Atiq Ur Rehman

• PhD
• Associate Professor at COMSATS University Islamabad

In this paper we investigate Chebyshev’s type inequalities for h-convex functions. These inequalities are obtained by imposing some convenient conditions on h-convex functions. Furthermore, the associated Chebyshev’s functional are estimated via mean value theorems.

In this article, the famous Giaccardi inequality is generalized for modified (h,m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(h,m)$\end{document}-convex functions u...

In this paper, a well-known inequality called Giaccardi inequality is established for isotonic linear functionals by applying s-convexity in the second sense, which leads to notable Petrović inequality. As a special case, discrete and integral versions of Giaccardi inequality are derived along with the Petrović inequality as a particular case. In a...

This paper investigates the metric dimensions of the polygonal networks, particularly, of subdivided honeycomb network as well as Aztec diamond network. The polygon is any two-dimensional shape formed by straight lines. Triangles, quadrilaterals, pentagons and hexagons are all representations of polygons. For instance, hexagons help us in many mode...

In this article, we consider the class of modified h − convex functions and derive the famous Giaccardi and Petrovi c ′ type inequalities for this class of functions. The mean value theorems for the functionals due to Giaccardi and Petrovi c ′ type inequalities are formulated. Some special cases are discussed by taking different examples of functio...

The aim of this paper is to study the fractional Hadamard inequalities for Caputo fractional derivatives of strongly convex functions. We obtain refinements of two known fractional versions of the Hadamard inequality for convex functions. By applying identities for Caputo fractional derivatives we get refinements of error bounds of these inequaliti...

The aim of this article is to present fractional versions of the Hadamard type inequalities for exponentially $(s, m)$-convex functions via $k$-analogue of Riemann-Liouville fractional integrals. The results provide generalizations of various known fractional integral inequalities. Some special cases are analyzed in the form of corollaries and rema...

In this article, we prove some fractional versions of Hadamard-type inequalities for strongly exponentially α,h-m-convex functions via generalized Riemann-Liouville fractional integrals. The outcomes of this paper provide inequalities of strongly convex, strongly m-convex, strongly s-convex, strongly α,m-convex, strongly s,m-convex, strongly h-m-co...

This paper comprises the introduction of weighted mean products of fuzzy graph structures (FGSs) to construct weighted mean fuzzy graph structures (WMFGSs) with the help of optimization parameter, and establish some novel results after validating with examples, accordingly. The notions of regular and mμk-regular FGSs are described, where m ∈]0,1] r...

The multi-criteria decision-making (MCDM) problem has a solution whose quality can be affected by the experts’ inclinations. Under essential conditions, the fuzzy MCDM method can provide more acceptable and efficient outcomes to select the best alternatives. This work consists of a consensus-based technique for selecting and evaluating suppliers in...

Abstract This research investigates the bounds of fractional integral operators containing an extended generalized Mittag-Leffler function as a kernel via several kinds of convexity. In particular, the established bounds are studied for convex functions and further connected with known results. Furthermore, these results applied to the parabolic fu...

In this paper, Petrović’s inequality is generalized for \(h−\)convex functions on coordinates with the condition that \(h\) is supermultiplicative. In the case, when \(h\) is submultiplicative, Petrović’s inequality is generalized for \(h−\) concave functions. Also particular cases for \(P−\) function, Godunova-Levin functions, \(s−\) Godunova-Le...

This paper is dedicated to Opial-type inequalities for arbitrary kernels using convex functions. These inequalities are further applied to a power function. Applications of the presented results are studied in fractional calculus via fractional integral operators by associating special kernels.

This paper studies the k-fractional analogue of the Caputo fractional derivatives, their properties, and applications. A convolution of two functions instead of the product is analyzed by means of Caputo k-fractional derivatives. By virtue of defined convolution Chebyshev type, inequalities have been investigated. Moreover, Hadamard Caputo fraction...

In this paper, Petrovi´cPetrovi´c's inequality is generalized for h−convex functions, when h is supermul-tiplicative function. It is noted that the case for h−convex functions does not lead the particular cases for P −function, Godunova-Levin functions, s−Godunova-Levin functions and s−convex functions due to the conditions imposed on h. To cover t...

We use generalized fractional integral operator containing the generalized Mittag-Leffler function to establish some new integral inequalities of Grüss type. A cluster of fractional integral inequalities have been identified by setting particular values to parameters involved in the Mittag-Leffler special function. Presented results contain several...

Fejér Hadamard inequality is generalization of Hadamard inequality. In this paper we prove certain Fejér Hadamard inequalities for k-fractional integrals. We deduce Fejér Hadamard-type inequalities for Riemann-Liouville fractional integrals. Also as special case Hadamard inequalities for k-fractional as well as fractional integrals are given. © 201...

In this paper some new general fractional integral inequalities for convex and m-convex functions by involving an extended Mittag-Leffler function are presented. These results produce inequalities for several kinds of fractional integral operators. Some interesting special cases of our main results are also pointed out.

Fractional integral and differential inequalities provide the bounds for the uniqueness of solutions of the fractional differential equations. In this paper we compute some new Hadamard and the Fejer-Hadamard fractional inequalities for harmonically convex functions via Caputo k-fractional derivatives. Also results for Caputo fractional derivatives...

In this paper the authors extend Giaccardi's inequality to coordinates in the plane. The authors consider the nonnegative associated functional due to Giaccardi's inequality in plane and discuss its properties for certain class of parametrized functions. Also the authors proved related mean value theorems.

In this paper, we give some fractional integral inequalities of Os-trowski type for s-Godunova-Levin functions via Riemann-Liouville k−fractional integrals. We deduce some known Ostrowski type fractional integral inequalities for Riemann-Liouville fractional integrals and we also prove results for p−functions and Godunova-Levin functions by taking...

In this paper we give certain Hadamard and Hadamard-type inequalities for relative convex functions via Caputo k-fractional derivatives. We also find results for Caputo fractional derivatives.

In this paper, we prove a version of the Hadamard inequality for function f such that f^(n) is convex via k-fractional Caputo derivatives. Using convexity of |f (n) |^q , q ≥ 1 we find the bounds of the difference of fractional differential inequality. Also we have found inequalities for Caputo fractional derivatives.

Fractional inequalities are useful in establishing the uniqueness of solution for partial differential equations of fractional order. Also they provide upper and lower bounds for solutions of fractional boundary value problems. In this paper we obtain some general integral inequalities containing generalized Mittag-Leffler function and some already...

In this paper we give generalizations of the Hadamard-type inequalities for fractional integrals. As special cases we derive several Hadamard type inequalities.

Based on the Grüss inequality and the Ostrowski inequality, we obtain some new versions of the Os-trowski and the Ostrowski-Grüss type inequalities.

In this paper, we give an order relation between J-divergence and generalized K-divergence. By using this order relation we give generalizations of the results related to an order relation between J-divergence and K-divergence given by J. Burbea and C. R. Rao. Also we construct class of m-exponentially convex functions introducing by nonnegative di...

In this paper we prove Hadamard-type inequalities for k-fractional Riemann-Liouville integrals and Hadamard-type inequalities for fractional Riemann-Liouville integrals are deduced. Also we deduced some well known results related to Hadamard inequality.

In this paper, the authors extend Petrović’s inequality to coordinates in the plane. The authors consider functionals due to Petrović’s inequality in plane and discuss its properties for certain class of coordinated log-convex functions. Also, the authors proved related mean value theorems.

In this paper, we give generalization of the Fejér-Hadamard inequality by using definition of convex functions on n-coordinates. Results given in [8, 12] are particular cases of results given here.

We give Fejer-Hadamard inequality for convex functions on coordinates in the rectangle from the plane. We define some mappings associated to it and discuss their properties.

We have discussed the generalization of Hermite-Hadamard inequality introduced by Lupaş for convex functions on coordinates defined in a rectangle from the plane. Also we define that mappings are related to it and their properties are discussed.

In this paper, we prove the Hadamard type inequalities for m-convex functions via fractional integrals and related inequalities. These results have some relationships with the Hadamard inequalities for fractional integrals and related inequalities.

We derive new mean value theorems for functionals associated with Hadamard inequality for convex functions on the coordinates. We present some Hadamard type inequalities and related results for m-convex functions on the coordinates.

AAAAAA. In this paper, we give Hermite–Hadamard's inequality for s-convex functions in first sense and second sense on n-coordinates.

In this paper, we obtain Ostrowski-type bounds for the weighted ˇ Cebyˇsev functional. Also we give bounds of weighted ˇ Cebyˇsev functional in the case of Steffesen’s generalization of ˇ Cebyˇsev inequality

We construct n-exponentially convex functions and exponentially convex functions using the functional defined as the difference of the right parts of the Hermite-Hadamard inequality, for different classes of functions. Applying these results to some starshaped functions, we derive non-symmetric means of Stolarsky type.

In this paper, the Giaccardi's difference is considered in some special cases. By utilizing two classes of the convex functions, the logarithmic convexity of the Giaccardi's difference is proved. The positive semi-definiteness of the matrix generated by Giac-cardi's difference is shown. Related means of Cauchy type are defined and monotonicity prop...

We give a proof of Giaccardi's inequality for convex-concave antisymmetric function.Also we give the exponential convexity of improvement of Giaccardi's inequality by using positive semidefinite matrix and related mean value theorems of Cauchy type.

We use parameterized class of increasing functions to give exponential convexity of the non-negative difference of certain inequality as a function of parameter in connection with power sums. We define new means of Cauchy type and give its relation to the means we defined earlier [J. Inequal. Appl. 2008, Article ID 389410 (2008; Zbl 1149.26021); J....

We consider functionals due to the difference in Petrović and related inequalities and prove the log-convexity and exponential convexity of these functionals by using different families of functions. We construct positive semi-definite matrices generated by these functionals and give some related results. At the end, we give some examples.

We give a simple proof of the Stolarsky means inequality as well as some related inequalities for similar means of Stolarsky type.

The first two chapters of the classical book [E. F. Beckenbach and R. Bellman, Inequalities. Berlin-Göttingen-Heidelberg: Springer-Verlag (1961; Zbl 0097.26502)] on inequalities are devoted to fundamental inequalities and positive definiteness. In this paper we obtain results which give a connection between fundamental inequalities and positive def...

In the paper “On logarithmic convexity for power sums and related results” (2008), we introduced means by using power sums and increasing function. In this paper, we will define new means of convex type in connection to power sums. Also we give integral analogs of new means.

We give a simple proof of the Stolarsky means inequality as well as some related inequalities for similar means of Stolarsky type.

We give some further consideration about logarithmic convexity for differences of power sums inequality as well as related mean value theorems. Also we define quasiarithmetic sum and give some related results.