Let and be two synchronous functions on.
Let (f,g:[a,b]rightarrow[0,1]) be two synchronous functions and C be a copula.
Let C be a copula, (f,g:[a,b]rightarrow[0,1]) be two synchronous functions, (p:[a,b]rightarrowmathbb{R}) be an integrable function.
Let (f,g:[a,b]rightarrow[0,1]) be two synchronous functions, and (p:[a,b]rightarrowmathbb{R}) be a nonnegative integrable function.
Belarbi and Dahmani [2] gave the following integral inequality, using the Riemann-Liouville fractional integral: if and are two synchronous functions on, then (1.1).
Moreover, the authors [2] proved a generalized form of (1.1), namely that if and are two synchronous functions on, then (1.2). for all,, and.
Dragomir and Crstici [7] established the relationship between two synchronous functions and Chebyshevian mappings.
This type of functionals is usually defined as T f,g)= frac{1}{b-a} int_{a}^{b} f(x) g(x),dx- biggl(frac{1}{b-a} int_{a}^{b} f(x),dx biggr) biggl( frac{1}{b-a} int_{a}^{b} g(x),dx biggr), (1.1) where f and g are two integrable functions which are synchronous on ([a, b]), i.e., bigl(f(x -f y) bigr) bigl(g(x -f y) bigr)geq0, (1.2) for any (x, y in[a, bigl
The well-known Grüss inequality [13] is defined by biglvert T f, g bigrvert leqfrac{ M-m)(N-n)}{4}, (1.3) where f and g are two integrableqfrac{ M-m which are synchronous on ([a, b]) and satisfy the following inequalities: m leq f(x)leq Mquad text{and}{uad n leq g(y)leq N, (1.4} for all (x, y in[a, b]) and for some (m, M, n, N inmathbb{R}).
These are two distinct functions.
There are three different functions.
