الامام ضیاء المقدسی و منھجہ فی کتابہ الاحادیث المختارۃ

Different scholars have compiled the books which contain a large numbers of authentic Ahadith (Ahadith Sahiha), to achieve this purpose, they introduced different hadith sciences to distinguish between the true and the fabricated hadith. The authentic Sunnah is contained within the vast body of Hadith literature. One of them is Imam Zia ul Maqdasi. Imam Zia Uddin Muhammad bin Abdul Wahid Maqdasi’s book “Al Ahadith al Makhtara” is one of the best books of its kind. Many Islamic scholars have declared it better than Imam Hakim’s book Al Mustadrak. Allama Iraqi, one of his contemporaries said that the Ahadith given in his book Al Ahadith al Makhtara were not ascertained to be authentic before. Only those Ahadith have been given in this book whose asaneed are correct but they have not been reported by Imam Bukhari and Imam Muslim. Also, one of the strengths of this book is that it reflects the glimpses of Muajam. Imam Maqdasi wrote this book in the manner of Masaneed that is to say that he mentioned the name of the companion of the Holy Prophet (SAW) and then reported his traditions. Sometimes he also indicates the factors responsible for the interruption in the authenticity of Ahadith. But, sadly, Imam Maqdasi passed away and could not complete this great book. In this article I will discuss the Imam Zia ul Maqdasi approach towards “Ahadith Sahiha” in his book Al Ahadith ul Mukhtara.

On Jensen’S and Related Inequalities

Inequalities are one of the most important instruments in many branches of mathe- matics such as functional analysis, theory of differential and integral equations, inter- polation theory, harmonic analysis, probability theory, etc. They are also useful in mechanics, physics and other sciences. A systematic study of inequalities was started in the classical book [31] and continued in [54, 55]. In the eighties and nineties of the last century an impetuous increase of interest in inequalities took place. One result of this fact was a great number of published books on inequalities (see e.g. [4, 5, 37, 39, 38]) and on their applications (see e.g. [2, 11]). Nowadays the theory of inequalities is still being intensively developed. This fact is confirmed by a great number of recent published books (see e.g. [6, 56]) and a huge number of articles on inequalities. Thus, the theory of inequalities may be regarded as an independent area of mathematics. This PhD thesis is devoted to special kind of inequalities, namely Jensen’s and some its related inequalities involving Hermite-Hadamard inequality, Hardy and its limit Polya-Knopp inequality. In the first chapter, called Introduction, some basic notions and results from theory of convex functions and theory of inequalities are being introduced along with classical results of convex functions. In the second chapter, The weighted Jensen’s Inequality for convex-concave anti- symmetric functions is proved and some applications are given. In the third chapter we have discussed the generalized form of Hermite-Hadamard inequality for integrable Convex functions. In the fourth chapter Some estimates of Hardy, strengthened Hardy-Knopp and multidimensional Hardy-Polya-Knopp type differences for p < 0 and 0 < p < 1 are calculated. In the fifth chapter we prove a new general one-dimensional inequality for convex functions and Hardy-Littlewood averages. Furthermore, we apply this result to unify and refine the so-called Boas’s inequality and the strengthened inequalities of the Hardy-Knopp-type, deriving their new refinements as special cases of the obtained general relation. In particular, we get new refinements of strengthened versions of the well-known Hardy and P ́olya-Knopp’s inequalities, while in the last chapter some measures of divergences between vectors in a convex set of n−dimensional real vector space are defined in terms of certain types of entropy functions, and their log-convexity properties with some applications in Information theory are discussed.