CUBO, A Mathematical Journal Vol.22, No¯ 01, (01–21). April 2020 http: // doi. org/ 10. 4067/ S0719-06462020000100001 Bounds for the Generalized (Φ, f)-Mean Difference Silvestru Sever Dragomir 1,2 1Mathematics, College of Engineering & Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia. sever.dragomir@vu.edu.au, http://rgmia.org/dragomir 2School of Computer Science & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa ABSTRACT In this paper we establish some bounds for the (Φ, f)-mean difference introduced in the general settings of measurable spaces and Lebesgue integral, which is a two functions generalization of Gini mean difference that has been widely used by economists and sociologists to measure economic inequality. RESUMEN En este art́ıculo establecemos algunas cotas para la (Φ, f)-diferencia media introdu- cida en el contexto general de espacios medibles e integral de Lebesgue, que es una generalización a dos funciones de la diferencia media de Gini que ha sido ampliamente utilizada por economistas y sociólogos para medir desigualdad económica. Keywords and Phrases: Gini mean difference, Mean deviation, Lebesgue integral, Expectation, Jensen’s integral inequality. 2010 AMS Mathematics Subject Classification: 26D15; 26D10; 94A17. http://doi.org/10.4067/S0719-06462020000100001 2 Silvestru Sever Dragomir CUBO 22, 1 (2020) 1. Introduction Let (Ω, A, ν) be a measurable space consisting of a set Ω, a σ -algebra A of subsets of Ω and a countably additive and positive measure ν on A with values in R ∪ {∞} . For a ν-measurable function w : Ω → R, with w (x) ≥ 0 for ν-a.e. (almost every) x ∈ Ω and ∫ Ω w (x) dν (x) = 1, consider the Lebesgue space Lw (Ω, ν) := {f : Ω → R, f is ν-measurable and ∫ Ω w (x) |f (x)| dν (x) < ∞}. Let I be an interval of real numbers and Φ : I → R a Lebesgue measurable function on I. For f : Ω → I a ν-measurable function with Φ ◦ f ∈ Lw (Ω, ν) we define the generalized (Φ, f)-mean difference RG (Φ, f; w) by RG (Φ, f; w) := 1 2 ∫ Ω ∫ Ω w (x) w (y) |(Φ ◦ f) (x) − (Φ ◦ f) (y)| dν (x) dν (y) (1.1) and the generalized (Φ, f)-mean deviation MD (Φ, f; w) by MD (Φ, f; w) := ∫ Ω w (x) |(Φ ◦ f) (x) − E (Φ, f; w)| dν (x) , (1.2) where E (Φ, f; w) := ∫ Ω (Φ ◦ f) (y) w (y) dν (y) the generalized (Φ, f)-expectation. If Φ = e, where e (t) = t, t ∈ R is the identity mapping, then we can consider the particular cases of interest, the generalized f-mean difference RG (f; w) := RG (e, f; w) = 1 2 ∫ Ω ∫ Ω w (x) w (y) |f (x) − f (y)| dν (x) dν (y) (1.3) and the generalized f-mean deviation MD (f; w) := MD (e, f; w) = ∫ Ω w (x) |f (x) − E (f; w)| dν (x) , (1.4) where E (f; w) := ∫ Ω f (y) w (y) dν (y) is the generalized f-expectation. If Ω = [−∞, ∞] and f = e then we have the usual mean difference RG (w) := RG (f; w) = 1 2 ∫ ∞ −∞ ∫ ∞ −∞ w (x) w (y) |x − y| dxdy (1.5) and the mean deviation MD (w) := MD (f; w) = ∫ Ω w (x) |x − E (w)| dx, (1.6) CUBO 22, 1 (2020) Bounds for the Generalized (Φ, f)-Mean Difference 3 where w : R →[0, ∞) is a density function, this means that w is integrable on R and ∫ ∞ −∞ w (t) dt = 1, and E (w) := ∫ ∞ −∞ xw (x) dx (1.7) denote the expectation of w provided that the integral exists and is finite. The mean difference RG (w) was proposed by Gini in 1912 [21], after whom it is usually named, but was discussed by Helmert and other German writers in the 1870’s (cf. H. A. David [13], see also [26, p. 48]). It has a certain theoretical attraction, being dependent on the spread of the variate-values among themselves and not on the deviations from some central value ([26, p. 48]). Further, its defining integral (1.5) may converge when that of the variance σ (w) , σ (w) := ∫ ∞ −∞ (x − E (w)) 2 w (x) dx, (1.8) does not. It is, however, more difficult to compute than the standard deviation. For some recent results concerning integral representations and bounds for RG (w) see [5], [6], [8] and [9]. For instance, if w : R →[0, ∞) is a density function we define by W (x) := ∫x −∞ w (t) dt, x ∈ R its cumulative function. Then we have [5], [6]: RG (w) = 2 Cov (e, W) = ∫ ∞ −∞ (1 − W (y)) W (y) dy = 2 ∫ ∞ −∞ xw (x) W (x) dx − E (w) = 2 ∫ ∞ −∞ (x − E (w)) (W (x) − γ) w (x) dx = 2 ∫ ∞ −∞ (x − δ) ( W (x) − 1 2 ) w (x) dx (1.9) for any γ, δ ∈ R and [6]: RG (w) = ∫ ∞ −∞ ∫ ∞ −∞ (x − y) (W (x) − W (y)) w (x) w (y) dxdy. (1.10) With the above assumptions, we have the bounds [5]: 1 2 MD (w) ≤ RG (w) ≤ 2 sup x∈R |W (x) − γ| MD (w) ≤ MD (w) , (1.11) 4 Silvestru Sever Dragomir CUBO 22, 1 (2020) for any γ ∈ [0, 1] , where W (·) is the cumulative distribution of w and MD (w) is the mean deviation. Consider the n-tuple of real numbers a = (a1, ..., an) and p = (p1, ..., pn) a probability distribution, i.e. pi ≥ 0 for each i ∈ {1, ..., n} with ∑n i=1 pi = 1, then by taking Ω = {1, ..., n} and the discrete measure, we can consider from (1.1) and (1.2) that (see [7]) RG (a; p) := 1 2 n∑ i=1 n∑ j=1 pipj |Φ (ai) − Φ (aj)| , (1.12) and MD (a; p) := 1 2 n∑ i=1 pi ∣ ∣ ∣ ∣ ∣ ∣ Φ (ai) − n∑ j=1 pjΦ (aj) ∣ ∣ ∣ ∣ ∣ ∣ (1.13) where a ∈ In := I × ... × I and Φ : I → R. The quantity RG (a; p) has been defined in [7] and some results were obtained. In the case when Φ = e, then we get the special case of Gini mean difference and mean deviation of an empirical distribution that is particularly important for applications, RG (a; p) := 1 2 n∑ i=1 n∑ j=1 pipj |ai − aj| , (1.14) and MD (a; p) := 1 2 n∑ i=1 pi ∣ ∣ ∣ ∣ ∣ ∣ ai − n∑ j=1 pjaj ∣ ∣ ∣ ∣ ∣ ∣ . (1.15) The following result incorporates an upper bound for the weighted Gini mean difference [7]: For any a ∈ Rn and any p a probability distribution, we have the inequality: 1 2 MD (a; p) ≤ RG (a; p) ≤ ı́nf γ∈R [ n∑ i=1 pi |ai − γ| ] ≤ MD (a; p) . (1.16) The constant 1 2 in the first inequality in (1.16) is sharp. For some recent results for discrete Gini mean difference and mean deviation, see [7], [11], [14] and [15]. CUBO 22, 1 (2020) Bounds for the Generalized (Φ, f)-Mean Difference 5 2. General Bounds We have: Theorem 1. Let I be an interval of real numbers and Φ : I → R a Lebesgue measurable function on I. If w : Ω → R is a ν-measurable function with w (x) ≥ 0 for ν-a.e. (almost every) x ∈ Ω and ∫ Ω w (x) dν (x) = 1 and if f : Ω → I is a ν-measurable function with Φ ◦ f ∈ Lw (Ω, ν) , then 1 2 MD (Φ, f; w) ≤ RG (Φ, f; w) ≤ I (Φ, f; w) ≤ MD (Φ, f; w) , (2.1) where I (Φ, f; w) := ı́nf γ∈R ∫ Ω w (x) |(Φ ◦ f) (x) − γ| dν (x) . (2.2) Demostración. Using the properties of the integral, we have RG (Φ, f; w) = 1 2 ∫ Ω ∫ Ω w (x) w (y) |(Φ ◦ f) (x) − (Φ ◦ f) (y)| dν (x) dν (y) ≥ 1 2 ∫ Ω w (x) ∣ ∣ ∣ ∣ (Φ ◦ f) (x) ∫ Ω w (y) dν (y) − ∫ Ω w (y) (Φ ◦ f) (y) dν (y) ∣ ∣ ∣ ∣ dν (x) = 1 2 ∫ Ω w (x) ∣ ∣ ∣ ∣ (Φ ◦ f) (x) − ∫ Ω w (y) (Φ ◦ f) (y) dν (y) ∣ ∣ ∣ ∣ dν (x) = 1 2 MD (Φ, f; w) and the first inequality in (2.1) is proved. By the triangle inequality for modulus we have |(Φ ◦ f) (x) − (Φ ◦ f) (y)| = |(Φ ◦ f) (x) − γ + γ − (Φ ◦ f) (y)| (2.3) ≤ |(Φ ◦ f) (x) − γ| + |(Φ ◦ f) (y) − γ| for any x, y ∈ Ω and γ ∈ R. 6 Silvestru Sever Dragomir CUBO 22, 1 (2020) Now, if we multiply (2.3) by 1 2 w (x) w (y) and integrate, we get RG (Φ, f; w) = 1 2 ∫ Ω ∫ Ω w (x) w (y) |(Φ ◦ f) (x) − (Φ ◦ f) (y)| dν (x) dν (y) ≤ 1 2 ∫ Ω ∫ Ω w (x) w (y) [|(Φ ◦ f) (x) − γ| + |(Φ ◦ f) (y) − γ|] dν (x) dν (y) = 1 2 ∫ Ω ∫ Ω w (x) w (y) |(Φ ◦ f) (x) − γ| dν (x) dν (y) + 1 2 ∫ Ω ∫ Ω w (x) w (y) |(Φ ◦ f) (y) − γ| dν (x) dν (y) = 1 2 ∫ Ω w (x) |(Φ ◦ f) (x) − γ| dν (x) + 1 2 ∫ Ω w (y) |(Φ ◦ f) (y) − γ| dν (y) = ∫ Ω w (x) |(Φ ◦ f) (x) − γ| dν (x) (2.4) for any γ ∈ R. Taking the infimum over γ ∈ R in (2.4) we get the second part of (2.1). Since, obviously I (Φ, f; w) = ı́nf γ∈R ∫ Ω w (x) |(Φ ◦ f) (x) − γ| dν (x) ≤ ∫ Ω w (x) ∣ ∣ ∣ ∣ (Φ ◦ f) (x) − ∫ Ω w (y) (Φ ◦ f) (y) dν (y) ∣ ∣ ∣ ∣ dν (x) = MD (Φ, f; w) , the last part of (2.1) is thus proved. By the Cauchy-Bunyakowsky-Schwarz (CBS) inequality, if (Φ ◦ f) 2 ∈ Lw (Ω, ν) , then we have [∫ Ω w (x) ∣ ∣ ∣ ∣ (Φ ◦ f) (x) − ∫ Ω w (y) (Φ ◦ f) (y) dν (y) ∣ ∣ ∣ ∣ dν (x) ]2 ≤ ∫ Ω w (x) [ (Φ ◦ f) (x) − ∫ Ω w (y) (Φ ◦ f) (y) dν (y) ]2 dν (x) = ∫ Ω w (x) (Φ ◦ f) 2 (x) dν (x) − 2 ∫ Ω w (y) (Φ ◦ f) (y) dν (y) ∫ Ω w (x) (Φ ◦ f) (x) dν (x) + [∫ Ω w (y) (Φ ◦ f) (y) dν (y) ]2 ∫ Ω w (x) dν (x) = ∫ Ω w (x) (Φ ◦ f) 2 (x) dν (x) − [∫ Ω w (x) (Φ ◦ f) (x) dν (x) ]2 . CUBO 22, 1 (2020) Bounds for the Generalized (Φ, f)-Mean Difference 7 By considering the generalized (Φ, f)-dispersion σ (Φ, f; w) := (∫ Ω w (x) (Φ ◦ f) 2 (x) dν (x) − [∫ Ω w (x) (Φ ◦ f) (x) dν (x) ]2 )1/2 , then we have MD (Φ, f; w) ≤ σ (Φ, f; w) (2.5) provided (Φ ◦ f) 2 ∈ Lw (Ω, ν). If there exists the constants m, M so that − ∞ < m ≤ Φ (t) ≤ M < ∞ for almost any t ∈ I (2.6) then by the reverse CBS inequality σ (Φ, f; w) ≤ 1 2 (M − m) , (2.7) by (2.1) and by (2.5) we can state the following result: Corollary 1. Let I be an interval of real numbers and Φ : I → R a Lebesgue measurable function on I satisfying the condition (2.6) for some constants m, M. If w : Ω → R is a ν-measurable function with w (x) ≥ 0 for ν -a.e. x ∈ Ω and ∫ Ω w (x) dν (x) = 1 and if f : Ω → I is a ν-measurable function with (Φ ◦ f) 2 ∈ Lw (Ω, ν) , then we have the chain of inequalities 1 2 MD (Φ, f; w) ≤ RG (Φ, f; w) ≤ I (Φ, f; w) ≤ MD (Φ, f; w) ≤ σ (Φ, f; w) ≤ 1 2 (M − m) . (2.8) We observe that, in the discrete case we obtain from (2.1) the inequality (1.16) while for the univariate case with ∫ ∞ −∞ w (t) dt = 1 we have 1 2 MD (w) ≤ RG (w) ≤ I (w) ≤ MD (w) ≤ σ (Φ, f; w) (2.9) where I (w) := ı́nf γ∈R ∫ ∞ −∞ w (x) |x − γ| dx. (2.10) If w is supported on the finite interval [a, b] , namely ∫b a w (x) dx = 1, then we have the chain of inequalities 1 2 MD (w) ≤ RG (w) ≤ I (w) ≤ MD (w) ≤ σ (Φ, f; w) ≤ 1 2 (M − m) . (2.11) 8 Silvestru Sever Dragomir CUBO 22, 1 (2020) 3. Bounds for Various Classes of Functions In the case of functions of bounded variation we have: Theorem 2. Let Φ : [a, b] → R be a function of bounded variation on the closed interval [a, b] . If w : Ω → R is a ν-measurable function with w (x) ≥ 0 for ν -a.e. x ∈ Ω and ∫ Ω w (x) dν (x) = 1 and if f : Ω → [a, b] is a ν-measurable function with Φ ◦ f ∈ Lw (Ω, ν) , then RG (Φ, f; w) ≤ 1 2 b ∨ a (Φ) , (3.1) where ∨b a (Φ) is the total variation of Φ on [a, b] . Demostración. Using the inequality (2.4) we have RG (Φ, f; w) ≤ ∫ Ω w (x) |(Φ ◦ f) (x) − γ| dν (x) (3.2) for any γ ∈ R. By the triangle inequality, we have ∣ ∣ ∣ ∣ (Φ ◦ f) (x) − 1 2 [Φ (a) + Φ (b)] ∣ ∣ ∣ ∣ ≤ 1 2 |Φ (a) − Φ (f (x))| + 1 2 |Φ (b) − Φ (f (x))| (3.3) for any x ∈ Ω. Since Φ : [a, b] → R is of bounded variation and d is a division of [a, b] , namely d ∈ D ([a, b]) := {d := {a = t0 < t1 < ... < tn = b}} , then b ∨ a (Φ) = sup d∈D([a,b]) n−1∑ i=0 |Φ (ti+1) − Φ (ti)| < ∞. Taking the division d0 := {a = t0 < t < t2 = b} we then have |Φ (t) − Φ (a)| + |Φ (b) − Φ (t)| ≤ b ∨ a (Φ) for any t ∈ [a, b] and then |Φ (f (x)) − Φ (a)| + |Φ (b) − Φ (f (x))| ≤ b ∨ a (Φ) (3.4) for any x ∈ Ω. CUBO 22, 1 (2020) Bounds for the Generalized (Φ, f)-Mean Difference 9 On making use of (3.3) and (3.4) we get ∣ ∣ ∣ ∣ (Φ ◦ f) (x) − 1 2 [Φ (a) + Φ (b)] ∣ ∣ ∣ ∣ ≤ 1 2 b ∨ a (Φ) (3.5) for any x ∈ Ω. If we multiply (3.5) by w (x) and integrate, then we obtain ∫ Ω w (x) ∣ ∣ ∣ ∣ (Φ ◦ f) (x) − 1 2 [Φ (a) + Φ (b)] ∣ ∣ ∣ ∣ ≤ 1 2 b ∨ a (Φ) . (3.6) Finally, by choosing γ = 1 2 [Φ (a) + Φ (b)] in (3.2) and making use of (3.6) we deduce the desired result (3.1). In the case of absolutely continuous functions we have: Theorem 3. Let Φ : [a, b] → R be an absolutely continuous function on the closed interval [a, b] . If w : Ω → R is a ν-measurable function with w (x) ≥ 0 for ν -a.e. x ∈ Ω and ∫ Ω w (x) dν (x) = 1 and if f : Ω → [a, b] is a ν-measurable function with Φ ◦ f ∈ Lw (Ω, ν) , then RG (Φ, f; w) ≤    ‖Φ′‖ [a,b],∞ RG (f; w) if Φ ′ ∈ L ∞ ([α, β]) , 1 21/p ‖Φ′‖ [a,b],p R 1/q G (f; w) if Φ ′ ∈ Lp ([α, β]) , p > 1, 1 p + 1 q = 1, (3.7) where the Lebesgue norms are defined by ‖g‖ [α,β],p :=    essupt∈[α,β] |g (t)| if p = ∞, (∫β α |g (t)| p dt )1/p if p ≥ 1 and Lp ([α, β]) := { g| g measurable and ‖g‖ [α,β],p < ∞ } , p ∈ [1, ∞] . Demostración. Since f is absolutely continuous, then we have Φ (t) − Φ (s) = ∫t s Φ′ (u) du for any t, s ∈ [a, b] . Using the Hölder integral inequality we have |Φ (t) − Φ (s)| = ∣ ∣ ∣ ∣ ∫t s Φ′ (u) du ∣ ∣ ∣ ∣ ≤    ‖Φ′‖ [a,b],∞ |t − s| if p = ∞, ‖Φ′‖ [a,b],p |t − s| 1/q if p > 1, 1 p + 1 q = 1 (3.8) 10 Silvestru Sever Dragomir CUBO 22, 1 (2020) for any t, s ∈ [a, b] . Using (3.8) we then have |(Φ ◦ f) (x) − (Φ ◦ f) (y)| ≤    ‖Φ′‖ [a,b],∞ |f (x) − f (y)| if p = ∞, ‖Φ′‖[a,b],p |f (x) − f (y)| 1/q if p > 1, 1 p + 1 q = 1 (3.9) for any x, y ∈ Ω. If we multiply (3.9) by 1 2 w (x) w (y) and integrate, then we get 1 2 ∫ Ω ∫ Ω w (x) w (y) |(Φ ◦ f) (x) − (Φ ◦ f) (y)| dν (x) dν (y) ≤    1 2 ‖Φ′‖[a,b],∞ ∫ Ω ∫ Ω w (x) w (y) |f (x) − f (y)| dν (x) dν (y) if p = ∞, 1 2 ‖Φ′‖ [a,b],p ∫ Ω ∫ Ω w (x) w (y) |f (x) − f (y)| 1/q dν (x) dν (y) if p > 1, 1 p + 1 q = 1. (3.10) This proves the first branch of (3.7). Using Jensen’s integral inequality for concave function Ψ (t) = ts, s ∈ (0, 1) we have for s = 1 q < 1 that ∫ Ω ∫ Ω w (x) w (y) |f (x) − f (y)| 1/q dν (x) dν (y) ≤ (∫ Ω ∫ Ω w (x) w (y) |f (x) − f (y)| dν (x) dν (y) )1/q , which implies that 1 2 ‖Φ′‖ [a,b],p ∫ Ω ∫ Ω w (x) w (y) |f (x) − f (y)| 1/q dν (x) dν (y) ≤ 1 2 ‖Φ′‖ [a,b],p (∫ Ω ∫ Ω w (x) w (y) |f (x) − f (y)| dν (x) dν (y) )1/q = ‖Φ′‖ [a,b],p ( 1 2q ∫ Ω ∫ Ω w (x) w (y) |f (x) − f (y)| dν (x) dν (y) )1/q = ‖Φ′‖ [a,b],p ( 1 2q−1 1 2 ∫ Ω ∫ Ω w (x) w (y) |f (x) − f (y)| dν (x) dν (y) )1/q = 1 2 q−1 q ‖Φ′‖ [a,b],p (RG (f; w)) 1/q = 1 21/p ‖Φ′‖ [a,b],p R 1/q G (f; w) and the second part of (3.7) is proved. The function Φ : [a, b] → R is called of r-H-Hölder type with the given constants r ∈ (0, 1] and H > 0 if |Φ (t) − Φ (s)| ≤ H |t − s| r CUBO 22, 1 (2020) Bounds for the Generalized (Φ, f)-Mean Difference 11 for any t, s ∈ [a, b] . In the case when r = 1, namely, there is the constant L > 0 such that |Φ (t) − Φ (s)| ≤ L |t − s| for any t, s ∈ [a, b] , the function Φ is called L-Lipschitzian on [a, b] . We have: Theorem 4. Let Φ : [a, b] → R be a function of r-H-Hölder type on the closed interval [a, b] . If w : Ω → R is a ν-measurable function with w (x) ≥ 0 for ν-a.e. x ∈ Ω and ∫ Ω w (x) dν (x) = 1 and if f : Ω → [a, b] is a ν-measurable function with Φ ◦ f ∈ Lw (Ω, ν) , then RG (Φ, f; w) ≤ 1 21−r HRrG (f; w) . (3.11) In particular, if Φ is L-Lipschitzian on [a, b] , then RG (Φ, f; w) ≤ LRG (f; w) . (3.12) Demostración. We have |(Φ ◦ f) (x) − (Φ ◦ f) (y)| ≤ H |f (x) − f (y)| r (3.13) for any x, y ∈ Ω. If we multiply (3.13) by 1 2 w (x) w (y) and integrate, then we get 1 2 ∫ Ω ∫ Ω w (x) w (y) |(Φ ◦ f) (x) − (Φ ◦ f) (y)| dν (x) dν (y) ≤ 1 2 H ∫ Ω ∫ Ω w (x) w (y) |f (x) − f (y)| r dν (x) dν (y) . (3.14) By Jensen’s integral inequality for concave functions we also have ∫ Ω ∫ Ω w (x) w (y) |f (x) − f (y)| r dν (x) dν (y) ≤ (∫ Ω ∫ Ω w (x) w (y) |f (x) − f (y)| dν (x) dν (y) )r . (3.15) Therefore, by (3.14) and (3.15) we get RG (Φ, f; w) ≤ 1 2 H (∫ Ω ∫ Ω w (x) w (y) |f (x) − f (y)| dν (x) dν (y) )r = 1 21−r H ( 1 2 ∫ Ω ∫ Ω w (x) w (y) |f (x) − f (y)| dν (x) dν (y) )r = 1 21−r HRrG (f; w) and the inequality (3.11) is proved. 12 Silvestru Sever Dragomir CUBO 22, 1 (2020) We have: Theorem 5. Let Φ, Ψ : [a, b] → R be continuos functions on [a, b] and differentiable on (a, b) with Ψ′ (t) 6= 0 for t ∈ (a, b) . If w : Ω → R is a ν-measurable function with w (x) ≥ 0 for ν -a.e. x ∈ Ω and ∫ Ω w (x) dν (x) = 1 and if f : Ω → [a, b] is a ν-measurable function with Φ ◦ f ∈ Lw (Ω, ν) , then ı́nf t∈(a,b) ∣ ∣ ∣ ∣ Φ′ (t) Ψ′ (t) ∣ ∣ ∣ ∣ RG (Ψ, f; w) ≤ RG (Φ, f; w) ≤ sup t∈(a,b) ∣ ∣ ∣ ∣ Φ′ (t) Ψ′ (t) ∣ ∣ ∣ ∣ RG (Ψ, f; w) . (3.16) Demostración. By the Cauchy’s mean value theorem, for any t, s ∈ [a, b] with t 6= s there exists a ξ between t and s such that Φ (t) − Φ (s) Ψ (t) − Ψ (s) = Φ′ (ξ) Ψ′ (ξ) . This implies that ı́nf τ∈(a,b) ∣ ∣ ∣ ∣ Φ′ (τ) Ψ′ (τ) ∣ ∣ ∣ ∣ |Ψ (t) − Ψ (s)| ≤ |Φ (t) − Φ (s)| ≤ sup τ∈(a,b) ∣ ∣ ∣ ∣ Φ′ (τ) Ψ′ (τ) ∣ ∣ ∣ ∣ |Ψ (t) − Ψ (s)| (3.17) for any t, s ∈ [a, b] . Therefore, we have ı́nfτ∈(a,b) ∣ ∣ ∣ ∣ Φ′ (τ) Ψ′ (τ) ∣ ∣ ∣ ∣ |Ψ (f (x)) − Ψ (f (y))| ≤ |Φ (f (x)) − Φ (f (y))| ≤ supt∈(a,b) ∣ ∣ ∣ ∣ Φ′ (τ) Ψ′ (τ) ∣ ∣ ∣ ∣ |Ψ (f (x)) − Ψ (f (y))| (3.18) for any x, y ∈ Ω. If we multiply (3.18) by 1 2 w (x) w (y) and integrate, we get the desired result (3.16). Corollary 2. Let Φ : [a, b] → R be a continuos function on [a, b] and differentiable on (a, b) . If w is as in Theorem 5, then we have ı́nf t∈(a,b) |Φ′ (t)| RG (f; w) ≤ RG (Φ, f; w) ≤ sup t∈(a,b) |Φ′ (t)| RG (f; w) . (3.19) We also have: Theorem 6. Let Φ : [a, b] → R be an absolutely continuous function on the closed interval [a, b] . If w : Ω → R is a ν-measurable function with w (x) ≥ 0 for ν -a.e. x ∈ Ω and ∫ Ω w (x) dν (x) = 1 CUBO 22, 1 (2020) Bounds for the Generalized (Φ, f)-Mean Difference 13 and if f : Ω → [a, b] is a ν-measurable function with Φ ◦ f ∈ Lw (Ω, ν) , then RG (Φ, f; w) ≤    ‖Φ′‖ [a,b],∞ M (f; w) if p = ∞, ‖Φ′‖ [a,b],p M 1/q (f; w) if p > 1, 1 p + 1 q = 1 ≤    1 2 (b − a) ‖Φ′‖ [a,b],∞ if p = ∞, 1 21/q (b − a) 1/q ‖Φ′‖ [a,b],p if p > 1, 1 p + 1 q = 1, (3.20) where M (f; w) is defined by M (f; w) := ∫ Ω w (x) ∣ ∣ ∣ ∣ f (x) − a + b 2 ∣ ∣ ∣ ∣ dν (x) . (3.21) Demostración. From the inequality (3.8) we have ∣ ∣(Φ ◦ f) (x) − Φ ( a+b 2 ) ∣ ∣ ≤    ‖Φ′‖ [a,b],∞ ∣ ∣f (x) − a+b 2 ∣ ∣ if p = ∞, ‖Φ′‖[a,b],p ∣ ∣f (x) − a+b 2 ∣ ∣ 1/q if p > 1, 1 p + 1 q = 1 (3.22) for any x ∈ Ω. Now, if we multiply (3.22) by w (x) and integrate, then we get ∫ Ω w (x) ∣ ∣ ∣ ∣ (Φ ◦ f) (x) − Φ ( a + b 2 ) ∣ ∣ ∣ ∣ dν (x) ≤    ‖Φ′‖ [a,b],∞ ∫ Ω w (x) ∣ ∣f (x) − a+b 2 ∣ ∣dν (x) if p = ∞, ‖Φ′‖ [a,b],p ∫ Ω w (x) ∣ ∣f (x) − a+b 2 ∣ ∣ 1/q dν (x) if p > 1, 1 p + 1 q = 1. (3.23) By Jensen’s integral inequality for concave functions we have ∫ Ω w (x) ∣ ∣ ∣ ∣ f (x) − a + b 2 ∣ ∣ ∣ ∣ 1/q dν (x) ≤ (∫ Ω w (x) ∣ ∣ ∣ ∣ f (x) − a + b 2 ∣ ∣ ∣ ∣ dν (x) )1/q . (3.24) On making use of (3.2), (3.23) and (3.24) we get the first inequality in (3.20). The last part of (3.20) follows by the fact that ∣ ∣ ∣ ∣ f (x) − a + b 2 ∣ ∣ ∣ ∣ ≤ 1 2 (b − a) for any x ∈ Ω. 14 Silvestru Sever Dragomir CUBO 22, 1 (2020) 4. Bounds for Special Convexity When some convexity properties for the function Φ are assumed, then other bounds can be derived as follows. Theorem 7. Let w : Ω → R be a ν-measurable function with w (x) ≥ 0 for ν -a.e. x ∈ Ω and ∫ Ω w (x) dν (x) = 1 and f : Ω → [a, b] be a ν-measurable function with Φ ◦ f ∈ Lw (Ω, ν) . Assume also that Φ : [a, b] → R is a continuous function on [a, b] . (i) If |Φ| is concave on [a, b] , then RG (Φ, f; w) ≤ |Φ (E (f; w))| , (4.1) (ii) If |Φ| is convex on [a, b] , then RG (Φ, f; w) ≤ 1 b − a [(b − E (f; w)) |Φ (a)| + (E (f; w) − a) Φ |(b)|] . (4.2) Demostración. (i) If |Φ| is concave on [a, b] , then by Jensen’s inequality we have ∫ Ω w (x) |(Φ ◦ f) (x)| dν (x) ≤ ∣ ∣ ∣ ∣ Φ (∫ Ω w (x) f (x) dν (x) ) ∣ ∣ ∣ ∣ . (4.3) From (3.2) for γ = 0 we also have RG (Φ, f; w) ≤ ∫ Ω w (x) |(Φ ◦ f) (x)| dν (x) . (4.4) This is an inequality of interest in itself. On utilizing (4.3) and (4.4) we get (4.1). (ii) Since |Φ| is convex on [a, b] , then for any t ∈ [a, b] we have |Φ (t)| = ∣ ∣ ∣ ∣ Φ ( (b − t) a + b (t − a) b − a ) ∣ ∣ ∣ ∣ ≤ (b − t) |Φ (a)| + (t − a) Φ |(b)| b − a . This implies that |(Φ ◦ f) (x)| ≤ (b − f (x)) |Φ (a)| + (f (x) − a) Φ |(b)| b − a (4.5) for any x ∈ Ω. If we multiply (4.5) by w (x) and integrate, then we get ∫ Ω w (x) |(Φ ◦ f) (x)| dν (x) ≤ 1 b − a [( b ∫ Ω w (x) dν (x) − ∫ Ω w (x) f (x) dν (x) ) |Φ (a)| + (∫ Ω w (x) f (x) dν (x) − a ∫ Ω w (x) dν (x) ) Φ |(b)| ] , which, together with (4.4), produces the desired result (4.2). CUBO 22, 1 (2020) Bounds for the Generalized (Φ, f)-Mean Difference 15 In order to state other results we need the following definitions: Definition 1 ([19]). We say that a function f : I → R belongs to the class P (I) if it is nonnegative and for all x, y ∈ I and t ∈ [0, 1] we have f (tx + (1 − t) y) ≤ f (x) + f (y) . It is important to note that P (I) contains all nonnegative monotone, convex and quasi convex functions, i.e. functions satisfying f (tx + (1 − t) y) ≤ máx {f (x) , f (y)} for all x, y ∈ I and t ∈ [0, 1] . For some results on P-functions see [19] and [28] while for quasi convex functions, the reader can consult [18]. Definition 2 ([3]). Let s be a real number, s ∈ (0, 1]. A function f : [0, ∞) → [0, ∞) is said to be s-convex (in the second sense) or Breckner s-convex if f (tx + (1 − t) y) ≤ tsf (x) + (1 − t) s f (y) for all x, y ∈ [0, ∞) and t ∈ [0, 1] . For some properties of this class of functions see [1], [2], [3], [4], [16], [17], [25], [27] and [29]. Theorem 8. Let w : Ω → R be a ν-measurable function with w (x) ≥ 0 for ν -a.e. x ∈ Ω and ∫ Ω w (x) dν (x) = 1 and f : Ω → [a, b] be a ν-measurable function with Φ ◦ f ∈ Lw (Ω, ν) . Assume also that Φ : [a, b] → R is a continuous function on [a, b] . (i) If |Φ| belongs to the class P on [a, b] , then RG (Φ, f; w) ≤ |Φ (a)| + Φ |(b)| ; (4.6) (ii) If |Φ| is quasi convex on [a, b] , then RG (Φ, f; w) ≤ máx {|Φ (a)| , Φ |(b)|} ; (4.7) (iii) If |Φ| is Breckner s-convex on [a, b] , then RG (Φ, f; w) ≤ 1 (b − a) s [ |Φ (a)| ∫ Ω w (x) (b − f (x)) s dν (x) +Φ |(b)| ∫ Ω w (x) (f (x) − a) s dν (x) ] ≤ 1 (b − a) s [ |Φ (a)| (b − E (f; w)) s dν (x) +Φ |(b)| (E (f; w) − a) s dν (x) ] . (4.8) 16 Silvestru Sever Dragomir CUBO 22, 1 (2020) Demostración. (i) Since |Φ| belongs to the class P on [a, b] , then for any t ∈ [a, b] we have |Φ (t)| = ∣ ∣ ∣ ∣ Φ ( (b − t) a + b (t − a) b − a ) ∣ ∣ ∣ ∣ ≤ |Φ (a)| + Φ |(b)| . This implies that |(Φ ◦ f) (x)| ≤ |Φ (a)| + Φ |(b)| (4.9) for any x ∈ Ω. If we multiply (4.9) by w (x) and integrate, then we get ∫ Ω w (x) |(Φ ◦ f) (x)| dν (x) ≤ |Φ (a)| + Φ |(b)| , (4.10) which, together with (4.4), produces the desired result (4.6). (ii) Goes in a similar way. (iii) By Breckner s-convexity we have |Φ (t)| = ∣ ∣ ∣ ∣ Φ ( (b − t) a + b (t − a) b − a ) ∣ ∣ ∣ ∣ ≤ ( b − t b − a )s |Φ (a)| + ( t − a b − a )s Φ |(b)| for any t ∈ [a, b] . This implies that |(Φ ◦ f) (x)| ≤ 1 (b − a) s [ (b − f (x)) s |Φ (a)| + (f (x) − a) s Φ |(b)| ] (4.11) for any x ∈ Ω. If we multiply (4.11) by w (x) and integrate, then we get ∫ Ω w (x) |(Φ ◦ f) (x)| dν (x) ≤ 1 (b − a) s [ |Φ (a)| ∫ Ω w (x) (b − f (x)) s dν (x) +Φ |(b)| ∫ Ω w (x) (f (x) − a) s dν (x) ] , (4.12) which, together with (4.4), produces the first part of (4.8). The last part follows by Jensen’s integral inequality for concave functions, namely ∫ Ω w (x) (b − f (x)) s dν (x) ≤ ( b − ∫ Ω w (x) f (x) dν (x) )s and ∫ Ω w (x) (f (x) − a) s dν (x) ≤ (∫ Ω w (x) f (x) dν (x) − a )s , where s ∈ (0, 1) . CUBO 22, 1 (2020) Bounds for the Generalized (Φ, f)-Mean Difference 17 5. Some Examples Let f : Ω → [0, ∞) be a ν-measurable function and w : Ω → R a ν-measurable function with w (x) ≥ 0 for ν -a.e. x ∈ Ω and ∫ Ω w (x) dν (x) = 1. We define, for the function Φ (t) = tp, p > 0, the generalized (p, f)-mean difference RG (p, f; w) by RG (p, f; w) := 1 2 ∫ Ω ∫ Ω w (x) w (y) |fp (x) − fp (y)| dν (x) dν (y) (5.1) and the generalized (p, f)-mean deviation MD (p, f; w) by MD (p, f; w) := ∫ Ω w (x) |fp (x) − E (p, f; w)| dν (x) , (5.2) where E (p, f; w) := ∫ Ω fp (y) w (y) dν (y) (5.3) is the generalized (p, f)-expectation. If f : Ω → [a, b] ⊂ [0, ∞) is a ν-measurable function, then by (3.1) we have RG (p, f; w) ≤ 1 2 (bp − ap) . (5.4) By (3.7) we have RG (p, f; w) ≤ pδp (a, b) RG (f; w) , (5.5) where δp (a, b) :=    bp−1 if p ≥ 1, ap−1 if p ∈ (0, 1) and RG (p, f; w) ≤ p 21/α [ bα(p−1)+1 − aα(p−1)+1 α (p − 1) + 1 ]1/α R 1/β G (f; w) , (5.6) where α > 1, 1 α + 1 β = 1. From (3.20) we also have RG (p, f; w) ≤    δp (a, b) M (f; w) , p ( bα(p−1)+1−aα(p−1)+1 α(p−1)+1 )1/α M1/β (f; w) if α > 1, 1 α + 1 β = 1 ≤    1 2 (b − a) δp (a, b) , 1 21/β (b − a) 1/β p ( b α(p−1)+1 −a α(p−1)+1 α(p−1)+1 )1/α if α > 1, 1 α + 1 β = 1, (5.7) 18 Silvestru Sever Dragomir CUBO 22, 1 (2020) where M (f; w) is defined by (3.21). If p ∈ (0, 1) , then the function |Φ (t)| = tp is concave on [a, b] ⊂ [0, ∞) and by (4.1) we have RG (p, f; w) ≤ E p (f; w) . (5.8) For p ≥ 1 the function |Φ (t)| = tp is convex on [a, b] ⊂ [0, ∞) and by (4.2) we have RG (p, f; w) ≤ 1 b − a [(b − E (f; w)) ap + (E (f; w) − a) bp] . (5.9) Let f : Ω → [0, ∞) be a ν-measurable function and w : Ω → R a ν-measurable function with w (x) ≥ 0 for ν -a.e. x ∈ Ω and ∫ Ω w (x) dν (x) = 1. We define, for the function Φ (t) = ln t, the generalized (ln, f)-mean difference RG (ln, f; w) by RG (ln, f; w) := 1 2 ∫ Ω ∫ Ω w (x) w (y) |ln f (x) − ln f (y)| dν (x) dν (y) (5.10) and the generalized (p, f)-mean deviation MD (ln, f; w) by MD (ln, f; w) := ∫ Ω w (x) |ln f (x) − E (ln, f; w)| dν (x) , (5.11) where E (ln, f; w) := ∫ Ω w (y) ln f (y) dν (y) (5.12) is the generalized (ln, f)-expectation. If f : Ω → [a, b] ⊂ [0, ∞) is a ν-measurable function, then by (3.1) we have RG (ln, f; w) ≤ 1 2 (ln b − ln a) . (5.13) By (3.7) we have RG (ln, f; w) ≤    1 a RG (f; w) , 1 21/p ( bp−1−ap−1 (p−1)bp−1ap−1 )1/p R 1/q G (f; w) if p > 1, 1 p + 1 q = 1. (5.14) By (3.20) we have RG (ln, f; w) ≤    1 a M (f; w) , ( bp−1−ap−1 (p−1)bp−1ap−1 )1/p M1/q (f; w) if p > 1, 1 p + 1 q = 1 ≤    1 2 ( b a − 1 ) , 1 21/q (b − a) 1/q ( b p−1 −a p−1 (p−1)bp−1ap−1 )1/p if p > 1, 1 p + 1 q = 1. (5.15) CUBO 22, 1 (2020) Bounds for the Generalized (Φ, f)-Mean Difference 19 Now, observe that the function |Φ (t)| = |ln t| is convex on (0, 1) and concave on [1, ∞). If f : Ω → [a, b] ⊂ (0, 1) is a ν-measurable function, then by (4.2) we have RG (ln, f; w) ≤ 1 b − a [(b − E (f; w)) |ln a| + (E (f; w) − a) |ln b|] (5.16) and if f : Ω → [a, b] ⊂ [1, ∞), then by (4.1) we have RG (ln, f; w) ≤ ln (E (f; w)) . (5.17) The interested reader may state similar bounds for functions Φ such as Φ (t) = exp t, t ∈ R or Φ (t) = t ln t, t > 0. We omit the details. Acknowledgement. The author would like to thank the anonymous referee for valuable suggestions that have been implemented in the final version of the paper. 20 Silvestru Sever Dragomir CUBO 22, 1 (2020) Referencias [1] M. Alomari and M. Darus, The Hadamard’s inequality for s-convex function. Int. J. Math. Anal. (Ruse) 2 (2008), no. 13-16, 639–646. [2] M. Alomari and M. Darus, Hadamard-type inequalities for s-convex functions. Int. Math. 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