Sesión AnálisisNew integral inequalities of the Hermite-Hadamard type for functions $(h, m)$-convex twice differentiable
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.
Convex functions have been generalized widely; highlighting the m-convex function, r-convex function, h-convex function, (h, m)-convex function, s-convex function and many others. Readers interested in going through many of these extensions and generalizations of the classical notion of convexity can consult [1].
For convex functions, the following inequality is known, undoubtedly one of the most famous in Mathematics, for its multiple connections and applications:
\[ \phi \left( \frac { a+b }{ 2 } \right) \leq \frac { 1 }{ b-a } \int _{{ a }}^{{ b }}\quad \phi (x)\, dx\leq \frac { \phi (a)+\phi (b) }{ 2 }, \]
this is called the Hermite–Hadamard inequality.
This inequality was published by Hermite in 1883 ( [2]) and independently by Hadamard in 1893 ( [3]). In the last 30 years especially, many researchers have focused their attention on this inequality and many results have appeared.
In [4] 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 \[ \psi \left( \tau \xi +m(1-\tau )\varsigma \right) \leq h^{s}(\tau )\psi (\xi)+m(1-h^{s}(\tau ))\psi \left(\varsigma\right) \label{e:hmc1} \] is fulfilled for all $\xi ,\varsigma \in I$ and $\tau \in [0,1]$, where $m\in [0,1]$, $s\in [-1,1]$. 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 \[ \psi \left( \tau \xi +m(1-\tau )\varsigma \right) \leq h^{s}(\tau )\psi (\xi)+m(1-h(\tau ))^{s}\psi\left(\varsigma\right) \label{e:hmc11} \] is fulfilled for all $\xi ,\varsigma \in I$ and $\tau \in [0,1]$, where $m\in [0,1]$, $s\in [-1,1]$. Then a function $\psi $ is called a $(h,m)$-convex modified of the second type on $I$.
Remark. The reader can verify, without much difficulty, that various functional classes are particular cases of these definitions, for the appropriate choice of $h,m,s$.
Next we present the weighted integral operators, which will be the basis of our work.
Let $\phi \in L\left( [a,b]\right) $ and let $w $ be a continuous and positive function, $w :[0,1]\rightarrow [0,+\infty )$, with second order derivatives integrable on $I$. Then the weighted fractional integrals are defined by (right and left respectively):
\[ J_{a^{+}}^{w }\phi (r)=\int_{a}^{b}w''\left(\frac{\sigma -a}{b-a}\right) \phi (\sigma)d\sigma \]
and
\[ J_{b^{-}}^{w }\phi (r)=\int_{a}^{b}w''\left(\frac{b-\sigma }{b-a}\right) \phi (\sigma)d\sigma. \]
In this work, we obtain different variants of the Hermite-Hadamard inequality, in the framework of the $(h,m)$-convex modified functions, via generalized operators of the Definitions presented before.
Referencias
[1] J. E. Nápoles Valdés, 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.
[2] C. Hermite, Sur deux limites d'une intégrale définie, Mathesis 3, 82 (1883).
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