 1 + ( 1 ε ) κ 1 ) | r 2 ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ | T λ ( ε ν 1 + ( 1 ε ) κ 1 ) | r 2 ( d ε ) λ 1 r 2 κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ [ ( 1 ε ) λ | T λ ( ν 1 ) | r 2 + ε λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ [ ( 1 ε ) λ | T λ ( ν 1 ) | r 2 + ε λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ [ ( 1 ε ) λ | T λ ( ν 1 ) | r 2 + ε λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ [ ( 1 ε ) λ | T λ ( ν 1 ) | r 2 + ε λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ [ ( 1 ε ) λ | T λ ( ν 1 ) | r 2 + ε λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ [ ( 1 ε ) λ | T λ ( ν 1 ) | r 2 + ε λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ [ ε λ | T λ ( ν 1 ) | r 2 + ( 1 ε ) λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ [ ε λ | T λ ( ν 1 ) | r 2 + ( 1 ε ) λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ [ ε λ | T λ ( ν 1 ) | r 2 + ( 1 ε ) λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ [ ε λ | T λ ( ν 1 ) | r 2 + ( 1 ε ) λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ [ ε λ | T λ ( ν 1 ) | r 2 + ( 1 ε ) λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 + 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ ( d ε ) λ 1 1 r 2 × 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ [ ε λ | T λ ( ν 1 ) | r 2 + ( 1 ε ) λ | T λ ( κ 1 ) | r 2 ] ( d ε ) λ 1 r 2 .
Hence, the desired relation ia acquired. □
Example 3.
If we take T ( δ ) = δ σ λ , ν 1 = 0 and κ 1 = 3 in Theorem 9. Then, 2D and 3D validations of Theorem 9 are
Figure 3a,b demonstrate the accuracy of Theorem 9.
Now, we develop a new counterpart for the remainder of Weddle’s quadrature rules involving Young’s inequality.
Theorem 10.
Suppose that all constraints of Lemma 2 are fulfilled. If there exist m , M R such that m T λ ( ε ) M for ε [ ν 1 , κ 1 ] . Then
W ( ν 1 , κ 1 ) Γ ( 1 + λ ) ( κ 1 ν 1 ) λ I κ 1 a T ( δ ) ( M m ) ( κ 1 ν 1 ) λ Γ ( 1 + λ ) 2 λ Γ ( 1 + 2 λ ) 26 225 λ ,
M > 0 .
Proof. 
From Lemma 2, we have
W ( ν 1 , κ 1 ) Γ ( 1 + λ ) ( κ 1 ν 1 ) λ I κ 1 ν 1 T ( δ ) = κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 ( ε ) T λ ( ( 1 ε ) ν 1 + ε κ 1 ) T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ = κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) m + M 2 ( d ε ) λ + 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) m + M 2 ( d ε ) λ + 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) m + M 2 ( d ε ) λ + 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) m + M 2 ( d ε ) λ + 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) m + M 2 ( d ε ) λ + 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) m + M 2 ( d ε ) λ + 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ m + M 2 T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ m + M 2 T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ m + M 2 T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ m + M 2 T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ m + M 2 T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ m + M 2 T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ .
By taking the modulus of the above equality, we obtain
W ( ν 1 , κ 1 ) Γ ( 1 + λ ) ( κ 1 ν 1 ) λ I κ 1 ν 1 T ( δ ) κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) m + M 2 ( d ε ) λ + 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) m + M 2 ( d ε ) λ + 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) m + M 2 ( d ε ) λ + 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) m + M 2 ( d ε ) λ + 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) m + M 2 ( d ε ) λ + 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) m + M 2 ( d ε ) λ + 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ m + M 2 T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ m + M 2 T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ m + M 2 T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ m + M 2 T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ m + M 2 T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ m + M 2 T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ .
As m T λ ( ε ) M and ε [ ν 1 , κ 1 ] , we have
T λ ( ε ν 1 + ( 1 ε ) κ 1 ) m + M 2 M m 2 ,
and
m + M 2 T λ ( ε ν 1 + ( 1 ε ) κ 1 ) M m 2 .
From inequality (6) and (7), we obtain
W ( ν 1 , κ 1 ) Γ ( 1 + λ ) ( κ 1 ν 1 ) λ I κ 1 ν 1 T ( δ ) κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ ( d ε ) λ + 1 6 1 3 ε 6 20 λ ( d ε ) λ + 1 3 1 2 ε 7 20 λ ( d ε ) λ + 1 2 2 3 ε 13 20 λ ( d ε ) λ + + 2 3 5 6 ε 14 20 λ ( d ε ) λ + + 5 6 1 ε 19 20 λ ( d ε ) λ ( M m ) .
This yields the required inequality. □
Example 4.
If we take T ( δ ) = δ σ λ , ν 1 = 0 and κ 1 = 3 in Theorem 10. Then, 2D and 3D validations of Theorem 10 are
Figure 4a,b illustrate the comparison between sides of Theorem 10.
Theorem 11.
Suppose that all constraints of lemma 2 are fulfilled and T λ satisfies the L-Lipschit property. Then
W ( ν 1 , κ 1 ) Γ ( 1 + λ ) ( κ 1 ν 1 ) λ I κ 1 a T ( δ ) L ( κ 1 ν 1 ) 2 λ 2 λ Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 109 450 λ Γ ( 1 + 2 λ ) Γ ( 1 + 3 λ ) 43 150 λ .
Proof. 
From Lemma 2, we have
W ( ν 1 , κ 1 ) Γ ( 1 + λ ) ( κ 1 ν 1 ) λ I κ 1 ν 1 T ( δ ) = κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 ( ε ) T λ ( ( 1 ε ) ν 1 + ε κ 1 ) T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ = κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ .
Since T λ is L-Lipschitzian functon, we obtain
W ( ν 1 , κ 1 ) Γ ( 1 + λ ) ( κ 1 ν 1 ) λ I κ 1 ν 1 T ( δ ) κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 6 1 3 ε 6 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 3 1 2 ε 7 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 1 2 2 3 ε 13 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 2 3 5 6 ε 14 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ + 1 Γ ( 1 + λ ) 5 6 1 ε 19 20 λ T λ ( ( 1 ε ) ν 1 + ε κ 1 ) T λ ( ε ν 1 + ( 1 ε ) κ 1 ) ( d ε ) λ κ 1 ν 1 2 λ 1 Γ ( 1 + λ ) 0 1 6 ε 1 20 λ ( d ε ) λ + 1 6 1 3 ε 6 20 λ ( d ε ) λ + 1 3 1 2 ε 7 20 λ ( d ε ) λ L ( 1 2 ε ) λ ( κ 1 ν 1 ) λ + 1 2 2 3 ε 13 20 λ ( d ε ) λ + 2 3 5 6 ε 14 20 λ ( d ε ) λ + + 5 6 1 ε 19 20 λ ( d ε ) λ L ( 2 ε 1 ) λ ( κ 1 ν 1 ) λ .
Finally, we get the required relation. □
Example 5.
If we take T ( δ ) = δ σ λ , ν 1 = 0 and κ 1 = 3 in Theorem 11. Then, 2D and 3D validations of Theorem 11 are
Figure 5a,b illustrate the comparison between sides of Theorem 11.

3. Applications

3.1. Applications to Means

This subsection offers some new inequalities associated with generalized means and their differences. First, we look at binary means.
1.
The generalized arithmetic mean:
A λ ( ν 1 , κ 1 ) = ν 1 λ + κ 1 λ 2 λ = ν 1 + κ 1 2 λ .
2.
The generalized weighted arithmetic mean:
A λ w ( ν 1 , κ 1 ; m 1 , m 2 ) = m 1 λ ν 1 λ + m 2 λ κ 1 λ ( ν 1 + κ 1 ) λ .
3.
The generalized log- r 1 -mean:
L λ , r 1 ( ν 1 , κ 1 ) = Γ ( 1 + r 1 λ ) Γ ( 1 + ( 1 + r 1 ) λ ) κ 1 ( r 1 + 1 ) λ ν 1 ( r 1 + 1 ) λ ( r 1 + 1 ) ( κ 1 ν 1 ) λ 1 r 1 , r 1 R { 1 , 0 } .
Proposition 1.
For ν 1 , κ 1 0 , the Theorem 10 results in the following inequality,
Γ ( 1 + ( σ 1 ) λ ) ( 90 ) λ Γ ( 1 + σ λ ) ( 2 ) λ A λ ( ν 1 σ , κ 1 σ ) + 5 λ A λ σ w ν 1 , κ 1 , 5 6 , 1 6 + w A λ σ ν 1 , κ 1 , 1 3 , 2 3 + 6 λ A λ σ ( ν 1 , κ 1 ) + w A λ σ ν 1 , κ 1 , 1 3 , 2 3 + 5 λ A λ σ w ν 1 , κ 1 , 1 6 , 5 6 Γ ( 1 + λ ) L λ , σ σ ( ν 1 , κ 1 ) ( M m ) ( κ 1 ν 1 ) λ Γ ( 1 + λ ) 2 λ Γ ( 1 + 2 λ ) 26 225 λ .
Proof. 
The proof follows directly by applying T ( δ ) = Γ ( 1 + ( σ 1 ) λ ) Γ ( 1 + σ λ ) δ σ λ , σ > 0 in Theorem 10. □

3.2. Error Bounds

To conclude composite generalized Weddle’s rules with the help of newly established results, we consider a partition of the interval [ ν 1 , κ 1 ] such as: P : ν 1 = δ 0 < δ 1 < δ 2 < . . . < δ σ = κ 1 , and h k = δ i + 1 δ i 6 ( k = 1 , 2 , 3 , σ 1 ) , where σ must be divisible by 6. Then
1 Γ ( 1 + λ ) ν 1 κ 1 T ( δ ) d δ = ε ( λ , T ) + R ¯ ( λ , T ) ,
where
ε ( λ , T ) = i = 0 σ 1 ( δ i + 1 δ i ) ( 20 ) λ T ( δ i ) + 5 λ T 5 δ i + δ i + 1 6 + T 2 δ i + δ i + 1 3 + T δ i + δ i + 1 2 + T δ i + 2 δ i + 1 3 + 5 λ T δ i + 5 δ i + 1 6 + T ( δ i + 1 ) .
where R ¯ ( λ , T ) is the error term.
Proposition 2.
From Theorem 7, we have
R ¯ ( λ , T ) i = 0 σ 1 ( δ i + 1 δ i ) λ Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 4 75 λ + Γ ( 1 + 2 λ ) Γ ( 1 + 3 λ ) 1 150 λ | T λ ( δ i ) | + Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 14 225 λ Γ ( 1 + 2 λ ) Γ ( 1 + 3 λ ) 1 150 λ | T λ ( δ i ) | .
Proof. 
To obtain the desired result, we apply Theorem 7 over the subinterval [ δ i , δ i + 1 ] and take the sum from i = 0 to i = σ . □
Proposition 3.
From Theorem 8, we have
R ¯ ( λ , T ) i = 0 σ 1 ( δ i + 1 δ i ) λ Γ ( 1 + λ r 1 ) Γ ( 1 + ( r 1 + 1 ) λ ) 7 60 ( r 1 + 1 ) λ + 1 20 ( r 1 + 1 ) λ 1 r 1 × Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 11 36 λ | T λ ( δ i ) | r 2 + 1 36 λ | T λ ( δ i + 1 ) | r 2 1 r 2 + Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 1 36 λ | T λ ( δ i ) | r 2 + 11 36 λ | T λ ( δ i + 1 ) | r 2 1 r 2 + Γ ( 1 + λ r 1 ) Γ ( 1 + ( r 1 + 1 ) λ ) 1 30 ( r 1 + 1 ) λ + 2 15 ( r 1 + 1 ) λ 1 r 1 × Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 1 4 λ | T λ ( δ i ) | r 2 + 1 12 λ | T λ ( δ i + 1 ) | r 2 1 r 2 + Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 1 12 λ | T λ ( δ i ) | r 2 + 1 4 λ | T λ ( δ i + 1 ) | r 2 1 r 2 + Γ ( 1 + λ r 1 ) Γ ( 1 + ( r 1 + 1 ) λ ) 3 20 ( r 1 + 1 ) λ + 1 60 ( r 1 + 1 ) λ 1 r 1 × Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 7 36 λ | T λ ( δ i ) | r 2 + 5 36 λ | T λ ( δ i + 1 ) | r 2 1 r 2 + Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 5 36 λ | T λ ( δ i ) | r 2 + 7 36 λ | T λ ( δ i + 1 ) | r 2 1 r 2 ,
where 1 r 1 + 1 r 2 = 1 .
Proof. 
To obtain the desired result, we apply Theorem 8 over the subinterval [ δ i , δ i + 1 ] and take the sum from i = 0 to i = σ . □

3.3. Applications to Probability

Let p : [ ν 1 , κ 1 ] [ 0 , 1 ] λ be a probability density mapping over convex set X. Then cumulative distribution is demonstrated as
P r λ ( X κ 1 ) = F λ ( κ 1 ) = 1 Γ ( 1 + λ ) ν 1 κ 1 p ( δ ) ( d δ ) λ .
Utilizing the fact that
E λ ( X ) = 1 Γ ( λ + 1 ) ν 1 κ 1 δ λ p ( δ ) ( d δ ) λ E λ ( X ) = κ 1 λ 1 κ 1 ν 1 ν 1 κ 1 F λ ( δ ) ( d δ ) λ .
Proposition 4.
Considering Theorem 7, we have
1 ( 20 ) λ P r λ ( ν 1 ) + 5 λ P r λ 5 ν 1 + κ 1 6 + P r λ 2 ν 1 + κ 1 3 + 6 λ P r λ ν 1 + κ 1 2 + P r λ ν 1 + 2 κ 1 3 + 5 λ P r λ ν 1 + 5 κ 1 6 + P r λ ( κ 1 ) κ 1 λ E λ ( X ) ( κ 1 ν 1 ) λ ( κ 1 ν 1 ) λ Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 4 75 λ + Γ ( 1 + 2 λ ) Γ ( 1 + 3 λ ) 1 150 λ | p ( ν 1 ) | + Γ ( 1 + λ ) Γ ( 1 + 2 λ ) 14 225 λ Γ ( 1 + 2 λ ) Γ ( 1 + 3 λ ) 1 150 λ | p ( κ 1 ) | .
Proof. 
We conclude this result employing probability density function on Theorem 7. □
Remark 1.
By using the proposed inequalities, we can prove several bounds for generalized expected values and moments.
  • It is a known fact that various definite integrals cannot be evaluated through analytical techniques neither in classical nor in Yang local fractional calculus. They are approximated through Newton–Cotes schemes. In the aforementioned sections, we have provided the error bounds of definite integrals approximated through generalized Weddle’s incorporated with various classes of mappings. Our results provide the upper bounds of error terms for first-order local differentiable mapping. Additionally, from these bounds one can compute the upper bounds of definite integrals. For this, we apply Theorem 7 for T ( x ) = exp ( x 2 α ) over [ 0 , 2 ] with α = 1 . Then we get 10.2302 < 0 2 exp ( x 2 ) d x < 22.8469 . Note that this bound can be refined by increasing the order of differentiability. Through the application of our proposed inequalities, one can establish the bounds and relation between special functions like q-digamma function, modified Bessel functions and Beta functions. For further detail, consult [29,30,31].
  • Adopting the technique of [32], several new iterative methods to solve non-linear equations can be obtained. To discuss the convergence and dynamic analysis of proposed methods will be a new challenge for researchers.
  • Also by following the technique of the papers [33,34] on our proposed results and generalized Hilbert transform defined in [35], various estimates for Hilbert transform can be investigated through the utilization of diverse function classes.

4. Conclusions

In this manuscript, we have presented the error analysis of Weddle’s procedure in the framework of Yang fractal calculus. Our approach involved the generation of inequalities through a local differentiable identity, generalized convexity, bounded mapping, and Lipschitz mapping. Our results unify what already exists in the literature. The confirmation of inequalities has been provided through various simulations. It is an interesting and new problem; we hope the new researchers will explore its implications across various mathematical domains. One can explore the generalized fractional calculus approach within fractal domain through other general classes of convexity especially strong convexity. In the future, we will also focus on tight bounds of fractal inequalities utilizing diverse function classes of convexity. Another interesting problem is the construction of error inequalities of general Newton–Cotes schemes up to seven points leveraging the fractal calculus.

Author Contributions

Conceptualization, U.A., M.Z.J. and M.U.A.; methodology, Y.W., U.A., M.Z.J., M.U.A., A.G.K., and M.B.-A.; software, Y.W., U.A., M.B.-A., and K.S.A.; validation, Y.W., M.Z.J., M.U.A., A.G.K., M.B.-A., and K.S.A.; formal analysis, Y.W., U.A., M.U.A., A.G.K., M.B.-A., and K.S.A.; investigation, Y.W., U.A., M.Z.J., M.U.A., A.G.K., M.B.-A., and K.S.A.; writing—original draft preparation, U.A., M.Z.J., M.U.A., A.G.K., M.B.-A., and K.S.A.; writing—review and editing, U.A., M.Z.J., M.U.A., M.B.-A., and K.S.A.; visualization, Y.W., M.Z.J., A.G.K., M.B.-A., and K.S.A.; supervision, M.U.A. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2504).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Acknowledgments

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2504).

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. (a) Three- and (b) two-dimensional simulations of Theorem 7 for λ [ 0.6 , 1 ] , λ = 4 5 and σ [ 3 , 5 ] , respectively.
Figure 1. (a) Three- and (b) two-dimensional simulations of Theorem 7 for λ [ 0.6 , 1 ] , λ = 4 5 and σ [ 3 , 5 ] , respectively.
Fractalfract 09 00662 g001
Figure 2. (a) Three- and (b) two-dimensional simulations of Theorem 8 for λ [ 0.6 , 1 ] , λ = 4 5 and σ [ 3 , 5 ] , respectively.
Figure 2. (a) Three- and (b) two-dimensional simulations of Theorem 8 for λ [ 0.6 , 1 ] , λ = 4 5 and σ [ 3 , 5 ] , respectively.
Fractalfract 09 00662 g002
Figure 3. (a) Three- and (b) two-dimensional simulations of Theorem 9 for λ [ 0.6 , 1 ] , λ = 4 5 and σ [ 3 , 5 ] , respectively.
Figure 3. (a) Three- and (b) two-dimensional simulations of Theorem 9 for λ [ 0.6 , 1 ] , λ = 4 5 and σ [ 3 , 5 ] , respectively.
Fractalfract 09 00662 g003
Figure 4. (a) Three- and (b) two-dimensional simulations of Theorem 10 for λ [ 0.6 , 1 ] , λ = 4 5 and σ [ 3 , 5 ] , respectively.
Figure 4. (a) Three- and (b) two-dimensional simulations of Theorem 10 for λ [ 0.6 , 1 ] , λ = 4 5 and σ [ 3 , 5 ] , respectively.
Fractalfract 09 00662 g004
Figure 5. (a) Three- and (b) two-dimensional simulations of Theorem 11 for λ [ 0.6 , 1 ] , λ = 4 5 and σ [ 3 , 5 ] , respectively.
Figure 5. (a) Three- and (b) two-dimensional simulations of Theorem 11 for λ [ 0.6 , 1 ] , λ = 4 5 and σ [ 3 , 5 ] , respectively.
Fractalfract 09 00662 g005
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Wang, Y.; Asif, U.; Awan, M.U.; Javed, M.Z.; Khan, A.G.; Bin-Asfour, M.; Albalawi, K.S. Local Fractional Perspective on Weddle’s Inequality in Fractal Space. Fractal Fract. 2025, 9, 662. https://doi.org/10.3390/fractalfract9100662

AMA Style

Wang Y, Asif U, Awan MU, Javed MZ, Khan AG, Bin-Asfour M, Albalawi KS. Local Fractional Perspective on Weddle’s Inequality in Fractal Space. Fractal and Fractional. 2025; 9(10):662. https://doi.org/10.3390/fractalfract9100662

Chicago/Turabian Style

Wang, Yuanheng, Usama Asif, Muhammad Uzair Awan, Muhammad Zakria Javed, Awais Gul Khan, Mona Bin-Asfour, and Kholoud Saad Albalawi. 2025. "Local Fractional Perspective on Weddle’s Inequality in Fractal Space" Fractal and Fractional 9, no. 10: 662. https://doi.org/10.3390/fractalfract9100662

APA Style

Wang, Y., Asif, U., Awan, M. U., Javed, M. Z., Khan, A. G., Bin-Asfour, M., & Albalawi, K. S. (2025). Local Fractional Perspective on Weddle’s Inequality in Fractal Space. Fractal and Fractional, 9(10), 662. https://doi.org/10.3390/fractalfract9100662

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