Weighted majorization inequalities for n-convex functions via extension of Montgomery identity using Green function

Arabian Journal of Mathematics, Nov 2017

New identities and inequalities are given for weighted majorization theorem for n-convex functions by using extension of the Montgomery identity and Green function. Various bounds for the reminders in new generalizations of weighted majorization formulae are provided using Čebyšev type inequalities. Mean value theorems are also discussed for functional related to new results.

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Weighted majorization inequalities for n-convex functions via extension of Montgomery identity using Green function

Weighted majorization inequalities for n -convex functions via extension of Montgomery identity using Green function Andrea Aglic´ Aljinovic´ 0 1 2 Asif R. Khan 0 1 2 Josip E. Pecˇaric´ 0 1 2 Mathematics Subject Classification 0 1 2 0 J. E. Pecˇaric ́ Faculty of Textile Technology, University of Zagreb , Prilaz Baruna Filipovic ́a 28A, 10000 Zagreb , Croatia 1 A. R. Khan Department of Mathematics, Faculty of Science, University of Karachi , University Road, Karachi 75270 , Pakistan 2 A. Aglic ́ Aljinovic ́ ( New identities and inequalities are given for weighted majorization theorem for n-convex functions by using extension of the Montgomery identity and Green function. Various bounds for the reminders in new generalizations of weighted majorization formulae are provided using Cˇ ebyšev type inequalities. Mean value theorems are also discussed for functional related to new results. - 26A51 · 26D15 · 26D20 1 Introduction Convex functions have got attention of most of the mathematicians of present era (see, e.g.) [ 7, 15–18, 20– 22, 24 ]. For a standard definition of convex function and some alternate forms, we refer the reader to the following books [ 12, 14, 19, 23 ]. Now, we commence the present article by recalling related concepts and definition of generalized convex function as in (see, e.g.) [ 2, 6, 9, 10 ] and [ 19 ]. Definition 1.1 The nth order divided difference of a real-valued function f defined on an interval I of R at distinct points ξi , ξi+1, . . . , ξi+n for i ∈ N is defined recursively by ξ j ; f = f ξ j , j ∈ {i, . . . , i + n} ξi , . . . , ξi+n ; f = ξi+1, . . . , ξi+n ; f − ξi , . . . , ξi+n−1; f ξi+n − ξi . Remark 1.2 Let us denote [ξi , . . . , ξi+n ; f ] by (n) f (ξi ). The value ξi , . . . , ξi+n ; f is independent of the order of the points ξi , ξi+1, . . . , ξi+n . We can extend this definition by considering the cases in which two or more points coincide by taking respective limits. Definition 1.3 A function f : I → distinct points ξi , . . . , ξi+n we have R is called convex of order n or n-convex if for all choices of (n + 1) (n) f (ξi ) ≥ 0. · · · ≤ ξ[ 2 ] ≤ ξ[ 1 ], If nth order derivative f (n) exists, then f is n-convex iff f (n) ≥ 0. For fixed m ≥ 2, let x = (ξ1, . . . , ξm ) and y = (η1, . . . , ηm ) denote two real m-tuples and ξ[m] ≤ ξ[m−1] ≤ η[m] ≤ η[m−1] ≤ · · · ≤ η[ 2 ] ≤ η[ 1 ] their ordered components. Definition 1.4 For x, y ∈ Rm , x ≺ y if k im=1 ξ[i] ≤ i=1 ξi = k mi=1 η[i], i=1 ηi , k ∈ {1, . . . , m − 1}, when x ≺ y, x is said to be majorized by y. This concept and notation of majorization were introduced by Hardy, Littlewood and Pólya [ 5 ]. Next, we state the well-known majorization theorem from [ 5 ]. Proposition 1.5 Let x, y ∈ Rm . The inequality m i=1 f (ξi ) ≤ f (ηi ) m i=1 holds for any continuous convex function f : R → R if and only if x ≺ y. The following weighted version of the majorization theorem was given by Fuchs in [ 4 ] (see also [11, p. 580] and [19, p. 323]). Proposition 1.6 Let p ∈ Rm and let x, y be two nonincreasing real m-tuples such that (1.1) (1.2) (1.3) (1.4) Then, for any continuous convex function f : R → R, the following inequality holds and k i=1 m i=1 pi ξi ≤ pi ξi = m i=1 k i=1 m i=1 pi ηi . m i=1 pi f (ξi ) ≤ pi f (ηi ). pi ηi , k ∈ {1, . . . , m − 1} Some integral inequalities in relation to majorization can be stated as follows. The following proposition is a consequence of Theorem 1 in [ 16 ] (see also [19, p. 328]). Proposition 1.7 Let ξ, η : [α0, β0] → R be two nonincreasing continuous functions and let p : [α0, β0] → be a continuous function. Then, if p (u) ξ(u) du ≤ p (u) η(u) du, for each γ ∈ (α0, β0), and p (u) ξ(u) du ≤ p (u) ξ(u) du = p (u) η(u) du, for each γ ∈ (α0, β0), p (u) η(u) du, then again inequality (1.7) holds. In this article, we will state our results for nonincreasing x and y satisfying the assumption of Proposition 1.7, but they are still valid for nondecreasing x and y satisfying the above condition see for example [11, p. 584]. From [ 1 ], we have extracted here an extension of Montgomery identity via Taylor’s formula which may be stated as follows. Proposition 1.9 Let n ∈ N, f : I → R be such that f (n−1) is absolutely continuous, I ⊂ R an open interval, a0, b0 ∈ I , a0 < b0. Then, the following identity holds and p (u) ξ(u) du = p (u) η(u) du, hold, then for every continuous convex function f : R → R the following inequality holds β0 α0 p (u) f (ξ(u)) du ≤ p (u) f (η(u)) du. β0 α0 Remark 1.8 Let ξ, η : [α0, β0] → R be two nondecreasing continuous functions and let p : [α0, β0] → be a continuous function. If (1.5) (1.6) (1.7) (1.8) (1.9) where In case n = 1, the sum (see for instance [ 13 ]), f (x) = b0 − a0 a0 1 b0 f (s) ds + n−2 f (k+1) (b0) (x − b0)k+2 − k=0 k! (k + 2) n−2 f (k+1) (a0) (x − a0)k+2 k=0 k! (k + 2) ⎪⎧ − n((xb−0−s)an0) + bx0−−aa00 (x − s)n−1 , a0 ≤ s ≤ x, Tn (x, s) = ⎨ (x−s)n ⎪⎩ − n(b0−a0) + bx0−−ba00 (x − s)n−1 , x < s ≤ b0. n−2 k=0 · · · is empty, so identity (1.8) reduces to the famous Montgomery identity where P (x, s) is the Peano kernel, defined by f (s) = b0 − a0 a0 1 b0 f (t) dt + b0 a0 P (s, t) f (t) dt ⎧ bt0−−aa00 , a0 ≤ t ≤ s, P (s, t) = ⎨ ⎩ bt0−−ba00 , s < t ≤ b0. Finally, we define of Green function G : [a0, b0] × [a0, b0] by G(s, t ) = (s−b0)(t−a0) b0−a0 (t−b0)(s−a0) b0−a0 for a0 ≤ t ≤ s, for s ≤ t ≤ b0. The function G is continuous and convex with respect to both t and s. For any function f : [a0, b0] → R, f ∈ C 2[a0, b0], we can obtain the following integral identity by simply using integration by parts b0 − x x − a0 f (b0) + f (x ) = b0 − a0 f (a0) + b0 − a0 b0 a0 G(x , s) f (s)ds, (1.11) where the function G is defined as above in (1.10) (see also [ 24 ]). The aim of the present article is to introduce new weighted majorization theorems for convex functions via extension of Montgomery identity. 2 Majorization type identities and inequalities via extension of Montgomery identity In this section, we state results related to weighted majorization identities and inequalities. First, we define notations in terms of positive linear functional: ( pi , ξi , ηi , f ) = f (b0) − f (a0) where pi , ξi , ηi and f are defined in Proposition 1.6 and G is defined in (1.10). Also Theorem 2.1 Let all the assumptions of Proposition 1.6 be valid. Also let f : I → R be a function such that f (n−1) (n ≥ 3) is absolutely continuous, I ⊂ R an open interval, a0, b0 ∈ I , a0 < b0. Then for all s ∈ [a0, b0] we have the following identity: (1.10) (2.1) (2.2) (2.3) (2.4) ds (2.5) where ⎧ 1 T˜n−2 (s, t ) = ⎨⎪⎪ b0−a0 and G(·, s) is as defined in (1.10). We also have the following identity: where Tn is defined in Proposition 1.9. Proof Using (1.11) in (2.1) and using linearity of ( pi , ξi , ηi , f ), we get Differentiating (1.8) twice with respect to s, we get b0 a0 f (s) = b0 b0 Similarly, using (2.9) in (2.7) and applying Fubini’s theorem, we get (2.6). ( pi , ξi , ηi , G(·, s))Tn−2(s, t )ds dt f k (a0)(s − a0)k−1 − f k (b0)(s − b0)k−1 T˜n−2(s, t ) f (n)(t )dt ds ( pi , ξi , ηi , f ) ≥ ( pi , ξi , ηi , f ) ≥ + + + + b0 a0 b0 a0 ( pi , ξi , ηi , id) b0 b0 a0 Theorem 2.2 Let all the assumptions of Theorem 2.1 be valid with additional condition where G is defined in (1.10) and T˜n in Theorem 2.1. Then for every n-convex function f : I → R, the following inequality holds: ( pi , ξi , ηi , G(·, s))T˜n−2(s, t )ds ≥ 0, ∀ t ∈ [a0, b0] (2.10) ds. (2.11) (2.12) ds. (2.13) (2.14) Proof Function f is n-convex, so we have f (n) ≥ 0. Using this fact and (2.10) in (2.5) we obtain required result. Theorem 2.3 Let all the assumptions of Theorem 2.1 hold with additional condition ( pi , ξi , ηi , G(·, s))Tn−2(s, t )ds ≥ 0, ∀ t ∈ [a0, b0] where G is as defined in (1.10) and Tn as in Proposition 1.9. Then for every n-convex function f : I → R, the following inequality holds f k (a0)(s − a0)k−1 − f k (b0)(s − b0)k−1 Proof Function f is n-convex, so we have f (n) ≥ 0. Using this fact and (2.12) in (2.6), we obtain required result. Next, we state an important consequence. Theorem 2.4 Let all the assumptions from Theorem 2.1 hold. If f is n-convex and n is even, then inequalities (2.11) and (2.13) hold. Proof Since Green function G(s, t ) is convex with respect to t for all s ∈ [a0, b0] and x = (ξ1, . . . , ξm ) and p = ( p1, . . . , pm ) satisfy condition (1.2) and (1.3). Therefore, from Proposition 1.6 (inequality (1.4)), we have ( pi , ξi , ηi , G(·, s)) ≥ 0 for s ∈ [a0, b0]. Also note that T˜n−2(s, t ) ≥ 0 (Tn−2(s, t ) ≥ 0) if n − 2 is even. Therefore combining this fact with (2.14), we get inequality (2.10) (inequality (2.12)). As f is n-convex, results follow from Theorem 2.2 (Theorem 2.3, respectively). Following is the integral version of our main results. Since proofs of these results are of similar nature we omit details. have the following identity Theorem 2.5 Let all the assumptions of Proposition 1.7 be valid. Let f : I → R be a function such that f (n−1) is absolutely continuous, I ⊂ R an open interval, a0, b0 ∈ I , a0 < b0. Then for all s ∈ [a0, b0] we where G is as defined in (1.10) and T˜n is defined in Theorem 2.1. Then, for every n-convex function f : I → R, the following inequality holds Theorem 2.6 Let all the assumptions of Theorem 2.5 hold with additional condition ( p, ξ, η, f ) ≥ b0 − a0 + + n−1 k=2 Theorem 2.7 Let all the assumptions of Theorem 2.5 hold with additional condition ( p, ξ, η, G(·, s))Tn−2(s, t)ds ≥ 0, ∀ t ∈ [a0, b0] 3 Bounds for identities related to generalized linear inequalities Let g, h : [a0, b0] → R be two Lebesgue integrable functions. We consider the Cˇebyšev functional defined by 1 T (g, h) = b0 − a0 a0 b0 g(u)h(u)du − 1 Proposition 3.1 Let g : [a0, b0] → R be a Lebesgue integrable function and let h : [a0, b0] → R be an absolutely continuous function with (· − a0)(b0 − ·)[h ]2 ∈ L[a0, b0]. Then we have the inequality 1 1 |T (g, h)| ≤ √2 b0 − a0 |T (g, g)| b0 a0 (u − a)(b − u)[h (u)]2du 1/2 . The constant √1 in (3.2) is the best possible. 2 Using aforementioned result, we are going to obtain generalizations of the result proved in previous section. Under the assumptions of Theorems 2.2, 2.3, 2.6 and 2.7, respectively, we define the following linear functionals 1(t) = 2(t) = 3(t) = 4(t) = b0 a0 b0 a0 b0 a0 b0 a0 ( pi , ξi , ηi , G(·, s))T˜n−2(s, t)ds ≥ 0, ∀ t ∈ [a0, b0] ( pi , ξi , ηi , G(·, s))Tn−2(s, t)ds ≥ 0, ∀ t ∈ [a0, b0] ( p, ξ, η, G(·, s))T˜n−2(s, t)ds, ∀ t ∈ [a0, b0] ( p, ξ, η, G(·, s))Tn−2(s, t)ds, ∀ t ∈ [a0, b0] Hence using these notations, we may define Cˇebyšev functional as follows (e.g., using ): 1 T ( , ) = b0 − a0 a0 b0 2(s) ds − 1 b0 ds (2.20) (3.2) (3.3) (3.4) (3.5) (3.6) Theorem 3.2 Let f : [a0, b0] → R be such that f (n) is an absolutely continuous function for n ∈ N with (· − a0)(b0 − ·)[ f (n+1)]2 ∈ L[a0, b0]. Let pi and ξi , i ∈ {1, . . . , m} satisfy the assumptions of Proposition 1.6 and the functions G, T and 1 be defined in (1.10), (3.1) and (3.3), respectively. Then, we have f k (a0)(s − a0)k−1 − f k (b0)(s − b0)k−1 1(s)ds + Rn1( f ; a0, b0), b0 a0 b0 where the remainder Rn1( f ; a0, b0) satisfies the estimation 1 |Rn1( f ; a0, b0)| ≤ (n − 3)! Therefore, we have 1 b0 (n − 3)! a0 1(t ) f (n)(t )dt = f (n−1)(b0) − f (n−1)(a0) where Rn1( f ; a0, b0) satisfies inequality (3.8). Now from identity (2.5), we obtain (3.7). Theorem 3.3 Let f : [a0, b0] → R be such that f (n) is an absolutely continuous function for n ∈ N with (· − a0)(b0 − ·)[ f (n+1)]2 ∈ L[a0, b0]. Let pi and ξi , i ∈ {1, . . . , m} satisfy the assumptions of Proposition 1.6 and the functions G, T and 2 be defined in (1.11), (3.1) and (3.4), respectively. Then, we have ds (3.7) (3.8) ds (3.9) 1/2 . (3.10) ( pi , ξi , ηi , f ) = where the remainder Rn2( f ; a0, b0) satisfies the estimation 1 |Rn2( f ; a0, b0)| ≤ (n − 3)! Finally, we obtain Ostrowski type inequalities for functions f whose n-th derivative belongs to L p spaces. The symbol L p [a0, b0] (1 ≤ p < ∞) denotes the space of p-power integrable functions on the interval [a0, b0] equipped with the norm and L∞ [a0, b0] denotes the space of essentially bounded functions on [a0, b0] with the norm φ p = b0 a0 |φ (t )| p dt ds ds (3.11) (3.12) ab00 |λ(t )|r dt 1/r , we find a function f for which the equality in f (n)(t ) = sgnλ(t ) · |λ(t )|1/(q−1). f (n)(t ) = sgnλ(t ). Now, we state some Ostrowski type inequalities related to the generalized majorization inequalities. Theorem 3.4 Let all the assumptions of Theorem 2.1 be valid. Furthermore, let (q, r ) be a pair of conjugate exponents and f (n) ∈ Lq [a0, b0] for some n ∈ N. Then, we have The constant on the R.H.S. of (3.11) is sharp for 1 < q ≤ ∞ and the best possible for q = 1. Proof Let us denote 1 Using the identity (2.5) and applying Hölder’s inequality, we obtain For q = ∞, take f such that The fact that (3.11) is the best possible for q = 1 can be proved as in [8, Thm 12]. Theorem 3.5 Let all the assumptions of Theorem 2.1 be valid. Furthermore, let (q, r ) be a pair of conjugate exponents. Let f (n) ∈ Lq [a0, b0] for some n ∈ N. Then, we have ( pi , ξi , ηi , G(·, s))Tn−2(s, t )ds The constant on the R.H.S. of (3.13) is sharp for 1 < q ≤ ∞ and the best possible for q = 1. Finally, we state all the integral analogous of results from this section. Since proofs are of similar nature, so we omit the details. Theorem 3.6 Let f : [a0, b0] → R be such that f (n) is an absolutely continuous function for n ∈ N with (· − a0)(b0 − ·)[ f (n+1)]2 ∈ L[a0, b0] and let p, x and y be as defined in Proposition 1.7 and let the functions G, T and 3 be defined in (1.10), (3.1) and (3.5), respectively. Then, we have where the remainder Rn3( f ; a0, b0) satisfies the estimation + + + + + + ( p, ξ, η, G(·, s))ds ( p, ξ, η, G(·, s))ds Theorem 3.7 Let all the assumptions of Theorem 3.6 valid. Then, we have where 4 is as defined in (3.6) and the remainder Rn4( f ; a0, b0) satisfies the estimation 1 |Rn4( f ; a0, b0)| ≤ (n − 3)! Now, we state some Ostrowski type inequalities related to the generalized majorization inequalities. Theorem 3.8 Let all the assumptions of Theorem 2.5 hold. Furthermore, let (q, r ) be a pair of conjugate exponents. Let f (n) ∈ Lq [a0, b0] for some n ∈ N. Then, we have The constant on the right hand side of (3.18) is sharp for 1 < q ≤ ∞ and the best possible for q = 1. Theorem 3.9 Let all the assumptions of Theorem 2.5 hold. Furthermore, let (q, r ) be a pair of conjugate exponents. Let f (n) ∈ Lq [a0, b0] for some n ∈ N. Then, we have The constant on the R.H.S. of (3.19) is sharp for 1 < q ≤ ∞ and the best possible for q = 1. 4 Mean value theorems In the last section, we give mean value theorems for new functionals related to our obtained results. Under the assumptions of Theorem 2.2 using (2.11), Theorem 2.3 using (2.13), Theorem 2.6 using (2.18) and Theorem 2.7 using (2.20), we define the following functionals, respectively: 1( f ) = b0 − a0 b0 ( pi , ξi , ηi , G(·, s))ds ( p, ξ, η, G(·, s)) Theorem 4.1 Let f ∈ Cn[a0, b0] and let k : Cn[a0, b0] → R for k ∈ {1, 2, 3, 4} be linear functionals as defined in ( A1), ( A2), ( A3) and ( A4), respectively. Then, there exists ξk ∈ [a0, b0] for k ∈ {1, 2, 3, 4} such that k ( f ) = f (n)(ξk ) k ( f0), k ∈ {1, 2, 3, 4} (4.5) Proof Let us define functions and F1(x) = M f0(x) − f (x) F2(x) = f (x) − L f0(x) f (n)([a0, b0]) = [L , M] L k ( f0) ≤ k ( f ) ≤ M k ( f0). where L and M are the minimum and maximum of the image of the n-th derivative of f [a0, b0], i.e., Then, F1 and F2 are n-convex. Hence, k (F1) ≥ 0 and k (F2) ≥ 0 and f (n)(ξk ). If k ( f0) = 0, then the statement obviously holds. If k ( f0) = 0, then kk((ff0)) ∈ [L , M] = f (n)([a0, b0]), so there exist ξk ∈ [a0, b0] such that kk((ff0)) = Applying Theorem 4.1 for function ω = k (h) f − k ( f )h, we get the following result. Theorem 4.2 Let f, h ∈ Cn[a0, b0] and let k : Cn[a0, b0] → R for k ∈ {1, 2, 3, 4} be linear functionals as defined in ( A1), ( A2), ( A3) and ( A4), respectively. Then, there exists ξk ∈ [a0, b0] for k ∈ {1, 2, 3, 4} such that k ( f ) k (h) = h(n)(ξk ) f (n)(ξk ) assuming that both the denominators are non-zero. Remark 4.3 If the inverse of hf ((nn)) exists, then from the above mean value theorems we can give generalized means ξk = f (n) −1 h(n) k ( f ) k (h) . (4.6) Remark 4.4 Using the same method as in [ 8 ], we can construct new families of exponentially convex functions and Cauchy type means (see also [ 2 ]). Also, using the idea described in [ 8 ] we can obtain the results for n-convex functions at point. 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Andrea Aglić Aljinović, Asif R. Khan, Josip E. Pečarić. Weighted majorization inequalities for n-convex functions via extension of Montgomery identity using Green function, Arabian Journal of Mathematics, 2017, 1-14, DOI: 10.1007/s40065-017-0188-y