Asian Research Thesis Index

Abstract

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.

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