Quantum Calculus: New Development and Applications
| dc.contributor.advisor | Prof. Dr. Prof. Dr. Chet Raj Bhatta | |
| dc.contributor.author | Tiwari, Pitamber | |
| dc.date.accessioned | 2024-08-25T04:29:32Z | |
| dc.date.available | 2024-08-25T04:29:32Z | |
| dc.date.issued | 2024-06 | |
| dc.description.abstract | The investigation of the calculus without the notion of limits is quantum calculus or q-calculus. This calculus is applied to estimate the discrete phenomena as well. The famous result on the theory of convex functions is the Hermite-Hadamard type integral inequality which provides a lower and an upper bounds for the integral average of any convex function defined on a compact interval involving the mid-point and the end points of the domain having a wider application for generalized means, information measures, quadrature rules, and many more. In this thesis, some new convexities related to harmonic convexity have been defined and for each new convexity, a corresponding Hermite-Hadamard inequality has been proved. The q-analogues of these inequalities have also been obtained. Apart from this, in this thesis we have generalized harmonically convex function to m-harmonically convex function and m-harmonically P-function and have obtained some Hermite-Hadamard type integral inequalities whose first order derivatives are m-harmonically convex functions. The integral mean of a convex function is connected with the Hermite-Hadamard inequality. From the results on the products of two classical convex functions as given by B. G. Pachpatte, in this thesis, we have been able to establish some new results on the products of generalized geometric convex functions, and these results are further extended to q-analogues which is a new paradigm in convexity theory and integral inequality. Diagnosing the idea of the classical convex functions, it is enhanced into m- convex functions which have helped to extend a convex function's equality of quantum estimate into an m-convex function. Using the resulted information, a few novel Hermite-Hadamard integral inequalities are established whose first order q-derivatives are m-convex functions, and these results are presented in q-analogues too. In this thesis, Hermite-Hadamard's inequalities via Riemann-Liouville fractional integral for the case of harmonically convex function as well as the products of two harmonically convex functions via Riemann-Liouville fractional integrals are also established. सिमा विनाको आधारमा खोज गरिएको calculus लाई quantum calculus वा q-calculus भनिन्छ । यो calculus को माध्यमबाट discrete (खण्डित) घट्नाहरूको पनि अनुमान गर्न मद्दत मिल्छ । Theory of convex functions र Theory of Inequality मा Hermite-Hadamard प्रकारको integral inequality एक अभिन्न inequality हो । यो inequality को मद्दतबाट कुनै पनि convex function को लागि माथिल्लो र तल्लो सिमा निर्धारण गर्न सकिन्छ । यस किसिमको inequality हरू generalized means, information measures, quadratures rules इत्यादिमा प्रयोग गरिन्छ । यो Thesis मा केहि नयां प्रकारका convexities लाई generalized form मा परिभाषित गरि तिनिहरूको Hermite-Hadamard प्रकारका integral inequality प्रमाणित गरिएका छन् । यसरी प्रमाणित गरिएका inequality लाई quantum calculus को स्वरूपमा पनि प्रमाणित गरिएका छन् । B. G. Pachpatte ले प्रस्तुत गरेको product of classical convex functions नतिजाको आधारमा अन्य प्रकारका convex functions जस्तै geometrically- arithmetic, harmonically – arithmetic, geometrically- geometric convex function हरूलाई generalized form मा प्रस्तुत गरी तिनिहरूका product form को नतिजाहरू प्रमाणित गरिएको छ । यसरी प्राप्त नतिजाहरूलाई q-calculus मा extension गरिएका छन् । यो Thesis मा harmonically- convex functions को लागि Riemann-Liouville fractional integral को माध्यमबाट Hermite-Hadamard integral inequality प्रमाणित गरी तिनिहरूको product form को नतिजा पनि प्रस्तुत गरिएका छन् । भविष्यमा अन्य अनुसन्धानकर्ताहरूले Theory of convex functions र Theory of inequalities मा बिभिन्न प्रकारका convex functions को Hermite- Hadamard प्रकारका integral inequality र तिनिहरूको quantum version को domain मा काम गर्न सक्नेछन् । | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14540/22719 | |
| dc.language.iso | en | |
| dc.publisher | Institue of Science & Technology | |
| dc.subject | Hermite-Hadamard inequality | |
| dc.subject | GA- convexity | |
| dc.subject | HA-convexity | |
| dc.subject | HG-convexity | |
| dc.subject | q-calculus | |
| dc.subject | fractional calculus | |
| dc.title | Quantum Calculus: New Development and Applications | |
| dc.type | Thesis | |
| local.academic.level | Ph.D. | |
| local.institute.title | Institute of Science & Technology |