Let f and g be two integrable functions on ([a, b]).
For (k>0), let f and g be two integrable functions on ([a,b]) and (rgeq0), (alpha,beta>0).
For (k>0), let f and g be two integrable functions on ([a,b]) and let (varphi_{1}), (varphi_{2}), (psi_{1}), and (psi_{2}) be four integrable functions satisfying the conditions (3) and (4) on ([a,b]).
Theorem 4 Let f, g : [ a, b ] → R be two integrable functions such that θ ≤ f ( x ) ≤ ϕ and γ ≤ g ( x ) ≤ Γ for all x ∈ [ a, b ], and let ϕ, θ, Γ, γ be constants.
Remark 5 Let f, g and q be three integrable functions, with var h ( g ) ≠ 0 and var h ( q ) ≠ 0. If we take the following function: w = f − cov h ( f, g ) var h ( g ) g − λ q, then we have the inequality 0 ≤ [ cov h ( f, q ) cov h ( g, q ) − cov h ( f, q ) var h ( g ) ] 2 var h ( g ) var h ( q ) ≤ var h ( f ) var h ( g ) − | cov h ( f, g ) | 2. (2.).
The final result uses the Chebyshev integral inequality [22, p. 40]: Suppose f and g are two integrable functions monotonic in the same sense (either both decreasing or both increasing).
Let us introduce Chebyshev functional M f,g,p)= int_{0}^{T} p(x),dx int_{0}^{T}p(x f(x g(x),dx- int_{0}^{T}p(x f(x), dx int_{0}^{T}p(x g(x),dx, where (T>0), f and g are two integrable functions on ([0,T]), and p is a non-negative and integrable function on ([0,T]).
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}).
From the integral form of the inequality from Consequence 1 (see [29]) we deduce if f, g : [ a, b ] → R + are two integrable functions, g a continuous function on [ a, b ], g ( x ) > 0, x ∈ [ a, b ] and f ( x ) ∈ ( 0, 1 ), x ∈ [ a, b ] the following inequality: M 1 [ f g r ( 1 − f ) ] ≥ 1 M 1 r [ g ] M 1 [ f ] 1 − M 1 [ f ], where r ∈ [ 0, 1 ).
Using the integral form of the reverse of inequality from Theorem 2.5 (see [27]) we obtain, for p ∈ ( − 1, 0 ), m ∈ ( − 1, 0 ) and m ≤ p, if f, g : [ a, b ] → R + are two integrable functions on [ a, b ] with g ( x ) > 0, x ∈ [ a, b ] a continuous function on [ a, b ], the inequality M 1 [ f m + 1 g p ] ≤ M 1 m + 1 [ f ] M 1 p [ g ].
