# Muhammad Aslam NoorCOMSATS University Islamabad | CUI · Department of Mathematics

Muhammad Aslam Noor

PhD

## About

1,408

Publications

264,034

Reads

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30,616

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Introduction

Additional affiliations

September 2001 - June 2005

**Etisalat College of Engineering, Sharja, United Arab Emirates**

Position

- Professor

Description

- Numerical and Convex Analysis, Nonlinear Optimization, Variational Inequalities, Equilibrium and Complementarity Problems, Iterative Methods, Dynamical Systems

## Publications

Publications (1,408)

In this paper, we consider the class of ϕ-convex functions, which was introduced and investigated by Noor [12] in 2006. We derive some quantum Hermite-Hadamard type inequalities for the ϕ-convex functions. Some special cases are discussed, which can be obtained from our results. The ideas and techniques of this paper may motivate further research i...

In this paper, we suggest and analyze some new iterative methods for solving nonlinear equation f (x) = 0 by using the variational iteration technique. We also give several examples to illustrate the efficiency of these methods. Comparison with other similar methods is also given. These new methods can be considered as alternative to the developed...

In this paper, we introduce a new class of convex functions which is called M T-harmonic convex functions. We establish some Hadamard-type inequalities for M T-harmonic convex functions. Results proved in this paper continue to hold for the special cases.

In this paper, we establish a new quantum integral identity for twice q-differentiable functions. We use this integral identity as an auxiliary result to derive some new quantum estimates for certain integral inequalities via geenralized preinvex functions. Some special cases are also discussed which can be deduced from our main results. Results ob...

In this paper, we introduce and consider a new class of harmonic convex functions, which is called harmonic nonconvex (p-convex) function. Several new Hermite-Hadamard type inequalities for harmonic nonconvex function are obtained. Some special cases are discussed. Results obtained in this paper continue to hold for these special cases. Our results...

The main objective of this paper is to obtain some new fractional estimates of Hermite-Hadamard type inequalities via h-convex functions. A new fractional integral identity for three times differentiable function is established. This result plays an important role in the development of new results. Several new special cases are also discussed. Some...

In this paper, we consider and investigate the relative harmonic preinvex functions,
which uniØes several new known classes of harmonic preinvex functions. We derive several
new integral inequalities such as Hermite-Hadamard, Simpson's, trapezoidal for the relative
harmonic preinvex functions. Since the relative harmonic preinvex functions include,...

In this paper, we introduce the concept of two dimensional pq-convex functions. We establish several new Hermite-Hadamard inequalities for two dimensional pq-convex functions. Some special cases are also discussed. Results obtained in this paper can be viewed as significant extensions of the previously known results.

In this paper, we derive several Hermite-Hadamard type integral inequalities for log φ-convex functions. Our results represent refinement and improvement of the previously known results. Several special cases are discussed.

In this article, we derive some new integral identities for differen-tiable functions. Then using these auxiliary results we obtain new Hermite-Hadamard type inequalities for differentiable p-convex functions. Some special cases are also discussed.

In this paper, we introduce a new class of harmonic preinvex functions, which are called harmonic h−preinvex functions. Several new Ostrowski-type inequalities for harmonic h-preinvex functions via Riemann-Liouville fractional integrals are established. Some special cases are also discussed, which appears to be new ones. The results obtained in thi...

In this paper, we introduce and study a new class of convex functions which is called strongly generalized harmonic convex functions. We establish some estimates involving the Euler Beta function and the hypergeometric functions of the integral b a (x − a) p (b − x) q f (x)dx for the class of functions whose certain powers of the absolute value are...

We introduce the concept of relative harmonic (s, η)-convex functions as a generalization of harmonic convex functions. Some basic inequalities related to relative harmonic (s, η)-convex functions are proved. We also investigate the famous Jensen, Hermite-Hadamard and Fejer type inequalities for this class of functions. The ideas and techniques of...

In this paper, we propose a new LQP method for solving nonlinear complementarity problems. The method uses a new descent direction which differs from the other existing LQP methods and another optimal step length is employed to reach substantial progress in each iteration. We prove the global convergence of the proposed method under the same assump...

In this paper, we consider a new class of variational inequalities, which is called the harmonic variational inequality. It
is shown that that the minimum of a differentiable harmonic convex function on the harmonic convex set can be characterized by the harmonic variational inequality. We use the auxiliary principle technique to discuss the existe...

In this paper, we introduce a new concept of harmonic (h,s)-convex functions in the second sense which generalizes the harmonic convex functions. Some Hermite-HadamardFejer type integral inequalities are derived. Some special cases also discussed. Results derived in this paper represent signiﬁcant reﬁnement and improvement of the known results.

In this paper, we use the auxiliary principle technique to suggest an implicit method for solving the harmonic variational inequalities. It is shown that the convergence of the proposed method
only needs pseudo monotonicity of the operator, which is a weaker condition than monotonicity.

In this paper, we establish a new quantum integral identity for twice q -differentiable functions. We use this integral identity as an auxiliary result to derive some new quantum estimates for certain integral inequalities via geenralized preinvex functions. Some special cases are also discussed which can be deduced from our main results. Results o...

In this paper, we introduce and investigate a new class of preinvex functions, which is called geometrically log-preinvex functions. Some new Hermite-Hadamard type inequalities are obtained. The idea and technique of this paper may stimulate further research in this ﬁeld.

In this paper, a new class of harmonic nonconvex function is introduced, which is called harmonic preinvex functions. The harmonic preinvex include harmonic functions as special case. Several new concepts are defined. New Hermite-Hadamard inequalities and their variant form are derived for harmonic preinvex functions. Some special cases are discuss...

In this paper, we introduce a new class of harmonic preinvex functions, which are called harmonic h−preinvex functions. Several new Ostrowski-type inequalities for harmonic h-preinvex functions via Riemann-Liouville fractional integrals are established. Some special cases are also discussed, which appears to be new ones. The results obtained in thi...

In this paper, we introduce and consider a new class of harmonic convex functions, which is called harmonic nonconvex (p-convex) function. Several new Hermite-Hadamard type inequalities for harmonic nonconvex function are obtained. Some special cases are discussed. Results obtained in this paper continue to hold for these special cases. Our results...

In this paper, we introduce an iterative sequence by using the hybrid generalized projection algorithm for approximating a solution of a general variational inequality in Banach spaces. Since the general variational inequality includes the classical variational inequality and complementarity problem as special cases, results obtained in this paper...

In this paper, we introduce and study a new class of quasi variational inequal-ities, known as multivalued extended general quasi variational inequalities. It is shown that the multivalued extended general quasi variational inequalities are equivalent to the fixed point problems. We use this alternative equivalent formulation to suggest and ana- ly...

In this paper, we introduce a new class of harmonic prein-vex functions, which is called harmonic M T-preinvex functions. Some new Hermite-Hadamard type inequalities for harmonic M T-preinvex functions are derived. Some special cases are also discussed. Results proved in this paper represent refinements and improvements of the known results.

The aim of this paper is to establish some new trapezoidal like inequalities for some classes of preinvex functions. Several special cases are also discussed.

In this paper, we establish a new quantum analogue of classical integral identity. Using this quantum integral identity, we derive some quantum analogues of Ostrowski type inequalities for q-differentiable convex functions.

In this paper, we consider the class of p-convex functions. We derive some new integral
inequalities of Hermite-Hadamard and Simpson type for differentiable p-convex functions using two new
integral identities. Some special cases are also discussed. Interested readers may find novel and innovative
applications of p-convex functions in various branc...

In this paper, we establish some new Hermite-Hadamard like integral inequalities involving harmonic log-convex functions.

In this paper, we consider the class of s-Godunova-Levin functions. We derive a new fractional integral identity for differentiable function. Using this new identity, we establish some new fractional Hermite-Hadamard type inequalities for the class of differentiable s-Godunova-Levin functions.

In this paper, we introduce the class of harmonically r-convex functions. We derive some Hermite-Hadamard type inequalities for this class of harmonic convex functions.

In this paper, we consider a newly introduced class of convex functions that is η-convex functions. We give some new quantum analogues for Hermite-Hadamard, Iynger and Ostrowski type inequalities via η-convex functions. Some special cases are also discussed.

We derive some new integral identities for differentiable functions. Then using these auxiliary results, we obtain new Hermite–Hadamard type inequalities for differentiable p-convex functions. Some special cases are also discussed.

In this paper, we discuss the existence of a solution of extended general quasi variational inequalities using projection technique. Using the technique of Noor, we suggest and analyze a new class of three-step iterative schemes for solving extended general quasi variational inequalities. Under some certain conditions on operators, we also discuss...

In this paper, we derive a new lemma including third-order derivative of a function via fractional integrals. Using this lemma, we establish some new fractional estimates for Hermite-Hadamard type inequalities for convex functions. Several special cases are also discussed. Some applications to special means of real numbers are also discussed. The i...

The concept of (Orlicz, Breckner) co-ordinated (s, σ)-convexity is introduced for two variables real functions. Upper and lower sharp integral boundary properties of Hermite-Hadamard type are proved on rectangles.

In this paper, we consider the class of $(p,h)$-convex functions. We
establish some new estimates for trapezoidal like inequalities via
differentiable $(p,h)$-convex functions. We also discuss some
special cases which can be deduced from our main results.

In this article, we derive some new integral identities for
differentiable functions. Then using these auxiliary results we
obtain new Hermite-Hadamard type inequalities for differentiable
$p$-convex functions. Some special cases are also discussed.

The main objective of this paper is to extend the notion of harmonic h-convex functions on rectangle, which is called as two dimensional harmonic h-convex function. We obtain some new inequalities for two dimensional harmonic h-convex function. Several new and known special cases which are inherent in our main results are also discussed. Results ob...

In this paper, we obtain a new integral identity for differentiable
function. Using this new integral identity as an auxiliary result,
we derive some new integral inequalities for differentiable harmonic
$h$-convex functions. We also discuss some new special cases which
can be deduced from our main results.

In this paper, we consider the class of harmonic $h$-convex
functions. We derive some new Simpson type inequalities for
differentiable harmonic $h$-convex functions. We also discuss some
new and known special cases which can be deduced from our main
results.

In this paper, we introduce and analyze two new methods for solving the NP-hard absolute value equations (AVE) Ax − |x| = b, where A is an arbitrary n × n real matrix and b ∈ R n , in the case, singular value of A exceeds 1. The comparison with other known methods is carried to show the effectiveness of the proposed methods for a variety of randoml...

In this chapter, we discuss quantum calculus and generalized convexity. We briefly discuss some basic concepts and results regarding quantum calculus. Some quantum analogues of derivatives and integrals on finite intervals are discussed. After this we move towards generalized convexity. Examples are given to illustrate the importance and significan...

In this paper, we suggest and analyze some unique recurrence relations which can generate different classes of iterative methods for solving nonlinear equations using the system of coupled equations together with decomposition technique by using an auxiliary function. Various numerical examples are given to illustrate the efficiency and performance...

In this paper, we derive some Newton's type of integral inequalities for geometrically relative convex functions. Some special cases are also discussed.

Various problems of pure and applied sciences can be studied in the unified frame work of the nonlinear equations. In this paper, a new family of iterative methods for solving nonlinear equations is developed by using a new decomposition technique. The convergence of the proposed methods is proved. It is shown that the new family contains several w...

In this paper, we consider a class of geometrically convex function which is called geometrically-arithmetically h-convex function. Some inequalities of Hermite-Hadamard type for geometrically-arithmetically h-convex functions are derived. Several special cases are discussed.

In this paper, we introduce a new class of convex functions, which is called nonconvex functions. We show that this class unifies several previously known and new classes of convex functions. We derive several new inequalities of Hermite-Hadamard type for nonconvex functions. Some special cases are also discussed. Results proved in this paper conti...

In this paper, we consider a new class of quasi variational inequalities involving three operators, which is called the extended general quasi variational inequality. It is shown that the extended general quasi variational inequalities are equivalent to the fixed problems and the extended general implicit Wiener-Hopf equations. These alternative fo...

Some Hermite-Hadamard type inequalities via Riemann-Liouville fractional integral for twice differentiable functions having some s-convexity of second kind properties are established. A class of s-affine of second kind functions is identified such as these inequalities are sharp.

In the paper, the authors introduce the notion ``logarithmically $h$-preinvex functions'', reveal that the class of $h$-preinvex functions include several new and known classes of preinvex functions, and establish several integral inequalities of Hermite--Hadamard type.

In this paper, we introduce a new class of convex functions which is called \({h_{\varphi}}\) -preinvex functions. We prove several Hermite–Hadamard inequalities for \({h_{\varphi}}\) -preinvex functions. Some special cases are also discussed. Results proved in this paper continue to hold for these special cases. Our results may stimulate further i...

In this paper, we derive a new lemma including third-order derivative of a function via fractional integrals. Using this lemma, we establish some new fractional estimates of Hermite-Hadamard type of inequalities via convex functions. Several special cases are also discussed. Some applications to special means of real numbers are also discussed. The...

In this paper, we use the system of coupled equations involving an auxiliary function together with decomposition technique to suggest and analyze some new classes of iterative methods for solving nonlinear equations. These new methods include the Halley method and its variant forms as special cases. Various numerical examples are given to illustra...

In this paper, we establish quantum analogue of classical integral identity. Using this identity, we derive some quantum estimates for Hermite–Hadamard inequalities for q-differentiable convex functions and q-differentiable quasi convex functions. Results obtained present refinement and improvement of the known results. The ideas and techniques of...

In this paper, we consider a very useful and significant class of convex sets and convex functions that is relative convex sets and relative convex functions which was introduced and studied by Noor [20]. Several new inequalities of Hermite-Hadamard type for relative convex functions are established using different approaches. We also introduce rel...

Using convolution and subordination concepts, we define some new subclasses of analytic functions and study their properties including inclusion relations and radius problems. Several applications related to conic domains and certain integral operators are also given. Results obtained in this paper may motivate further research in this dynamic and...

In this paper, we use fractional derivative operator to define certain classes of analytic functions related to conic domains. Using differential subordination and convolution techniques, we prove coefficient and inclusion results. It is also shown that these classes are closed under convolution with convex functions. Some interesting applications...

In this paper, using the projection operator, we introduce two new dynamical systems for extended general quasi variational inequalities. These dynamical systems are called extended general implicit projected dynamical system and extended general implicit Wiener-Hopf dynamical system. We prove that these new dynamical systems converge globally expo...

In this paper, we suggest and analyze a new two-step iterative method for solving nonlinear equations, which modified Noor method [6] without second derivatives for nonlinear equation. This Modified method has a fourth-order convergence and efficiency index of this method is ∛4 ≈ 1.5874. Several examples are given to illustrate the efficiency and t...

In this paper, we consider and investigate geometrically nonconvex functions. We derive several Hermite-Hadamard type inequalities for geometrically (GG) nonconvex (relative convex) function and geometrically arithmetically (GA) nonconvex (relative convex) functions. We also obtain some fractional Hermite-Hadamard type inequalities. It is shown tha...

In this paper, we introduce and investigate a new class of harmonically convex functions, which is called harmonically h-convex function. It is shown that this class unifies several new and known classes of harmonically convex functions. We derive some new Hermite-Hadamard like inequalities for harmonically h-convex functions. Harmonically s-convex...

In this paper, we introduce and investigate a new class of harmonically convex functions, which is called harmonically h-convex function. It is shown that this class unifies several new and known classes of harmonically convex functions. We derive some new Hermite-Hadamard like inequalities for harmonically h-convex functions. Harmonically s-convex...

It is known that the general variational inclusions involving diﬀerence of two (or more) monotone operators are equivalent to the resolvent equations. We use this equivalent form to develop the sensitivity analysis for general variational inclusions. This technique does not require the diﬀerentiability of the given data. Some special cases are disc...

In this paper, we suggest and analyze a two-step method for solving the variational inequalities on Hadamard manifold using the auxiliary principle technique. The convergence of this new method requires only the partially relaxed strongly monotonicity, which is a weaker condition than monotonicity. Results can be viewed as refinement and improvemen...

In this paper, we use the variational iteration technique to suggest some iterative methods for solving the nonlinear equations involving an auxiliary function. For appropriate and suitable choice of the auxiliary function,one can obtain a wide class of iterative methods for solving the nonlinear equations, which is a novel aspect of this technique...

In this paper, we introduce the notion of coordinated harmonically convex functions. We derive some new integral inequalities of Hermite–Hadamard type for coordinated Harmonically convex functions. The interested readers are encouraged to find the applications of harmonically convex functions in pure and applied sciences.

We prove that the set of second degree cumulative frontier gaps, via fractional integrals of positive orders, of twice differentiable functions having generalized convexity at the level of the second derivative is upper bounded. A sharp Hermite-Hadamard type inequality via fractional integrals leads to an evaluation of this upper bound, which does...

It is known that the general variational inclusions involving difference of two (or more) monotone operators are equivalent to the resolvent equations. We use this equivalent form to develop the sensitivity analysis for general variational inclusions. This technique does not require the differentiability of the given data. Some special cases are di...

In this paper, we consider a new class of quasi variational inequalities involv- ing three operators, which is called the extended general quasi variational inequality. It is shown that the extended general quasi variational inequalities are equivalent to the fixed problems and the extended general implicit Wiener-Hopf equations. These alternative...

In this paper, we introduce some new classes of s-Godunova-Levin functions, which are called (s,m)-Godunova-Levin functions of first and second kind. We show that these classes contain previously known classes of convex functions. Finally, we establish some new Ostrowski inequalities for (s,m)- Godunova-Levin functions via fractional integrals, Som...

In this paper, we define and study a new class of analytic functions by using the concept of generalized close-to-convexity. Coefficient results, Hankel determinant problem and some other interesting properties of this class are investigated. Results proved in this paper may stimulate further research in this area.

A new Hermite-Hadamard inequality for p-convex(nonconvex) functions is obtained , which can be viewed as a refinement of known results. We derive some new inequalities of functions whose derivatives in absolute value are nonconvex. Results obtained in this paper continue to hold for special cases. Techniques and ideas of this paper may stimulate fu...

In this paper, we discuss the existence of a solution of extended general quasi variational inequalities using projection technique. Using the technique of Noor, we suggest and analyze a new class of three-step iterative schemes for solving extended general quasi variational inequalities. Under some certain conditions on operators, we also discuss...

In this paper, we derive a new refinement of Hermite-Hadamard inequality for the class of h-convex functions. We also discuss some new and known special cases which can be obtained from our main results. Results obtained in this paper may be viewed as significant improvement of the known results.

In this article, we derive a new refinement of Hermite-Hadamard inequality for the class of h-convex functions. We also discus some new and known special cases, which can be obtained from our min results. Results obtained in this paper may be viewed as significant improvement of the known results.

. A new Hermite-Hadamard inequality for p -convex(nonconvex) functions is ob- tained, which can be viewed as a refinement of known results. We derive some new inequali- ties for functions whose derivatives in absolute value are nonconvex.Results obtained in this paper continue to hold for special cases. Techniques and ideas of this paper may stimul...

In this paper, we introduce the notion of coordinated harmonically convex functions. We derive some new integral inequalities of Hermite–Hadamard type for coordinated Harmonically convex functions. The interested readers are encouraged to find the applications of harmonically convex functions in pure and applied sciences.

In this paper, we use the modified decomposition method for solving the system of third-order nonlinear boundary value problems associated with obstacle problems. Some examples of system of third-order nonlinear boundary value problems are given. The comparison of the results obtained by modified decomposition method with modified variation of para...

In this chapter, we investigate some unified iterative methods for solving the general equilibrium problems using the auxiliary principle technique. The convergence of the proposed methods is analyzed under some suitable conditions. As special cases, we obtain a number of known and new classes of equilibrium and variational inequality problems. Res...

In this paper, some new classes of Godunova-Levin functions are introduced and investigated. Several new fractional Hermite-Hadamard inequalities are derived for these new classes of Godunova-Levin functions. The ideas and techniques of this paper may stimulate and inspire further research.

a b s t r a c t In this paper, we consider a new system of extended general quasi variational inequalities involving six nonlinear operators. Using projection operator technique, we show that the system of extended general quasi variational inequalities is equivalent to a system of fixed point problems. Using this alternative equivalent formulation...

In this paper, we use a new decomposition technique to suggest and consider some new iterative methods for solving system of linear equations. We prove that these iterative methods are similar to the iterative methods derived by using homotopy perturbation method and Adomian decomposition method. We consider the elliptic partial differential equati...

In this paper, we suggest and analyze some new derivative free iterative methods for solving nonlinear equation f (x) = 0 using a suitable transformation. We also give several examples to illustrate the efficiency of these methods. Comparison with other similar method is also given. These new methods can be considered as alternative to the develope...

In this paper we define and study a class of analytic functions which map the open unit disk onto same conic regions and are related to Bazilevic and higher order close-to-convex functions. We investigate rate of growth of coefficients, Hankel determinant problem, inclusion result and establish univalence criterion. Some other interesting propertie...

In this paper, we suggest and analyze a new family of iterative methods for finding zeros of multiplicity of nonlinear equations by using the variational iteration technique. These new methods include the Halley method and its variants forms as special cases. We also give several examples to illustrate the efficiency of these methods. Comparison wi...

Inspired by the very recent work by Noor and Noor [9] and given a closed convex set-valued mapping C, we propose a split algorithm for solving the problem of finding an element x * which is a zero of a given maximal monotone operator T such that its image, Ax * , under a linear operator, A, is in a closed convex set C(x *). Then, we present two str...