مشکل الحدیث کے حل میں ملا علی قاری ؒ کا منہج۔ایک تجزیاتی مطالعہ

The Problematic narration has always been under the special focus of the commentators of Hadith. This important branch of Hadith sciences, in fact, removes all objections that arise on the text of an authentic narration of the Holy Prophet (S.A.W). Mulla Ali Al-qari, being a famous commentator has opted for a comprehensive pattern in solving such problematic narrations in his famous commentar0y on Mishqat Al-masabih named Mirqat Al-mafatih. This article is an effort to explore his style by presenting ten examples from this voluminous commentary. Qari has at first, investigated the authenticity of such narration. He has tried to present the views and interpretations of his predecessor scholars such as Nawavi, Ibne-Hajar, Khattabi, etc. He seems to owe a clear viewpoint about this kind of narration that prophetic sayings after being confirmed and authentic as per principles set in Hadith Sciences, must be interpreted in a way that testifies the sanctity of that narration. This research concludes that problematic narrations have been interpreted by Muslim scholars of every age according to the knowledge they possessed. In this modern age of Science and technology, if any such narration has multi interpretations only one may be preferred which is supported by the available modern research It will surely make non-believers inclined to Islam and its eternal teachings.

On the Averages of Convex Functions

“Behind every theorem lies an inequality”. Mathematical inequalities play an impor- tant role in almost all branches of mathematics as well as in other areas of science. The basic work ”Inequalities” by Hardy, Littlewood and Polya appeared 1934 [37]and the books ”Inequalities” by Beckenbach and Bellman published in 1961 [9] and ”An- alytic inequalities” by Mitronovic published in 1970 made considerable contribution to this field and supplied motivation, ideas, techniques and applications. This theory in recent years has attached the attention of large number of researchers, stimulated new research directions and influenced various aspect of mathematical analysis and applications. Since 1934 an enormous amount of effort has been devoted to the dis- covery of new types of inequalities and the application of inequalities in many part of analysis. The usefulness of Mathematical inequalities is felt from the very be- ginning and is now widely acknowledged as one of the major deriving forces behind the development of modern real analysis. This dissertation deals with the inequali- ties for Jensen inqualites involving average of convex functions, Hermite-Hadamard inequalities. Chapter 1 offers an overview of the basic results contains a survey of basic concepts, indications and results from theory of convex functions and theory of inequalities used in subsequent chapters to which we refer as the known facts. Chapter 2 we give proofs of convexity and Schur convexity of the generalized inte- gral and weighted integral quasi-arithmetic mean. An overview of assorted proofs of schur convexity of integral arithmetic mean is discussed. In a detailed proof, discrete Jensen inequality for integral arithmetic mean is derived. Also integral version of Jensen inequality for integral arithmetic mean is proved. Motivated by discrete and viiviii integral Jensen inequalities functionals are defined. Two different method is given for constructing new examples of exponentially convex functions from non trivial gen- erating families of functions. Mean value theorem are proved. Different classes of monotonically increasing Cauchy means are created. Chapter 3 gives us convexity and Schur convexity of functions connected to Hermite- Hadamrd inequality as well as Schur convexity of differences of Hermite-Hadamrd inequality and Hammar-Bullen inequality by different proofs. Applying assorted gen- eralizations of Hermite-Hadamard inequality and Hammer-Bullen inequality on some special families of functions from varied classes, n-exponentially convex functions are generated by quite new method. Lyponuve, Dresher and Gramm’s type inequalities are developed. Pretty different Stolarsky type means are derives preserving inherited monotonically increasing property. Chapter 4 deals with inequalities of higher order convexity and divided difference. Two of them use majorization results and others are related to Jensen inequalities and Hermite-Hadamrd inequality. Integral Jensen inequality for divided difference is proved. Applications of averages of 3-convex functions as first order divided difference of convex functions are acquired. Method of producing n-exponentially convex func- tions is applied using divided differences. Produced functions are used in studying Stolarsky type means In the fifth chapter results about averages values of convex func- tions with variable limits and average values of composition functions is given. Study functionals for inequalities proved by D.E. Wulbert ( call them Wulbert’s inequalities for convenience) for convex and three convex functions. Extensions, improvements are accomplished. Variety of Stolarsky type means of a concave (convex) functions are obtained.