Some Chebyshev type inequalities for generalized Riemann–Liouville operator

Abstract

In this paper we are interested in the famous inequality introduced by Chebyshev. This inequality has several generalizations and applications in different fields of mathematics and others. In particular it is important for us the applications of fractional calculus for the different integral Chebyshev type inequalities. We establish and prove some theorems and corollaries relating to fractional integral, by applying more general fractional integral operator than Riemann-Liouville one: $$ K^{\alpha,\beta}_{u,v}=\frac{v(x)}{\Gamma(\alpha)}\int\limits^{x}_{0}(x-t)^{\alpha -1}\left[\ln\left(\frac{x}{t}\right)\right]^{\beta-1}f(t) u(t)dt, \quad x>0 $$ where $\alpha>0$, $\beta\geq 1$, $u$ and $v$ locally integrable non-negative weight functions, $\Gamma $ is the Euler Gamma-function. First, fractional integral Chebyshev type inequalities are obtained for operator $K^{\alpha,\beta}_{u,v}$ with two synchronous or two asynchronous functions and by induction for several functions. Second, we consider an extended Chebyshev functional \begin{align*} T(f,g,p,q):=&\int\limits_{a}^{b} q(x) dx \int\limits_{a}^{b}p(x) f(x) g(x) dx + \int\limits_{a}^{b} p(x)dx\int\limits_{a}^{b}q(x)f(x)g(x)dx \\ &- \left(\int\limits_{a}^{b} q(x) f(x) dx\right)\left(\int\limits_{a}^{b} p(x) g(x)dx\right) \\ &- \left(\int\limits_{a}^{b} p(x) f(x) dx\right) \left(\int\limits_{a}^{b} q(x) g(x) dx\right), \end{align*} where $p$, $q$ are positive integrable weight functions on $[a,b]$. In this case fractional integral weighted inequalities are established for two fractional integral operators $K^{\alpha_{1},\beta_{1}}_{u_{1},v_{1}}$ and $K^{\alpha_{2},\beta_{2}}_{u_{2},v_{2}}$, with two synchronous or asynchronous functions, where $\alpha_ {1} \neq \alpha_{2}$, $\beta _{1} \neq \beta_{2}$ and $u_{1} \neq u_{2}$, $v_{1} \neq v_{2}$. In addition, a fractional integral Hölder type inequality for several functions is established using the operator $K^{\alpha,\beta}_{u,v}$. At the end, another fractional integral Chebyshev type inequality is given for increasing function $f$ and differentiable function $g$.