الفهرس 
Some Fractional Hardy?s InequalitiesOnly 14 pages are availabe for public view
 |  | from | 113 |  |  |
| | | from | 113 | | |
Abstract
Mathematical analysis has been considered the main part of mathematics for the last
three centuries. The study of inequalities can be considered one of basics of mathematical
analysis. In recent decades, many inequalities were developed and explored which were used
together with their proofs in many applications in the Öelds of pure mathematics, applied
mathematics and mathematical physics. An assembly of some of the proved and published
inequalities are represented in Hardy, Littlewood and G. PÚlya [32] known as ìInequalitiesî
which was published in 1934. We refer to ([13], and [54]), and the references cited there in
[55] for more information.
Fractional calculus which can be deÖned as the studying of di§erentiation and integra-
tion of fractional order is one of resulting Öelds of the evolution of mathematical analysis,
see ([35], [52], and [83]). There are many deÖnitions of fractional di§erentiation and integra-
tion operators like RiemannñLiouville deÖnition, Caputo deÖnition, and Gr¸nwald-Letnikov
deÖnition. The fractional calculus appeared in signal processing, temperature Öeld problem
in oil strata, hydraulics of dams and di§usion problems, see ([16], [19], and [80]). In [38],
the authors introduced a deÖnition of the fractional derivative called conformable fractional
derivative. Later, in [1], the author proved some basic concepts of the conformable fractional
calculus. Fractional inequalities generalized by many authors by using the conformable frac-
tional calculus, like Ste§ensenís inequality [77], Copson and Converses Copsonís inequality
[68], Hardyís inequality [71] and Hermite-Hadamardís inequalities ([4], [39], [40], [78], [81],
and [82]).
vi
vii
In this thesis, we will prove some fractional inequalities by utilizing conformable frac-
tional calculus and consequently, we derive some classical integral inequalities as special
cases of our main results. The thesis consists of four chapters arranged as follows:
Chapter 1 contains an introduction to conformabe fractional calculus and some basic
deÖnitions, lemmas and the necessary results of the conformable fractional calculus that are
required for obtaining the main results in the next chapters.
Chapter 2 consists of four sections. In the Örst section, we will introduce continuous
Hardyís inequality and some of its generalizations. In the second section, we will present
some generalizations of Yang and Hwangís and Pachpatteís inequalities. In the third section,
we will present some new fractional Hardyís inequalities. In the fourth section, we present
some applications of Hardyís inequality by using the conformable fractional calculus.
Chapter 3 consists of three sections. In the Örst section, we will present an introduction
to Opialís inequality and some of its extensions. In the second section, we will present some
generalizations of the weighted fractional Opialís inequalities. In the third section, we will
deduce some new fractional inequalities of Opialís inequalities.
Chapter 4 consists of three sections. In the Örst section, we will provide an introduction
to the Leindlerís inequality and some of its extensions. In the second section, we will present
some new fractional Leindlerís inequalities by using conformable fractional calculus. In the
third section, we will prove some new some new fractional reverse Leindlerís inequalities.