generalizations of steffensen s inequality via some euler type identities

Acta Univ Sapientiae, Mathematica, 8, (2016) 103–126 DOI: 10.1515/ausm-2016-0007 Generalizations of Steffensen’s inequality via some Euler-type identities Josip Peˇcari´c Faculty of Textile Technology, University of Zagreb, Croatia email: pecaric@element.hr Anamarija Peruˇsi´c Pribani´c Ksenija Smoljak Kalamir Faculty of Civil Engineering, University of Rijeka, Croatia email: anamarija.perusic@gradri.uniri.hr Faculty of Textile Technology, University of Zagreb, Croatia email: ksmoljak@ttf.hr Abstract Using Euler-type identities some new generalizations of Steffensen’s inequality for n−convex functions are obtained Moreover, the Ostrowski-type inequalities related to obtained generalizations are given ˇ Furthermore, using inequalities for the Cebyˇ sev functional in terms of the first derivative some new bounds for the remainder in identities related to generalizations of Steffensen’s inequality are proven Introduction Firstly, we recall the well-known Steffensen inequality which reads (see [11]): Theorem Suppose that f is nonincreasing and g is integrable on [a, b] with b ≤ g ≤ and λ = a g(t)dt Then we have b b f(t)dt ≤ b−λ a+λ f(t)g(t)dt ≤ a f(t)dt (1) a 2010 Mathematics Subject Classification: 26D15; 26D20 Key words and phrases: Steffensen’s inequality, Euler-type identities, Bernoulli polynoˇ mials, Ostrowski-type inequalities, Cebyˇ sev functional 103 Unauthenticated Download Date | 2/15/17 5:13 AM 104 J Peˇcari´c, A Peruˇsi´c Pribani´c, K Smoljak Kalamir The inequalities are reversed for f nondecreasing Mitrinovi´c stated in [8] that the inequalities in (1) follow from the identities b a+λ f(t)g(t)dt f(t)dt − a a a+λ = (2) b [f(t) − f(a + λ)][1 − g(t)]dt + a [f(a + λ) − f(t)]g(t)dt a+λ and b b f(t)g(t)dt − a f(t)dt b−λ b−λ = (3) b [f(t) − f(b − λ)]g(t)dt + a [f(b − λ) − f(t)][1 − g(t)]dt b−λ In [4] Dedi´c, Mati´c and Peˇcari´c derived Euler-type identities which extend the well known formula for the expansion of an arbitrary function in Bernoulli polynomials Theorem Let f : [a, b] → R be such that f(n−1) is continuous function of bounded variation on [a, b] for some n ≥ Then for every x ∈ [a, b] we have f(x) = b b−a f(t)dt + Tn (x) + R1n (x) (4) f(t)dt + Tn−1 (x) + R2n (x), (5) a and f(x) = b−a b a where T0 (x) = 0, and for ≤ m ≤ n m Tm (x) = k=1 R1n (x) = − R2n (x) = − (b − a)k−1 Bk k! (b − a)n−1 n! B∗n [a,b] a)n−1 (b − n! x−a b−a B∗n [a,b] f(k−1) (b) − f(k−1) (a) , x−t b−a x−t b−a df(n−1) (t), − Bn x−a b−a df(n−1) (t) Unauthenticated Download Date | 2/15/17 5:13 AM Generalizations via some Euler-type identities 105 Here, Bk (x), k ≥ are the Bernoulli polynomials, Bk , k ≥ are the Bernoulli numbers and B∗k (x), k ≥ are periodic functions of period one, related to the Bernoulli polynomials as B∗k (x) = Bk (x), 0≤x For ε small enough we define fε (t) by   a ≤ t ≤ t0 , 0, n fε (t) = ε n! (t − t0 ) , t0 ≤ t ≤ t0 + ε,  1 n−1 , t + ε ≤ t ≤ b n! (t − t0 ) Then for ε small enough b t0 +ε C(t)f(n) (t)dt = a t0 1 C(t) dt = ε ε t0 +ε C(t)dt t0 Now from the inequality (24) we have ε t0 +ε t0 +ε C(t)dt ≤ C(t0 ) t0 t0 dt = C(t0 ) ε Since, t0 +ε ε→0 ε lim C(t)dt = C(t0 ) t0 the statement follows In the case C(t0 ) < 0,  n−1   n! (t − t0 − ε) , , fε (t) = − εn! (t − t0 − ε)n ,   0, we define fε (t) by a ≤ t ≤ t0 , t0 ≤ t ≤ t0 + ε, t0 + ε ≤ t ≤ b, and the rest of the proof is the same as above Using the identity (11) we obtain the following result Theorem 10 Suppose that all assumptions of Theorem hold Assume (p, q) is a pair of conjugate exponents, that is ≤ p, q ≤ ∞, 1/p + 1/q = Let p f(n) : [a, b] → R be an R-integrable function for some n ≥ Then we have a+λ b f(t)dt − a f(t)g(t)dt a n−1 + k=1 b (b − a)k−2 (k − 1)! (b − a)n−2 (n) ≤ f (n − 1)! − Bn−1 x−a b−a G1 (x)Bk−1 a p b b a q a dx dt x−a b−a G1 (x) B∗n−1 dx [f(k−1) (b) − f(k−1) (a)] (25) x−t b−a q Unauthenticated Download Date | 2/15/17 5:13 AM Generalizations via some Euler-type identities 113 The constant on the right-hand side of (25) is sharp for < p ≤ ∞ and the best possible for p = Similarly, we obtain the following Ostrowski-type inequalities related to results given in Theorems and Theorem 11 Suppose that all assumptions of Theorem hold Assume (p, q) is a pair of conjugate exponents, that is ≤ p, q ≤ ∞, 1/p + 1/q = Let p f(n) : [a, b] → R be an R-integrable function for some n ≥ Then we have b b f(t)g(t)dt − f(t)dt b−λ a n + k=1 b (b − a)k−2 (k − 1)! G2 (x)Bk−1 a (b − a)n−2 (n) ≤ f (n − 1)! p b b a a x−a b−a G2 (x)B∗n−1 dx [f(k−1) (b) − f(k−1) (a)] x−t b−a q dx dt (26) q The constant on the right-hand side of (26) is sharp for < p ≤ ∞ and the best possible for p = Theorem 12 Suppose that all assumptions of Theorem hold Assume (p, q) is a pair of conjugate exponents, that is ≤ p, q ≤ ∞, 1/p + 1/q = Let p f(n) : [a, b] → R be an R-integrable function for some n ≥ Then we have b b f(t)g(t)dt − f(t)dt b−λ a n−1 + k=1 b (b − a)k−2 (k − 1)! (b − a)n−2 (n) ≤ f (n − 1)! − Bn−1 x−a b−a G2 (x)Bk−1 a p b b a q a dx dt x−a b−a G2 (x) B∗n−1 dx [f(k−1) (b) − f(k−1) (a)] (27) x−t b−a q The constant on the right-hand side of (27) is sharp for < p ≤ ∞ and the best possible for p = Unauthenticated Download Date | 2/15/17 5:13 AM 114 J Peˇcari´c, A Peruˇsi´c Pribani´c, K Smoljak Kalamir Generalizations related to the bounds for the ˇ Cebyˇ sev functional Let f, h : [a, b] → R be two Lebesgue integrable functions By T (f, h) we ˇ denote the Cebyˇ sev functional b−a T (f, h) := b f(t)h(t)dt − a b−a b f(t)dt · a b b−a h(t)dt a ˇ In [3] Cerone and Dragomir proved the following bound for the Cebyˇ sev functional Theorem 13 Let f : [a, b] → R be a Lebesgue integrable function and h : [a, b] → R be an absolutely continuous function with (·−a)(b−·)[h ]2 ∈ L[a, b] Then we have the inequality 1 |T (f, h)| ≤ √ [T (f, f)] √ b−a The constant √1 b (x − a)(b − x)[h (x)] dx (28) a in (28) is the best possible Also, Cerone and Dragomir [3] proved the following inequality of Gră uss type Theorem 14 Assume that h : [a, b] → R is monotonic nondecreasing on [a, b] and f : [a, b] → R is absolutely continuous with f ∈ L∞ [a, b] Then we have the inequality |T (f, h)| ≤ The constant f 2(b − a) b ∞ (x − a)(b − x)dh(x) (29) a in (29) is the best possible ˇ In the sequel we use the aforementioned bound for the Cebyˇ sev functional to obtain generalizations of the results proved in Section Firstly, let us denote b H1 (t) = a G1 (x)B∗n−1 x−t b−a dx Unauthenticated Download Date | 2/15/17 5:13 AM (30) Generalizations via some Euler-type identities 115 Theorem 15 Let f : [a, b] → R be such that f(n) is absolutely continuous function for some n ≥ with (· − a)(b − ·)[f(n+1) ]2 ∈ L[a, b] and let g be an b integrable function on [a, b] Let λ = a g(t)dt and let the functions G1 and H1 be defined by (7) and (30) Then a+λ b f(t)dt − a n + k=1 − f(t)g(t)dt a k−2 a) (b − (k − 1)! b x−a b−a G1 (x)Bk−1 a dx [f(k−1) (b) − f(k−1) (a)] (31) b (b − a)n−3 [f(n−1) (b) − f(n−1) (a)] (n − 1)! H1 (t)dt = S1n (f; a, b) a where the remainder S1n (f; a, b) satisfies the estimation (b − a)n− [T (H1 , H1 )] S1n (f; a, b) ≤ √ 2(n − 1)! b (t − a)(b − t)[f(n+1) (t)]2 dt a (32) Proof Applying Theorem 13 for f → H1 and h → f(n) we obtain b−a b H1 (t)f(n) (t)dt − a b b−a 1 ≤ √ [T (H1 , H1 )] √ b−a H1 (t)dt · a b−a b f(n) (t)dt a (n+1) (t − a)(b − t)[f (33) b (t)] dt a Hence, if we subtract (b − a)n−1 · (n − 1)! b−a b H1 (t)dt · a = b−a a)n−3 b f(n) (t)dt a (b − [f(n−1) (b) − f(n−1) (a)] (n − 1)! b H1 (t)dt a from both side of the identity (8) and use the inequality (33) we obtain the representation (31) Similarly, using the identity (11) we obtain the following result Let us denote b x−a x−t − Bn−1 dx (34) Φ1 (t) = G1 (x) B∗n−1 b−a b−a a Unauthenticated Download Date | 2/15/17 5:13 AM 116 J Peˇcari´c, A Peruˇsi´c Pribani´c, K Smoljak Kalamir Theorem 16 Let f : [a, b] → R be such that f(n) is absolutely continuous function for some n ≥ with (· − a)(b − ·)[f(n+1) ]2 ∈ L[a, b] and let g be an b integrable function on [a, b] Let λ = a g(t)dt and let the functions G1 and Φ1 be defined by (7) and (34) Then a+λ b f(t)dt − a n−1 + k=1 − f(t)g(t)dt a b (b − a)k−2 (k − 1)! G1 (x)Bk−1 a x−a b−a dx [f(k−1) (b) − f(k−1) (a)] (35) b (b − a)n−3 [f(n−1) (b) − f(n−1) (a)] (n − 1)! Φ1 (t)dt = S2n (f; a, b) a where the remainder S2n (f; a, b) satisfies the estimation b (b − a)n− [T (Φ1 , Φ1 )] S2n (f; a, b) ≤ √ 2(n − 1)! (t − a)(b − t)[f(n+1) (t)]2 dt a We continue with the results related to the identities (13) and (14) Let us denote b x−t H2 (t) = G2 (x)B∗n−1 dx (36) b−a a and b Φ2 (t) = G2 (x) B∗n−1 a x−t b−a − Bn−1 x−a b−a dx (37) Theorem 17 Let f : [a, b] → R be such that f(n) is absolutely continuous function for some n ≥ with (· − a)(b − ·)[f(n+1) ]2 ∈ L[a, b] and let g be an b integrable function on [a, b] Let λ = a g(t)dt and let the functions G2 , H2 and Φ2 be defined by (12), (36) and (37) respectively Then (i) b b f(t)g(t)dt − a n + k=1 − f(t)dt b−λ b k−2 a) (b − (k − 1)! a G2 (x)Bk−1 x−a b−a (b − a)n−3 [f(n−1) (b) − f(n−1) (a)] (n − 1)! dx [f(k−1) (b) − f(k−1) (a)] (38) b H2 (t)dt = S3n (f; a, b) a Unauthenticated Download Date | 2/15/17 5:13 AM Generalizations via some Euler-type identities 117 where the remainder S3n (f; a, b) satisfies the estimation (b − a)n− [T (H2 , H2 )] S3n (f; a, b) ≤ √ 2(n − 1)! b (t − a)(b − t)[f(n+1) (t)]2 dt a (ii) b b f(t)dt f(t)g(t)dt − b−λ a n−1 + k=1 − b (b − a)k−2 (k − 1)! x−a b−a G2 (x)Bk−1 a dx [f(k−1) (b) − f(k−1) (a)] (39) b (b − a)n−3 [f(n−1) (b) − f(n−1) (a)] (n − 1)! Φ2 (t)dt = S4n (f; a, b) a where the remainder S4n (f; a, b) satisfies the estimation S4n (f; a, b) (b − a)n− [T (Φ2 , Φ2 )] ≤ √ 2(n − 1)! b (n+1) (t − a)(b − t)[f (t)] dt a Proof Similar to the proof of Theorem 15 The following Gră uss type inequalities also hold Theorem 18 Let f : [a, b] → R be such that f(n) (n ≥ 2) is absolutely continuous function and f(n+1) ≥ on [a, b] Let the function H1 be defined by (30) Then we have the representation (31) and the remainder S1n (f; a, b) satisfies the bound S ! 1n (f; a, b) ≤ (b − a)n−1 H1 (n − 1)! ∞ f(n−1) (b) + f(n−1) (a) − a, b; f(n−2) (40) Proof Applying Theorem 14 for f → H1 and h → f(n) we obtain b−a b H1 (t)f(n) (t)dt − a ≤ H 2(b − a) 1 b−a b H1 (t)dt · a b−a b f(n) (t)dt a b ∞ (n+1) (t − a)(b − t)f (t)dt a Unauthenticated Download Date | 2/15/17 5:13 AM (41) 118 J Peˇcari´c, A Peruˇsi´c Pribani´c, K Smoljak Kalamir Since b b (t − a)(b − t)f(n+1) (t)dt = a [2t − (a + b)]f(n) (t)dt a (n−1) = (b − a) f (b) + f(n−1) (a) − f(n−2) (b) − f(n−2) (a) Using the representation (8) and the inequality (41) we deduce (40) Theorem 19 Let f : [a, b] → R be such that f(n) (n ≥ 2) is absolutely continuous function and f(n+1) ≥ on [a, b] Let H1 , Φ1 and Φ2 be defined by (30), (34) and (37), respectively Then we have the representations (35), (38) and (39) where the remainders Sin (f; a, b), i = 2, 3, satisfy the bounds S2n (f; a, b) ≤ (b − a)n−1 Φ1 (n − 1)! ∞ f(n−1) (b) + f(n−1) (a) − a, b; f(n−2) S3n (f; a, b) ≤ (b − a)n−1 H2 (n − 1)! ∞ f(n−1) (b) + f(n−1) (a) − a, b; f(n−2) (b − a)n−1 Φ2 (n − 1)! ∞ f(n−1) (b) + f(n−1) (a) − a, b; f(n−2) , and S4n (f; a, b) ≤ Mean value theorems Motivated by inequalities (16), (18), (20) and (22), under the assumptions of Theorems and we define the following linear functionals: a+λ L1 (f) = b f(t)dt − a n + k=1 (b − a)k−2 (k − 1)! b G1 (x)Bk−1 a a+λ L2 (f) = + k=1 x−a b−a (42) (k−1) dx [f (k−1) (b) − f (a)] b f(t)dt − a n−1 f(t)g(t)dt a (b − a)k−2 (k − 1)! f(t)g(t)dt a b G1 (x)Bk−1 a x−a b−a (43) (k−1) dx [f (k−1) (b) − f (a)] Unauthenticated Download Date | 2/15/17 5:13 AM Generalizations via some Euler-type identities b b b−λ a n + k=1 b (b − a)k−2 (k − 1)! G2 (x)Bk−1 a k=1 (44) (k−1) dx [f (k−1) (b) − f (a)] f(t)dt f(t)g(t)dt − b−λ a + x−a b−a b b L4 (f) = n−1 f(t)dt f(t)g(t)dt − L3 (f) = 119 b (b − a)k−2 (k − 1)! G2 (x)Bk−1 a x−a b−a (45) dx [f (k−1) (b) − f (k−1) (a)] Remark We have Li (f) ≥ 0, i = 1, , for all n−convex functions f Now, we give the Lagrange-type mean value theorem related to defined functionals Theorem 20 Let f : [a, b] → R be such that f ∈ Cn [a, b] If the inequalities in (15) (i = 1), (17) (i = 2), (19) (i = 3) and (21) (i = 4) hold, then there exist ξi ∈ [a, b] such that Li (f) = f(n) (ξi )Li (ϕ), where ϕ(x) = xn n! i = 1, , (46) and Li , i = 1, , are defined by (42)-(45) Proof Let us denote m = f(n) (x) and x∈[a,b] M = max f(n) (x) x∈[a,b] For a given function f ∈ Cn [a, b] we define the functions F1 , F2 : [a, b] → R with F1 (x) = Mϕ(x) − f(x) and F2 (x) = f(x) − mϕ(x) (n) Now F1 (x) = M − f(n) (x) ≥ 0, so from Remark we conclude Li (F1 ) ≥ 0, i = (n) 1, , and then Li (f) ≤ M·Li (ϕ) Similarly, from F2 (x) = f(n) (x)−m ≥ we conclude m · Li (ϕ) ≤ Li (f) Hence, m · Li (ϕ) ≤ Li (f) ≤ M · Li (ϕ), i = 1, , If Li (ϕ) = 0, then (46) holds for all ξi ∈ [a, b] Otherwise, m≤ Li (f) ≤ M, Li (ϕ) i = 1, Unauthenticated Download Date | 2/15/17 5:13 AM 120 J Peˇcari´c, A Peruˇsi´c Pribani´c, K Smoljak Kalamir Since f(n) is continuous on [a, b] there exist ξi ∈ [a, b], i = 1, , such that (46) holds and the proof is complete We continue with the Cauchy-type mean value theorem Theorem 21 Let f, F : [a, b] → R be such that f, F ∈ Cn [a, b] and F(n) = If the inequalities in (15) (i = 1), (17) (i = 2), (19) (i = 3) and (21) (i = 4) hold, then there exist ξi ∈ [a, b] such that Li (f) f(n) (ξ) = (n) , Li (F) F (ξ) i = 1, , (47) where Li , i = 1, , are defined by (42)-(45) Proof We define functions φi (x) = f(x)Li (F) − F(x)Li (f), i = 1, , According to Theorem 20 there exist ξi ∈ [a, b] such that (n) Li (φi ) = φi (ξi )Li (ϕ), i = 1, , Since Li (φi ) = it follows that f(n) (ξi )Li (F) − F(n) (ξi )Li (f) = and (47) is proved n−exponential convexity Let us begin by recalling some definitions and results related to n−exponential convexity For more details see e.g [2], [6] and [9] Definition A function ψ : I → R is said to be n-exponentially convex in the Jensen sense on I if n ξi ξj ψ i,j=1 xi + xj ≥ 0, holds for all choices of ξi ∈ R and xi ∈ I, i = 1, , n A function ψ : I → R is said to be n-exponentially convex if it is nexponentially convex in the Jensen sense and continuous on I Remark It is clear from the definition that 1-exponentially convex functions in the Jensen sense are in fact nonnegative functions Also, n-exponentially convex functions in the Jensen sense are k-exponentially convex in the Jensen sense for every k ∈ N, k ≤ n Unauthenticated Download Date | 2/15/17 5:13 AM Generalizations via some Euler-type identities 121 Definition A function ψ : I → R is said to be exponentially convex in the Jensen sense on I if it is n-exponentially convex in the Jensen sense for all n ∈ N A function ψ : I → R is said to be exponentially convex if it is exponentially convex in the Jensen sense and continuous Remark It is known that ψ : I → R is log-convex in the Jensen sense if and only if x+y α2 ψ(x) + 2αβψ + β2 ψ(y) ≥ 0, holds for every α, β ∈ R and x, y ∈ I It follows that a positive function is log-convex in the Jensen sense if and only if it is 2-exponentially convex in the Jensen sense A positive function is log-convex if and only if it is 2-exponentially convex Proposition If f is a convex function on I and if x1 ≤ y1 , x2 ≤ y2 , x1 = x2 , y1 = y2 , then the following inequality is valid f(x2 ) − f(x1 ) f(y2 ) − f(y1 ) ≤ x2 − x1 y2 − y1 If the function f is concave, the inequality is reversed We use defined functionals Li , i = 1, , to construct exponentially convex functions An elegant method of producing n− exponentially convex and exponentially convex functions is given in [9] In the sequel the notion log denotes the natural logarithm function Theorem 22 Let Ω = {fp : p ∈ J}, where J is an interval in R, be a family of functions defined on an interval I in R such that the function p → [x0 , , xm ; fp ] is n−exponentially convex in the Jensen sense on J for every (m + 1) mutually different points x0 , , xm ∈ I Let Li , i = 1, , be linear functionals defined by (42) − (45) Then p → Li (fp ) is n−exponentially convex function in the Jensen sense on J If the function p → Li (fp ) is continuous on J, then it is n−exponentially convex on J Proof For ξj ∈ R and pj ∈ J, j = 1, , n, we define the function n Ψ(x) = ξj ξk f pj +pk (x) j,k=1 Unauthenticated Download Date | 2/15/17 5:13 AM 122 J Peˇcari´c, A Peruˇsi´c Pribani´c, K Smoljak Kalamir Using the assumption that the function p → [x0 , , xm ; fp ] is n-exponentially convex in the Jensen sense, we have n ξj ξk [x0 , , xm ; f pj +pk ] ≥ 0, [x0 , , xm , Ψ] = j,k=1 which in turn implies that Ψ is a m-convex function on J, so Li (Ψ) ≥ 0, i = 1, , Hence n ξj ξk Li f pj +pk j,k=1 ≥ We conclude that the function p → Li (fp ) is n-exponentially convex on J in the Jensen sense If the function p → Li (fp ) is also continuous on J, then p → Li (fp ) is nexponentially convex by definition As an immediate consequence of the above theorem we obtain the following corollary: Corollary Let Ω = {fp : p ∈ J}, where J is an interval in R, be a family of functions defined on an interval I in R, such that the function p → [x0 , , xm ; fp ] is exponentially convex in the Jensen sense on J for every (m + 1) mutually different points x0 , , xm ∈ I Let Li , i = 1, , 4, be linear functionals defined by (42)-(45) Then p → Li (fp ) is an exponentially convex function in the Jensen sense on J If the function p → Li (fp ) is continuous on J, then it is exponentially convex on J Corollary Let Ω = {fp : p ∈ J}, where J is an interval in R, be a family of functions defined on an interval I in R, such that the function p → [x0 , , xm ; fp ] is 2-exponentially convex in the Jensen sense on J for every (m + 1) mutually different points x0 , , xm ∈ I Let Li , i = 1, , be linear functionals defined by (42)-(45) Then the following statements hold: (i) If the function p → Li (fp ) is continuous on J, then it is 2-exponentially convex function on J If p → Li (fp ) is additionally strictly positive, then it is also log-convex on J Furthermore, the following inequality holds true: [Li (fs )]t−r ≤ [Li (fr )]t−s [Li (ft )]s−r , i = 1, , for every choice r, s, t ∈ J, such that r < s < t Unauthenticated Download Date | 2/15/17 5:13 AM Generalizations via some Euler-type identities 123 (ii) If the function p → Li (fp ) is strictly positive and differentiable on J, then for every p, q, u, v ∈ J, such that p ≤ u and q ≤ v, we have µp,q (Li , Ω) ≤ µu,v (Li , Ω), where    µp,q (Li , Ω) = Li (fp ) Li (fq )   exp p−q , d L (f ) dp i p Li (fp ) (48) p = q, (49) , p = q, for fp , fq ∈ Ω Proof (i) This is an immediate consequence of Theorem 22 and Remark (ii) Since p → Li (fp ) is positive and continuous, by (i) we have that p → Li (fp ) is log-convex on J, that is, the function p → log Li (fp ) is convex on J Applying Proposition we get log Li (fp ) − log Li (fq ) log Li (fu ) − log Li (fv ) ≤ , p−q u−v (50) for p ≤ u, q ≤ v, p = q, u = v Hence, we conclude that µp,q (Li , Ω) ≤ µu,v (Li , Ω) Cases p = q and u = v follow from (50) as limit cases Remark Results from the above theorem and corollaries still hold when two of the points x0 , , xm ∈ I coincide, say x1 = x0 , for a family of differentiable functions fp such that the function p → [x0 , , xm ; fp ] is n-exponentially convex in the Jensen sense (exponentially convex in the Jensen sense, log-convex in the Jensen sense), and furthermore, they still hold when all m + points coincide for a family of n differentiable functions with the same property The proofs use (6) and suitable characterization of convexity Unauthenticated Download Date | 2/15/17 5:13 AM 124 J Peˇcari´c, A Peruˇsi´c Pribani´c, K Smoljak Kalamir Applications to Stolarsky type means In this section, we present some families of functions which fulfil the conditions of Theorem 22, Corollary 1, Corollary and Remark This enables us to construct a large families of functions which are exponentially convex For a discussion related to this problem see [5] Example Let us consider a family of functions Ω1 = {fp : R → R : p ∈ R} defined by fp (x) = epx pn , xn n! , p = 0, p = dn f Since dxnp (x) = epx > 0, the function fp is n-convex on R for every p ∈ R dn f and p → dxnp (x) is exponentially convex by definition Using analogous arguing as in the proof of Theorem 22 we also have that p → [x0 , , xn ; fp ] is exponentially convex (and so exponentially convex in the Jensen sense) Now, using Corollary we conclude that p → Li (fp ), i = 1, , 4, are exponentially convex in the Jensen sense It is easy to verify that this mapping is continuous (although the mapping p → fp is not continuous for p = 0), so it is exponentially convex For this family of functions, µp,q (Li , Ω1 ), i = 1, , 4, from (49), becomes  Li (fp ) p−q   , p = q,   Li (fq ) L (id·f ) µp,q (Li , Ω1 ) = exp iLi (fp )p − np , p = q = 0,     exp Li (id·f0 ) , p = q = 0, n+1 Li (f0 ) where id is the identity function By Corollary µp,q (Li , Ω1 ), i = 1, , are monotonic functions in parameters p and q Since dn fp dxn dn fq dxn p−q (log x) = x, using Theorem 21 it follows that: Mp,q (Li , Ω1 ) = log µp,q (Li , Ω1 ), i = 1, , satisfy a ≤ Mp,q (Li , Ω1 ) ≤ b, i = 1, , So, Mp,q (Li , Ω1 ), i = 1, , are monotonic means Unauthenticated Download Date | 2/15/17 5:13 AM Generalizations via some Euler-type identities 125 Example Let us consider a family of functions Ω2 = {gp : (0, ∞) → R : p ∈ R} defined by     xp , p∈ / {0, 1, , n − 1}, p(p − 1) · · · (p − n + 1) gp (x) = xj log x   , p = j ∈ {0, 1, , n − 1}  (−1)n−1−j j!(n − − j)! dn g Since dxnp (x) = xp−n > 0, the function gp is n−convex for x > and p → dn gp dxn (x) is exponentially convex by definition Arguing as in Example we get that the mappings p → Li (gp ), i = 1, , are exponentially convex Hence, for this family of functions µp,q (Li , Ω2 ), i = 1, , 4, from (49), is equal to   p−q L (g )  p i   , p = q,   Li (gq )    n−1   Li (g0 gp )  n−1  exp (−1) (n − 1)! + ,   Li (gp ) k−p   k=0 µp,q (Li , Ω2 ) = p=q∈ /{0, 1, , n − 1},      n−1      , (−1)n−1 (n − 1)! Li (g0 gp ) +  exp     2Li (gp ) k − p  k=0   k=p    p = q ∈ {0, 1, , n − 1} Again, using Theorem 21 we conclude that a≤ Li (gp ) Li (gq ) p−q ≤ b, i = 1, , (51) So, µp,q (Li , Ω2 ), i = 1, , are means and by (48) they are monotonic Acknowledgements The research of Josip Peˇcari´c and Ksenija Smoljak Kalamir has been fully supported by Croatian Science Foundation under the project 5435 and the research of Anamarija Peruˇsi´c Pribani´c has been fully supported by University of Rijeka under the project 13.05.1.1.02 Unauthenticated Download Date | 2/15/17 5:13 AM 126 J Peˇcari´c, A Peruˇsi´c Pribani´c, K Smoljak Kalamir References [1] M Abramowitz, I A Stegun (Eds), Handbook of mathematical functions with formulae, graphs and mathematical tables, National Bureau of Standards, Applied Math Series 55, 4th printing, Washington, 1965 [2] S N Bernstein, Sur les fonctions absolument monotones, Acta Math 52 (1929), 1–66 [3] P Cerone, S S Dragomir, Some new Ostrowski-type bounds for the ˇ Cebyˇ sev functional and applications, J Math Inequal., (1) (2014), 159– 170 [4] L J Dedi´c, M Mati´c, J Peˇcari´c, On generalizations of Ostrowski inequality via some Euler-type identities, Math Inequal Appl., (3) (2000), 337–353 [5] W Ehm, M G Genton, T Gneiting, Stationary covariance associated with exponentially convex functions, Bernoulli, (4) (2003), 607–615 [6] J Jakˇseti´c, J Peˇcari´c, Exponential convexity method, J Convex Anal., 20 (1) (2013), 181–197 [7] V I Krylov, Approximate Calculation of Integrals, Macmillan, New YorkLondon, 1962 [8] D S Mitrinovi´c, The Steffensen inequality, Univ Beograd Publ Elektrotehn Fak Ser Mat Fiz., 247–273 (1969), 1–14 [9] J Peˇcari´c, J Peri´c, Improvements of the Giaccardi and the Petrovi´c inequality and related results, An Univ Craiova Ser Mat Inform 39 (1) (2012), 65–75 [10] J E Peˇcari´c, F Proschan, Y L Tong, Convex functions, partial orderings, and statistical applications, Mathematics in science and engineering 187, Academic Press, 1992 [11] J F Steffensen, On certain inequalities between mean values and their application to actuarial problems, Skand Aktuarietids., (1918), 82–97 Received: October 14, 2014 Unauthenticated Download Date | 2/15/17 5:13 AM