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Generalized Ostrowski Type Inequalities With Applications

Thesis Info

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Author

Irshad, Nazia

Program

PhD

Institute

University of Karachi

City

Karachi

Province

Sindh

Country

Pakistan

Thesis Completing Year

2019

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/12588/1/nazia%20irshad%20maths%202019%20uok%20karachi%20prr.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726261165

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In almost everyeld of science, inequalities play an important role. Although it is very vast discipline but our focus is mainly on Ostrowski type inequalities. Our aim is to discuss some weighted and non-weighted Ostrowski type inequalities with parameters and related results including Gr uss,Ceby sev and Ostrowski-Gr uss inequalities along with applications. Firstly, we generalize Ostrowski inequality for di erentiable functions in three di erent cases, that are bounded functions, bounded below functions and bounded above functions. Secondly, we introduce generalized inequalities of Ostrowski type for mappings on Lp spaces and of bounded variation. A weighted Ostrowski type inequality for twice di erentiable functions with bounded second derivatives and absolutely continuousrst derivatives are also discussed. Moreover, we generalize weighted Ostrowski-Gr uss type inequality with parameter for di erentiable functions by using weighted Korkine''s identity. Furthermore, we provide generalizations of Ostrowski type integral inequality with bounded derivatives by using the Riemann-Liouville fractional integral. We would like to give some generalizations of Ostrowski type inequality with better bounds. In addition, we modifyCeby sev inequality for two independent variables involving parameters. In the process of generalizing the Ostrowski type and its associated inequalities, we achieve new Montgomery identities with parameters for fractional, weighted and second order di erentiable functions of two variables. We will also discuss applications of our results in probability density function and numerical quadrature rules fornding error bounds of midpoint, trapezoidal, 1 3 Simpson''s, 3 8 Simpson''s and some non-standard quadrature rules. Our dissertation retrieves many established results. At places we will also obtain better bounds of di erent inequalities.
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شاعر صدیقی کی متفرق شاعری

شاعرؔ صدیقی کی متفرق شاعری
حمد
یہ شعری ادب کی وہ صنفِ سخن ہے جس میں اللہ تعالیٰ کی تعریف بیان کی جائے ۔عطاء الرحمٰن نوری اپنی کتاب ’’اْردو اصناف ادب‘‘ میں حمد کی تعریف کرتے ہوئییوں لکھتیہیں:
’’حمد ایک عربی لفظ ہے جس کے معنی ’’تعریف‘‘ کے ہے۔اللہ کی تعریف میں کہی جانے والی نظم کو’’حمد‘‘کہتے ہیں‘‘(۱)
اردو کی دیگر اصنافِ سخن میں حمد کو بہت اہمیت حاصل ہے۔ حمد کا تعلق چوں کہ براہِ راست اللہ تعالیٰ کی ذاتِ گرامی سے ہے اس لیے اردو زبان کے شعرا نے حمد کو بڑی اہمیت دی ہے۔ اردو کے دیگر شعرا کی طرح شاعرؔ صدیقی نے بھی حمدیہ شاعری کی ہے۔ اُن کی حمدیہ شاعری اْن کی مذہبی عقیدت مندی اورجذبات کی عکاسی کرتی ہے۔اگر چہ اْن کا حمدیہ کلام اتنا زیادہ نہیں ہے تاہم جوبھی ہے وہ معیار کے لحاظ اپنی مثال آپ ہے۔شاعرؔنے خالق حقیقی کی ثنا وتوصیف بیان کرنے میں ہنر مندی کے ساتھ ساتھ اپنے عجز اور انکساری کا مظاہر ہ بھی کیا ہے۔شاعرؔکے حمدیہ اشعار خالق کائنات پرکامل یقین واعتماد کے مظہر ہیں۔ اْنہوں نے بڑی عقیدت مندی کیساتھ اپنے رب کی تعریف وتوصیف بیان کی ہے۔اللہ سے بے پناہ محبت وعقیدت کی یہ جھلک محض نظموں تک محدود نہیں بلکہ دیگر اصنافِ سخن یعنی قطعہ ،باعی،اور دوہے میں بھی پوری شان کے ساتھ نمایاںہے۔حمد لکھنے کے لیے جو علمیت اور مطالعہ اسلام درکا ہوتا ہے شاعرؔصدیقی اس سے خوب بہرور ہے۔
مناجات
کلیات کے آغازمیں مناجات شامل ہیں جس میں شاعرؔنے اللہ سے دل کی وسعت اور فکر کی گہرائی مانگنے کے ساتھ ایسی بینائی مانگنے کی دْعا کی ہے جو اللہ کے جلوؤں کو دیکھنے کی صلاحیت رکھتی ہو:
دل میں وسعت دے میرے فکرکو گہرائی دے
ساتھ جو دے تیرے جلووں کا وہ بینائی دے
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