مولانا شاہ حسن مثنیٰ ندوی
یہ بھی پھلواری کے علمی و دینی خانوادے کے چشم و چراغ اورمولانا شاہ سلیمان پھلواروی کے پوتے تھے جو ہندوستان کے مشہور عالم و واعظ اور تحریک ندوۃ العلماء کے خاص رکن تھے، تقسیم کے بعد شاہ حسن مثنیٰ صاحب پاکستان چلے گئے اور وہاں سے مہرنیمروز کے نام سے ایک مذہبی و ادبی رسالہ نکالا، اس میں وہ ’’چہ دلاورست و زدے کہ بکف چراغ دار‘‘ کے عنوان سے ایک مستقل کالم لکھتے جس میں ادیبوں کی چوریوں کا ذکر ہوتا جو بہت پسند کیا جاتا، ادب سے ان کو بڑا شغف تھا اور ان کے ادبی و تنقیدی مضامین ان کے جوہر قابل ہونے کے غماز تھے۔ ان کا انتقال مارچ ہی میں ہوگیا تھا، لیکن پاکستان کی خبریں یہاں دیر سے پہنچتی ہیں، اﷲ تعالیٰ ان کو اپنی رحمت کاملہ سے نوازے، آمین! (ضیاء الدین اصلاحی، مئی ۱۹۹۸ء)
No society is safe from crimes hence with the passage of time, crimes amplify along with alteration in its nature. As the approaches of investigation and finding the crime develop, the ratio of crimes also increases and the casualties occur with new devices and techniques. On the other hand individual and collective endeavors are being made to stop it. The concerned authorities try to finish or decrease these crimes by formulating various new rules. The rules that the Creator and the real Owner of the whole world had bestowed upon us in the form of Islam, it includes the right and basic techniques to control the crimes. As the modern technology has facilitated us with many facilities, it has also facilitated us in finding a culprit or proofs against him that helps in the stoppage of crimes and finding the criminals. As this modern technology has brought a great reduction in the casualties and crimes, on the other hand we have also to face some legal and Islamic issues. One of these issues is the case of medical test for witness that whether the test of clinical laboratory can be accepted as witness
Inequalities lie at the heart of a great deal of mathematics. G. H. Hardy reported Harald Bohr as saying ‘all analysts spend half their time hunting through the literature for inequalities which they want to use but cannot prove’. Inequalities involving means open many doors for analysts e.g generalization of mixed means fallouts the refinements to the important inequalities of Holder and Minkowski. The well known Jensen’s inequality asserts a remarkable relation among the mean and the mean of function values and any improvement or refinements of Jensen’s inequality is a source to enrichment of monotone property of mixed means. our aim is to utilize all known refinements of Jensen’s inequality to give the re- finements of inequality among the power means by newly defined mixed symmetric means. In this context, our results not only ensures the generalization of classical but also speak about the most recent notions (e.g n-exponential convexity) of this era. In first chapter we start with few basic notions about means and convex functions. Then the classical Jensen’s inequality and the historical results about refinements of Jensen’s inequality are given from the literature together with their applications to the mixed symmetric means. In second chapter we consider recent refinements of Jensen’s inequality to refine inequality between power means by mixed symmetric means with positive weights under more comprehensive settings of index set. A new refinement of the classical Jensen’s inequality is also established. The Popovicui type inequality is generalized using green function. Using these refinements we define various versions of linear functionals that are positive on convex functions. This step ultimately leads us to viiviii the important and recently revitalized area of exponential convexity. Mean value theorems are proved for these functionals. Some non-trivial examples of exponential convexity and some classes of Cauchy means are given. These examples are further used to show monotonicity in defining parameters of constructed Cauchy means. In third chapter we develop the refinements of discrete Jensen’s inequality for con- vex functions of several variables which causes the generalizations of Beck’s results. The consequences of Beck’s results are given in more general settings. We also gen- eralize the inequalities of H ̈older and Minkowski by using the Quasiarithmetic mean function. In forth chapter we investigate the class of self-adjoint operators defined on a Hilbert space, whose spectra are contained in an interval. We extend several re- finements of the discrete Jensen’s inequality for convex functions to operator convex functions. The mixed symmetric operator means are defined for a subclass of positive self-adjoint operators to give the refinements of inequality between power means of strictly positive operators. In last chapter, some new refinements are given for Jensen’s type inequalities in- volving the determinants of positive definite matrices. Bellman-Bergstrom-Fan func- tionals are considered. These functionals are not only concave, but superlinear which is a stronger condition. The results take advantage of this property.