متابعتِ رسولﷺ (تکمیلِ ایمان) کے سات درجے: مکتوبات امام ربانی مجدد الفِ ثانی کی روشنی میں

The digit seven has great importance in our life. Seven rounds of  Holy Kabah, seven heavens, seven layers of earth, seven levels of  hell, seven recitation of Holy Quran, seven interior and exterior (meanings) of holy Quran, seven stages of human life, etc. Hazrat Mujadid Alf Sani mentioned the seven degrees and their secrets of the obedience of the Holy Prophet: Say: "if you do love (obey) Allah, then follow me, Allah will (love) save you". Actually the perfect following of the Holy Prophet is the source of the completion of faith. As we adopt the following of the Holy Prophet, so and so our faith will reach to the perfection. In this article, the introduction of seven degrees of the following of the Holy Prophet and their secrets are described, in the light of 54th writing in book II. So books so that every Muslim after seeing his faith, could be able to complete the degrees of the perfection of faith and could get the nearness of God.

Harmonically S, M -Convex Functions and Related Inequalities

The theory developed about convex functions, arising from intuitive geometrical observations, may be readily applied to topics in real analysis and economics.Convexity is a simple and natural notion which can be traced back to Archimedes (circa 250 B.C.), in connection with his famous estimate of the value of π (using inscribed and circumscribed regular polygons). He noticed the important fact that the perimeter of a convex figure is smaller than the perimeter of any other convex figure surrounding it. In modern era, there occurs a rapid development in the theory of convex functions. There are sereval reasons behind it: firstly, many areas in modern analysis directly or indirectly involve the applications of convex functions; secondly, convex functions are closely related to the theory of inequalities and many important inequalities are consequences of the applications of convex functions (see [64, 47]). Inequalities play a important role in almost all fields of mathematics. Several applications of inequalities are found in various area of sciences such as, physical, natural, engineering sciences. In numerical analysis, inequalities play a main role in error estimates of different important integrals whom analytic solutions could not be found. In recent years, a number of authors have considered extensions/generalizations of convex functions in various aspects and also tried to build several relations like HemiteHadamard’s inequalities. We introduce different types of convexities and drive more general Hermite-Hadamard’s inequalities. Further, we derive Ostrowski, Hermite-Hadamard and Simpson, Fejer type inequalities. Also, we discuss applications of these classes such that we can estimate the integrals like Rab x ex n dx; Rab sin xnxdx for n ≥ 1 and a, b ∈ (0, ∞) etc without using numerical analysis. In first chapter, we give information about convex functions, Log-convex functions, Quasi-convex functions, (s, m)-convex functions in second sense and Preinvex functions. In second chapter, we consider the class of harmonically convex functions and investigate some relations between harmonically convex and classical convex functions. We define class of harmonically (s, m)-convex functions which unify the different harmonic convexities and establish Ostrowski, Hemite-Hadamard and Simpson, and Fejer type inequalities for this class of functions. In third chapter, we define the classes of the harmonically p-convex functions which is a generalization of convex functions and harmonically convex functions, and harmonically p-quasiconvex functions, and harmonically logarithmic p-convex functions. Further, we investigate relationship between harmonically p-convex, p-convex and classical convex functions. Also, we give a characterization about the relation between harmonically p-convex and harmonically convex functions. Finally, we establish Hermite-Hadamard type inequalities for harmonically p-convex functions, and inequalities for the product of harmonically p-convex functions, and inequalities for harmonically logarithmic p-convex functions. In fourth chapter, we define the class of p-preinvex functions which is generalization of preinvex and harmonically preinvex functions. We also define the notion of p-prequasiinvex and logarithmic p-preinvex functions. Moreovere, we establish Hermite-Hadamard type inequalities when the power of the absolute value of the derivative of the integrand is p-preinvex and we give results for product of two ppreinvex, and logarithmic p-preinvex functions, and Ostrowski’s type for the class of p-preinvex functions. In fifth chapter, we define harmonically (p, h, m)-preinvex functions which is generalization of harmonically preinvex functions such that preinvex and harmonically p-convex functions are its special cases. Next, we introduce the concept of harmonically logarithmic p-preinvex and harmonically p-quasipreinvex functions. Finally, we establish important and interesting results related to these classes of functions.