Browsing by Author "Dr. Saad Ihsan Butt"

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Face Labeling of Graph Embedded on the Surface of Klein Bottle
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2016) Sharafat Ali; SP15-RMT-004; LHR TP 7004; Dr. Saad Ihsan Butt
We deal with the problem of labeling the Klein Grid’s Km n vertices, edges and faces with mn squares by the consecutive positive integers from 1 up to jV(Km n )j + jE(Km n )j + jF(Km n )j in such a way that if we add up all the labels of a 4-sided face that becomes the weight of that face. With common difference d these weight of the faces form an arithmetic progression. We will ﬁnd out the existence of such type of labelings for several differences of d.
Fractional Hermite-Hadamard Type Inequalities in Information Theory via Interval Calculus
(Library Information Services COMSATS University Islamabad Lahore Campus, 2025) Raheem Khan (FA23-RMT-032); Dr. Saad Ihsan Butt; LHR TP 9778
This thesis applies interval-valued analysis to information theory by using interval calculus to obtain fractional versions of information inequalities for convex functions. A novel center radius (cr)-order relation is introduced. The P-superquadratic variants of Hermite- Hadamard, Jensen’s, Jensen-Mercer inequalities along with their fractional extensions are further investigated and represented in interval-valued form. By proposing modifications to HH ¡-divergence inequalities for convex functions, information inequalities are also linked to contemporary studies. These efforts provide interval-valued estimates for a range of divergence metrics including Kullback-Leibler, Hellinger and c2.
Fractional Integral Inequalities via Multiplicative Caputo-Fabrizio Integral operator
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Muhammad Qais Ali (FA20-BSM-055) : Raheel Abbas (FA20-BSM-052); Dr. Saad Ihsan Butt; LHR TP 9919
Fractional integral operator generalize the classical concept of integration to non-integer or ders.In our work, first we establish the Hermite-Hadamard’s inequality via Multiplicative Caputo-Frabrizio operator then using this inequality we establish Mid-point, Trapezoidal equalities. After that we establish our main lemma by using this lemma we establish dif ferent result like, Power mean inequality, Holder inequality and absolute value inequality. BYusing these inequalities we obtain new error estimates.
Generalizations of Cyclic Refinements of Jensen’s Inequalities
(Library Information Services COMSATS University Islamabad Lahore Campus, 2020) Nasir Mehmood; FA14-PMATH-005; LHR TP 7451; Dr. Saad Ihsan Butt
In recent years, the concept of convex functions has been generalized extensively. Applications of convex functions are widely seen in many areas of modern analysis. Convex functions also have significant relation with the theory of inequalities and many useful inequalities are the result of the applications of convex functions. The Jensen's inequality has tremendous implications in many fields of modern analysis. It helps computing useful upper bounds for several entropic measures used in information theory. We consider discrete and continuous cyclic refinements of Jensen's inequality and extend them from convex to higher order convex function by using new Green functions introduced by us and employing different interpolating polynomials and identities. We formulate monotonicity of the linear functionals for nconvex functions at a point. We calculate some new Grüss and Ostrowski type bounds. As an application of our obtained results we give new bounds for Shannon, Relative and Zipf-Mandelbrot entropies.
Generalized Fractal–Fractional Integral Inequalities on Fractal Sets with Applications
(Library Information Services COMSATS University Lahore Campus, 2023-03-13) Hafiz Muhammad Umer Yasin; SP22-RMT-026; LHR TP 8731; Dr. Saad Ihsan Butt
In this study, we establish a set of novel Bullen-type inequalities applicable to differentiable convex functions within the framework of extended fractional integrals in a fractal domain. The key benefit of employing these inequalities and associated operators lies in their ver satility, allowing the conversion of these inequalities into established results for Riemann integrals. Additionally, they give rise to new inequalities applicable to Riemann-Liouville fractional integral inequalities, as well as generalized Riemann-Liouville fractional integral inequalities. To bolster the relevance of the conclusions, we also present the applications of recently developed results regarding the probability density functions, the quadrature formulae and the special means
Generalized Hermite--Jensen--Mercer-Type Inequalities in Fractal Sense
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Saba Yousaf; CIIT/FA19-RMT-007/LHR; Dr. Saad Ihsan Butt; LHR TP 7445
The most notable inequality pertaining convex functions is Jensen’s inequality which has tremendous applications in several ﬁelds. Mercer introduced an important variant of Jensen’s inequality called as Jensen-Mercer’s inequality. Fractal sets are useful tools for describing the accuracy of inequalities in convex functions. We establish a generalized Jensen–Mercer inequality and generalized Hermite-Hadamard–Mercer inequalities for a generalized con- vex function on a real linear fractal set Rς ( 0 < ς ≤ 1). Further, we also demonstrate some generalized Jensen–Mercer type inequalities by employing local fractional calculus. We establish two new lemmas involving local fractional integrals. By using these lemmas, we obtain several results related to generalized Hermite–Hadamard–Mercer type integral in- equalities for local differentiable generalized convex functions on real linear fractal space. Lastly, some applications related to Jensen–Mercer inequality, ς -type special means, and probability density functions are given. The present approach is efﬁcient, reliable, and may motivate further research in this area.
Hybrid Fractional Estimations for Quadrature Inequalities
(Library Information Services COMSATS University Lahore Campus, 2024-03-17) Saira Arif; FA22-RMT-012; Dr. Saad Ihsan Butt; LHR TP 9338
We study new fractional integral operators that involve linear combinations of Riemann and Caputo fractional integral operators. Thus, under the fundamentals of generalized con vexities, new fractional integral inequalities are explored and investigated for such hybrid fractional integral operators. This enables us to obtain several interfusing cases for frac tional parameter α ≥ 0. In order to derive fractional quadrature-type inequalities, some hybrid quadrature-type integral identities i.e. Simpson’s and Newton’s type in fractional calculus are derived for differentiable functions. Thus, by employing convexities of first and twice differentiable functions, several estimations of quadrature fractional integral in equalities are obtained. Finally, a number of fractional outcomes are provided related to special mean quadrature inequalities, q-digamma functions and Bessel functions
New Generalized Fractional Variants of Integral Inequalities
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2022) Iram Javed; FA20-RMT-041; LHR TP 7942; Dr. Saad Ihsan Butt
The graphical elegance of fractal theory takes into account the development’s achievability and exceptionalism. Due to its fascinating existence in the mathematical ﬁelds of sci- ences, there is a clear association between fractal sets and convexity. In this proposal, we will present generalized convexity and related integral inequalities on a fractal set Rv ( 0 < v 1). In the context of the Beta function, this research presents a new class of gener- alized Hermite-Hadamard type inequalities. This research contributes signiﬁcant results of novel versions of fractal H¨ older’s and Young’s inequalities. We derive some general con- clusions that capture novel results under investigation. One more remarkable contribution of the study is that two novel auxiliary results along with Trapezoidal and Midpoint type inequalities are provided. Hence, these new results will lead us to generalization of prior results.
New Generalized Ostrowski, Simpson and Boole’s Type Inequalities
(Library Information Services COMSATS University Islamabad Lahore Campus, 2025) Muhammad Mehtab (FA23-RMT-028); Dr. Saad Ihsan Butt; LHR TP 9774
In this proposal firstly, we introduce a parametric identity for generalized differentiable functions using a generalized fractal-fractional integral operators. Based on this identity, we establish several variants of parameterized inequalities for functions whose local fractional derivatives in absolute value satisfy generalized convexity conditions. Furthermore, we demonstrate that our main results reduce to well-known Ostrowski and Simpson type inequalities by selecting suitable parameters. These inequalities contribute to finding tight bounds for various integrals over fractal spaces. By comparing the classical H¨older and Power mean inequalities with their new generalized versions, we show that the improved forms yield sharper and more refined upper bounds. In particular, we illustrate that the generalizations of H¨older and Power mean inequalities provide better results when applied to fractal integrals, with their tighter bounds supported by graphical representations. Finally, a series of applications are discussed, including generalized special means, generalized probability density functions and generalized quadrature formulas, which highlight the practical significance of the proposed results in fractal analysis.
New Multiplicative Mid-Point and Trapezoidal Fractional Integral Inequalities
(Library Information Services COMSATS University Lahore Campus, 2024-03-17) Huma Tanveer; SP23-RMT-011; Dr. Saad Ihsan Butt; LHR TP 9584
In this thesis the mid-point fractional integral inequalities and multiplicative trapezoidal inequalities are thoroughly examined. We construct new lemmas and provide their pre cise estimations for differentiable and twice differentiable functions using fractional and k-fractional integrals. Furthermore, new bounds via Holder inequalities are obtained to ¨ demonstrate the strength, flexibility and usefulness of the suggested conclusions in the context of fractional calculus. A variety of examples are analyzed and their graphical rep resentations drawn for easier interpretation in order to validate the theoretical findings. In addition to contributing to our understanding of fractional integral inequalities, this re search provides new opportunities for future investigation into fractional calculus and its uses in science and engineering
New Perspectives of Solution of Heat Equation Using Neural Networks and Jensen’s Inequality
(Library Information Services COMSATS University Islamabad Lahore Campus, 2025) Rafay Ahmed (FA23-RMT-031); Dr. Saad Ihsan Butt; LHR TP 9777
In this study, we used Physics-Informed Neural Networks (PINNs) to solve the 1D, 2D, and 3D heat equation. To enhance training stability and accuracy, we replaced Jensen’s inequality as the loss function with more traditional methods such as Mean Square Error (MSE). We aimed to demonstrate that it is possible for PINNs to efficiently solve the heat equation with minimal data. In order to implement the model, we used a number of libraries, such as TensorFlow and Keras for creating the neural networks and NumPy, SciPy, and Matplotlib for managing the data and displaying the outcomes.
Results For Generalized Steffensen' S Inequality
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Adil Naseer; FA17-RMT-003; Dr. Saad Ihsan Butt; LHR TP 5642
This thesis present a study of some of the most notable and primal integral inequalities developed for n- convex functions. Thus, allowing us to assemble new developments in this research area under a specified frame. We construct new bounds of Gr¨uss and Ostrwoski type inequality associated to generalized Steffensen’s inequality by using ˇ Cebyˇsev functional. Finally, we construct new linear functionals for 2n-convex func tions and investigate their properties.