Sesión AnálisisNew weighted integral inequalities and fractional consequences
JUAN EDUARDO Nápoles Valdés
UNNE-FaCENA, UTN-FRRE, Argentina - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
In Mathematics, the notion of convex function plays a very prominent role due to its multiple applications and its theoretical overlaps with various other areas of science (see [10] for more information).
One of the most important inequalities for convex functions is the well-known Hermite-Hadamard inequality (see [4,5] and [9] for additional details):
$ \psi \left( \frac { \nu_1+\nu_2 }{ 2 } \right) \leq \frac { 1 }{ \nu_2-\nu_1 } \int _{\nu_1}^{\nu_2}\psi (x)dx\leq \frac { \psi (\nu_1)+\psi (\nu_2) }{ 2 }. $
In the last 25 years, we have witnessed a great growth in the number of researchers and their productions, interested in the Hermite-Hadamard Inequality. These productions have focused on the following work directions:
1) Using different notions of convexity.
2) Refinement of the mesh used (there is a crucial issue in this direction of work, suppose we use instead of $a$ and $b$, the ends of the interval, the points $a$, $\frac{a+b}{2}$ and $ b$, then we must ensure that at the midpoint, the integral operator used, does not have a jump, since the result would not be guaranteed in all $[a,b]$).
3) Improved estimates of the left and right members of Hermite-Hadamard inequality.
4) Using new generalized and fractional integral operators.
In [2] we presented the following definitions.
Let $h:[0,1]\rightarrow \mathbb{R}$ be a nonnegative function, $h\neq 0$ and $\psi :I=[0,+\infty )\rightarrow \lbrack 0,+\infty )$. If inequality $ \psi \left( \tau \xi +m(1-\tau )\varsigma \right) \leq h^{s}(\tau )\psi (\xi)+m(1-h^{s}(\tau ))\psi \left(\varsigma\right) $ is fulfilled for all $\xi ,\varsigma \in I$ and $\tau \in \lbrack 0,1]$, where $m\in \lbrack 0,1]$, $s\in \lbrack -1,1]$. Then a \ function \ $\psi $ is called a $(h,m)$-convex modified of the first type on $I$.
Let $h:[0,1]\rightarrow \mathbb{R}$ nonnegative functions, $ h\neq 0$ and $\psi :I=[0,+\infty )\rightarrow \lbrack 0,+\infty )$. If inequality $ \psi \left( \tau \xi +m(1-\tau )\varsigma \right) \leq h^{s}(\tau )\psi (\xi)+m(1-h(\tau ))^{s}\psi\left(\varsigma\right) $ is fulfilled for all $\xi ,\varsigma \in I$ and $\tau \in \lbrack 0,1]$, where $m\in \lbrack 0,1]$, $s\in \lbrack -1,1]$. Then a \ function \ $\psi $ is called a $(h,m)$-convex modified of the second type on $I$.
Interested readers can verify that the previous definitions contain many of the known notions of convexity.
A new way to define an integral operator, and take a first step in generalizing the known results, is to consider a certain weight in the definition of the operator integral, as follows:
(see [2]) Let $\phi \in L_{1}[a_1,a_2]$ and let $w$ be a continuous and positive function, $w:I\rightarrow \mathbb{R}^+$, with first derivative integrables on $ I° $. Then the weighted fractional integrals are defined by (right and left respectively):
$ I_{ a_1 + }^w \phi (t)= \int _{ a_1 }^{ t }{ w'''\left( \frac { a_2 -t }{ a_{ 2 }-a_{ 1 } } \right) } \phi (t)\, dt,\quad t \gt a_{ 1 } \\ I_{ a_ 2- }^w \phi (t)= \int _{ t }^{ a_2 }{ w'''\left( \frac { t-a_1 }{ a_{ 2 }-a_{ 1 } } \right) } \phi (t)\, dt,\quad t \lt a_{ 2 }. $
The consideration of the third derivative of the weight function $w$ is given by the nature of the problem to be solved, it can also be considered the first and second derivative.
To have a clearer idea of the amplitude of the previous Definition, let's consider some particular cases of the weight $w'''$:
a) Putting $w'''(t)\equiv \ 1$, we obtain the classical Riemann integral.
b) If $w'''(t)=\frac{t^{(\alpha - 1)}}{\Gamma(\alpha) }$, then we obtain the Riemann-Liouville fractional integral.
c) With convenient weight choices $w'''$ we can get the $k$-Riemann-Liouville fractional integral right and left, the right-sided fractional integrals of a function ${\psi }$ with respect to another function $h$ on $[a,b]$ (see [1]), the right and left integral operator of [6], the right and left sided generalized fractional integral operators and the integral operators of [7] and [8], can also be obtained from above Definition by imposing similar conditions to $w'$.
d) Of course there are other known integral operators, fractional or not, that can be obtained as particular cases of the previous one, but we leave it to interested readers.
In $2015$, Caputo and Fabrizio proposed the following operator (see [3]:
Let $0 \lt \alpha \leq 1$, $f \in AC^1[\nu_1,\nu_2]$. The right-sided and left-sided Caputo-Fabrizio fractional derivative of order $\alpha$ are defined as follows:
$ \left (^{C}D_{\nu_1+}^\alpha f \right )(t)=\frac{B(\alpha )}{1-\alpha} \int_{\nu_1}^{t}f'(x) e^{-\frac{\alpha (t-x)^\alpha }{1-\alpha }}dx, t \gt \nu_1 \\ \left (^{C}D_{\nu_2-}^\alpha f \right )(t)=-\frac{B(\alpha )}{1-\alpha} \int_{t}^{\nu_2}f'(x) e^{-\frac{\alpha (x-t)^\alpha }{1-\alpha }}dx, t \lt \nu_2, $
where $B(\alpha )$ is a normalization function such that $B(0)=B(1)=1$.
Their corresponding integral operators given by:
Let $0 \lt \alpha \leq 1$, $f \in AC^1[\nu_1,\nu_2]$. The right-sided and left-sided Caputo-Fabrizio integral of order $\alpha$ are defined as follows:
$ \left (^{CF}I_{\nu_1+}^\alpha f \right )(t)= \frac{1-\alpha }{B(\alpha )} f(t)+\frac{\alpha }{B(\alpha )} \int_{\nu_1}^{t}f(y) dy, t \gt \nu_1 \\ \left (^{CF}I_{\nu_2-}^\alpha f \right )(t)= \frac{1-\alpha }{B(\alpha )} f(t)+\frac{\alpha }{B(\alpha )} \int_{t}^{\nu_2}f(y) dy, t \lt \nu_2, $
where $B(\alpha )$ is a normalization function such that $B(0)=B(1)=1$.
In this paper we obtain new integral inequalities, within the framework of $(h,m)$-convex functions modified of second type, using weighted integrals. Various consequences for fractional integrals of type CF are presented throughout the work.
Referencias
[1] A. Akkurt, M. E. Yildirim, H. Yildirim, On some integral inequalities for $(k,h)$-RiemannLiouville fractional integral, NTMSCI 4, No. 1, 138-146 (2016) http://dx.doi.org/10.20852/ntmsci.2016217824
[2] B. Bayraktar, J. E. N\'apoles V., A note on Hermite-Hadamard integral inequality for $(h,m)$-convex modified functions in a generalized framework, submited.
[3] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl. 1 (2) (2015) 73-85
[4] J. Hadamard, \'Etude sur les propri\'et\'es des fonctions enti\'eres et en particulier d'une fonction consid\'er\'ee par Riemann, J. Math. Pures App. 9, 171-216 (1893).
[5] C. Hermite, Sur deux limites d'une int\'{e}grale d\'{e}finie, Mathesis3, 82 (1883).
[6] F. Jarad, T. Abdeljawad, T. Shah, ON THE WEIGHTED FRACTIONAL OPERATORS OF A FUNCTION WITH RESPECT TO ANOTHER FUNCTION, Fractals, Vol. 28, No. 8 (2020) 2040011 (12 pages) DOI: 10.1142/S0218348X20400113
[7] F. Jarad, E. Ugurlu, T. Abdeljawad, D. Baleanu, On a new class of fractional operators, Adv. Differ. Equ. 2017, 2017, 247.
[8] T. U. Khan, M. A. Khan, Generalized conformable fractional integral operators, J. Comput. Appl. Math. 2019, 346, 378-389.
[9] J. E. N\'apoles Valdes, A Review of Hermite-Hadamard Inequality, Partners Universal International Research Journal (PUIRJ), Volume: 01 Issue: 04 October-December 2022, 98-101 DOI:10.5281/zenodo.7492608
[10] J. E. N\'apoles Vald\'es, F. Rabossi, A. D. Samaniego, Convex functions: Ariadne's thread or Charlotte's spiderweb?, Advanced Mathematical Models \& Applications Vol.5, No.2, 2020, pp.176-191