امام بخاریؒ کا صحیح بخاری میں مؤطا امام مالک پراعتماد؛ تجزياتى مطالعہ

In every period, hadith experts have contributed to the preservation of the Hadith. Imam Malik composed M’ūaṭa to safeguard the hadith, and Imam al-Bukhari afterwards turned to al-Mu'tah for assistance. The research methodology of this paper is an analytical study of the reliance of Imām Bukhārī on Al-M’ūaṭa Imam Mālik in Ṣaḥiḥ Bukhārī. In our article, we came to the conclusion that Imam Bukhari recounted a total of six hundred sixty-eight narrations from Imam Malik in his Sahih, of which six hundred four are scriptural narrations that he took from ten versions of Al- M’ūaṭa. As a result, these narrations make up more than ninety percent of all narrations. Contrasted with in his Sahih, Imam Muslim narrates three hundred eighty-nine narrations from Imam Malik, three hundred twelve of which are scriptural narrations that he has derived from eight of narrations of al M’ūaṭa's. These narrations so account for more than eighty percent of all narrations.

Opial, Hardy and Related Inequalities Involving Kernels

The theory of inequalities has developed rapidly in last few decades. It now occupies a central position in analysis and will no doubt, continue to play an essential role in mathematics as a whole. It certainly reflected in the vast literature that exists on the subject. Inequalities are useful tools to obtain optimal results in different areas of mathematics. In particular, inequalities are found to be versatile while deciding about extrema of various functions. The theory of inequalities is in a process of unbroken onward development and have also become a very useful and commanding instrument for studying a wide range of problems in different fields of mathematics. Furthermore, the importance of fractional calculus in mathematical inequalities is enormous. Opial, Hardy and related inequalities are very famous and play significant part in mathematical inequalities. Many mathematicians gave generalizations, improvements and applications of the said inequalities and they used fractional integral and derivative operators to derive new integral inequalities. The present study has considered integral operators with non-negative kernel on measure spaces with positive σ-finite measure. This research studies the enhancement of Opial, Hardy and related inequalities for global fractional integral and derivative operators with respect to the convex, monotone convex and superquadratic functions. Different weights have been used to obtain fresh consequences of the said inequalities. The thesis is planned in the subsequent mode: The first chapter includes the essential concepts and notions from the theory of convex functions, superquadratic functions, fractional calculus and the theory of inequalities. Some constructive lemmas related to fractional integrals and derivatives are incorporated which have been frequently use in next chapters to establish results. The second chapter comprises of two sections. First section deals with some Opialtype inequalities for two functions with general kernels related to a particular class of functions U(f, k), which admits the representation: |g(t)| = | xZ a k(x, t) f(t) dt| ≤ xZ a k(x, t) |f(t)| dt, where f is a continuous function and k is an arbitrary non-negative kernel such that f(t) > 0 implies g(x) > 0 for every x ∈ [a, b]. We also assume that all integrals under consideration exist and that they are finite. At the end of this part, we provide the discrete description of the said inequalities. In the second section, we include the multiple Opial-type inequalities for general kernels by considering the monotonicity and boundedness of the weight functions. In continuation of our general results, we provide their applications for Widder’s derivative and linear differential operators. Chapter three consists of three sections. In first section, we present Hardy-type inequalities and their refinements for fractional integral and derivative operators using convex and increasing functions under certain conditions. The second section restrains the applications of refined Hardy-type inequalities for Hilfer fractional derivative and the generalized fractional integral operator involving generalized Mittag-Leffler function in its kernel via convex and monotone convex functions. In the third section, we offer the applications of refined Hardy-type inequalities for linear differential operator, Widder’s derivative and generalized fractional integral operator involving Hypergeometric function in its kernel.