نبی کریم ﷺ کے تعدد ازواج کے سماجی اثرات

The seerah of the Holy Prophet (SAW) is a diversified combination of various traits. Among hundreds of the aspects of seerah if analysed various such dimensions appear before us in accordance with the educational and cultural evolution and criticality of time. One of these many is the sociological aspect of the holy seerah of the Prophet (SAW). According to the teachings of the Holy Prophet SAW Islam n Society are quite compatible to each other where marital element holds foremost importance in social circles. There is complete guidance about it in the seerah. The Prophet of Islam himself provided practical model of polygamy which was subjected to severe censure by the non-believers on account of their prejudice, ignorance and dishonesty and which the research scholars of seerah responded to n refuted on logical, convincing and solid grounds. The objective of this thesis is to highlight various positive effects of the polygamy of the Prophet SAW on society and its value in eradicating a number of social evils. Among manifold positive effects of this practice of polygamy include such benefits as the well-being and social elevation of widows, the eradication of the frequent custom of adoption, the extinction of social distinction and discrimination, the removal of tribal n social enmities, the following of the Prophet's model of women's education, the recognition of social work and social workers, and the upbringing of orphans.

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.