ایڈیٹر برہان کو صدمہ
قارئین برہان کو یہ معلوم کرکے افسوس ہوگاکہ ۱۰؍جولائی کومولانا سعیداحمد اکبرآبادی مدیر برہان کاجواں سال بیٹا ’’عمر سعید‘‘ عمر۴۰ سال مختصر علالت کے بعد انتقال کرگیا۔ اِنَّالِلّٰہِ وَاِنَّااِلَیْہِ راجعون۔
مولانا کو مفتی صاحب ؒ کی وفات کاصدمۂ جانکاہ ابھی تازہ ہی تھا کہ ناگاہ ایک یہ حادثہ بھی پیش آگیا۔مولانا کواس قدرسخت صدمہ ہے کہ انھوں نے لکھنا پڑھنا سب ترک کردیا ہے اور ان پرایک عالم گمشدگی طاری ہے۔ قارئین سے درخواست ہے کہ وہ مرحوم کے لیے دعائے مغفرت کریں اورمولانا کے لیے بھی دعا فرمائیں کہ اﷲ تعالیٰ انھیں صبر جمیل کی توفیق عطا فرمائے ۔
[عمید الرحمن عثمانی(منیجر)، اگست ۱۹۸۴ء]
In this article the role played by the Muslim religious scholars in the Pakistan Movement has also been discussed. The most prominent among such scholars were Molana Mazharuddin Malik, Molana Shabeer Ahmad Usmani, Molana Ashraf Ali Thanvi, Molana Zafar Ahmad Ansari, Mufti Muhammad Shafee, Molana Ikram Khan Bengali, Molana Ahmad Raza Khan Brailvi, Molana Naeem Uddin Muradabadi, Molana Azad Subhani, Molana Abdul Hamid Badauni, and MolanaAbul Ala Maududi. At the end, an analysis of the ideology of Pakistan has been presented in the light of the excerpts taken from various speeches and statements made by the Quaid during 1938 and 1948. It shows that the Quaid wanted to make Pakistan an Islamic state governed by the teachings of Allah Taala. He wanted to make it a model Islamic state to convince others to realize that the commandments of Allah are practicable and are a means of salvation from hurdles and hardships.
It is well known that the theory of inequalities is considered as one of the central areas of mathematical analysis. It has many important applications in numerous scientific fields. In recent years, considerable attention has been given to this field in order to find the generalizations, variations and applications of different inequalities. The aim of this thesis is to prove several inequalities involving some special functions in terms of a new parameter k > 0. We can call these functions as special k-functions. Here, we do work on k-analogue gamma, beta and psi functions. This research work consists of eight chapters. In first chapter, we prove the inequalities involving k and q, k-analogue of gamma and psi functions. In chapter 2, We derive some classical inequalities of Chebyshev, H¨older and Gr¨uss type involving gamma and beta k-functions. In chapter 3, we discuss some basic properties, recurrence relation and special cases of incomplete gamma and beta functions in terms of the parameter k > 0. Some inequalities involving the incomplete beta k-functions are also given. In chapter 4, we prove some Gr¨uss type integral inequalities involving the generalized RiemannLiouville fractional integral in terms of parameter k > 0. In Chapter 5, the Ostrowski type inequalities involving the left and right-sided Riemann-Liouville fractional integrals are established in terms of the parameter k > 0. From our results, the classical Ostrowski inequalities can be deduced as some special cases. In chapter 6, we introduce the k-analogue of Hadamard fractional integral with some properties. We prove different types of inequalities involving this newly defined k-fractional integral. In chapter 7, we define the k-deformation of the fractional integral of a function with respect to another function and the inequalities involving the newly defined k-fractional integrals are also be proved. In last chapter, we introduce some properties of beta k-distribution and present some inequalities involving beta k-distribution via some classical inequalities, like Chebyshev’s inequality for synchronous (asynchronous) mappings and H¨older’s inequality. Also, we discuss the inequalities for harmonic mean, variance and coefficient of variation of βk random variable involving the parameter k > 0. Finally, we give conclusion of the present study and recommendations for the future work. Additionally, published work of the author has also been attached at the end of the thesis.