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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 128486, 14 pages doi:10.1155/2009/128486 Research Article Some New Results Related to Favard’s Inequality ˇ ´ ´ Naveed Latif,1 J Pecaric,1, and I Peric3 Abdus Salam School of Mathematical Sciences, GC University, Lahore 54000, Pakistan Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia Faculty of Food Technology and Biotechnology, University of Zagreb, 10000 Zagreb, Croatia Correspondence should be addressed to Naveed Latif, sincerehumtum@yahoo.com Received 31 July 2008; Revised 17 January 2009; Accepted February 2009 Recommended by A Laforgia Log-convexity of Favard’s difference is proved, and Drescher’s and Lyapunov’s type inequalities for this difference are deduced The weighted case is also considered Related Cauchy type means are defined, and some basic properties are given Copyright q 2009 Naveed Latif et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Introduction and Preliminaries Let f and p be two positive measurable real valued functions defined on a, b ⊆ R with b p x dx From theory of convex means cf 1, , the well-known Jensen’s inequality a gives that for t < or t > 1, b p x f x dx ≥ t a t b p x f x dx , 1.1 a and reverse inequality holds for < t < In , Simic considered the difference b Ds Ds a, b, f, p a He has given the following p x f s x dx − s b p x f x dx a 1.2 Journal of Inequalities and Applications Theorem 1.1 Let f and p be nonnegative and integrable functions on a, b , with then for < r < s < t, r, s, t / 1, one has t−r Ds s s−1 ≤ t−s Dr r r−1 Dt t t−1 b p a x dx 1, s−r 1.3 Remark 1.2 For an extension of Theorem 1.1 see Let us write the well-known Favard’s inequality Theorem 1.3 Let f be a concave nonnegative function on a, b ⊂ R If q > 1, then 2q q q b 1 b−a ≥ f x dx a b−a b f q x dx 1.4 a If < q < 1, the reverse inequality holds in 1.4 Note that 1.4 is a reversion of 1.1 in the case when p x 1/ b − a Let us note that Theorem 1.3 can be obtained from the following result and also obtained by Favard cf 4, page 212 Theorem 1.4 Let f be a nonnegative continuous concave function on a, b , not identically zero, and let φ be a convex function on 0, 2f , where f b−a b f x dx 1.5 a Then 2f 2f φ y dy ≥ Karlin and Studden cf 5, page 412 b−a b φ f x dx 1.6 a gave a more general inequality as follows Theorem 1.5 Let f be a nonnegative continuous concave function on a, b , not identically zero; f is defined in 1.5 , and let φ be a convex function on c, 2f − c , where c satisfies < c ≤ fmin (where fmin is the minimum of f) Then 2f − 2c For φ y 2f−c c φ y dy ≥ b−a b φ f x dx a yp , p > 1, we can get the following from Theorem 1.5 1.7 Journal of Inequalities and Applications Theorem 1.6 Let f be continuous concave function such that < c ≤ fmin ; f is defined in 1.5 If p > 1, then 2f − 2c p 2f − c p − cp ≥ 1 b−a b f p x dx 1.8 a If < p < 1, the reverse inequality holds in 1.8 In this paper, we give a related results to 1.3 for Favard’s inequality 1.4 and 1.8 We need the following definitions and lemmas Definition 1.7 It is said that a positive function f is log-convex in the Jensen sense on some interval I ⊆ R if f s f t ≥ f2 s t 1.9 holds for every s, t ∈ I We quote here another useful lemma from log-convexity theory cf Lemma 1.8 A positive function f is log-convex in the Jensen sense on an interval I ⊆ R if and only if the relation u2 f s 2uwf s t w2 f t ≥ 1.10 holds for each real u, w and s, t ∈ I Throughout the paper, we will frequently use the following family of convex functions on 0, ∞ : ⎧ s ⎪ x ⎪ ⎪ ⎪s s − , ⎨ ⎪− log x, ⎪ ⎪ ⎪ ⎩x log x, ϕs x s / 0, 1; s s 1.11 0; The following lemma is equivalent to the definition of convex function see 4, page Lemma 1.9 If φ is convex on an interval I ⊆ R, then φ s1 s3 − s2 φ s2 holds for every s1 < s2 < s3 , s1 , s2 , s3 ∈ I Now, we will give our main results s1 − s3 φ s3 s2 − s1 ≥ 1.12 Journal of Inequalities and Applications Favard’s Inequality In the following theorem, we construct another interesting family of functions satisfying the Lyapunov inequality The proof is motivated by Theorem 2.1 Let f be a positive continuous concave function on a, b ; f is defined in 1.5 , and ⎧ s b b ⎪ 2s 1 ⎪ s ⎪ ⎪ ⎪ s s − s b − a f x dx − b − a f x dx , ⎪ ⎪ a a ⎪ ⎪ ⎪ ⎨ b Δs f : log f x dx, − log − log f ⎪ b−a a ⎪ ⎪ ⎪ ⎪ ⎪ b ⎪ ⎪ ⎪ ⎪log 2f f log f − f − f x log f x dx, ⎩ b−a a s / 0, 1; s 0; s 2.1 Then Δs f is log-convex for s ≥ 0, and the following inequality holds for ≤ r < s < t < ∞: Δt−r f ≤ Δt−s f Δs−r f s r t 2.2 Proof Let us consider the function defined by φ x where r s u2 ϕs x w2 ϕt x , 2uwϕr x 2.3 t /2, ϕs is defined by 1.11 , and u, w ∈ R We have φ x u2 xs−2 uxs/2−1 2uwxr−2 wxt/2−1 w2 xt−2 ≥ 0, 2.4 x > Therefore, φ x is convex for x > Using Theorem 1.4, 2f 2f u2 ϕs y ≥ b−a 2uwϕr y w2 ϕt y dy 2.5 b a u2 ϕs f x 2uwϕr f x w2 ϕt f x dx, Journal of Inequalities and Applications or equivalently u2 2f 2f ϕs y dy − 2uw w2 2f 2f b−a b ϕs f x dx a 2f 2f ϕr y dy − ϕt y dy − b b−a ϕr f x dx b b−a 2.6 a ϕt f x dx ≥ a Since Δs f 2f 2f ϕs y dy − b−a b ϕs f x dx, 2.7 a we have u2 Δs f w2 Δt f ≥ 2uwΔr f 2.8 By Lemma 1.8, we have Δs f Δt f ≥ Δ2 f r Δ2s t /2 f , 2.9 that is, Δs f is log-convex in the Jensen sense for s ≥ Note that Δs f is continuous for s ≥ since lim Δs f s→0 Δ0 f and lim Δs f s→1 Δ1 f 2.10 This implies Δs f is continuous; therefore, it is log-convex Since Δs f is log-convex, that is, s → log Δs f is convex, by Lemma 1.9 for ≤ r < s < t < ∞ and taking φ s log Δs f , we get log Δt−r f ≤ log Δt−s f s r log Δs−r f , t 2.11 which is equivalent to 2.2 Theorem 2.2 Let f, Δs f be defined as in Theorem 2.1, and let t, s, u, v be nonnegative real numbers such that s ≤ u, t ≤ v, s / t, and u / v Then Δt f Δs f 1/ t−s ≤ Δv f Δu f 1/ v−u 2.12 Journal of Inequalities and Applications Proof An equivalent form of 1.12 is ϕ y2 − ϕ y1 ϕ x2 − ϕ x1 ≤ , x2 − x1 y2 − y1 2.13 where x1 ≤ y1 , x2 ≤ y2 , x1 / x2 , and y1 / y2 Since by Theorem 2.1, Δs f is log-convex, we can set in 2.13 : ϕ x log Δx f , x1 s, x2 t, y1 u, and y2 v We get log Δv f − log Δu f log Δt f − log Δs f ≤ , t−s v−u 2.14 from which 2.12 trivially follows The following extensions of Theorems 2.1 and 2.2 can be deduced in the same way from Theorem 1.5 Theorem 2.3 Let f be a continuous concave function on a, b such that < c ≤ fmin ; f is defined in 1.5 , and ⎧ s ⎪ b ⎪ 2f − c cs 1 ⎪ ⎪ ⎪ f s x dx , s / 0, 1; − − ⎪s s − ⎪ b−a a ⎪ 2f − 2c s 2f − 2c s ⎪ ⎪ ⎪ ⎪ ⎪ b ⎪ 1 ⎪ ⎪ ⎪ log f x dx, s 0; 2f c log c − 2c − 2f − c log 2f − c ⎨ b−a a 2f − 2c Δs f : ⎪ ⎪ ⎪ ⎪ ⎪ 2f − c log 2f − c − 2f 2cf − c2 log c 2c ⎪ ⎪ ⎪ 2f − 2c ⎪ ⎪ ⎪ ⎪ ⎪ b ⎪ ⎪ ⎪ − ⎪ f x log f x dx, s ⎩ b−a a 2.15 Then Δs f is log-convex for s ≥ 0, and the following inequality holds for ≤ r < s < t < ∞: Δt−r f ≤ Δt−s f Δs−r f s r t 2.16 Theorem 2.4 Let f, Δs f be defined as in Theorem 2.3, and let t, s, u, v be nonnegative real numbers such that s ≤ u, t ≤ v, s / t, and u / v, one has Δt f 1/ t−s Δs f ≤ Δv f Δu f 1/ v−u Weighted Favard’s Inequality The weighted version of Favard’s inequality was obtained by Maligranda et al in 2.17 Journal of Inequalities and Applications Theorem 3.1 Let f be a positive increasing concave function on a, b Assume that φ is a convex function on 0, ∞ , where fi b f a b−a b a t w t dt t − a w t dt 3.1 Then b b−a φ f t w t dt ≤ a φ 2r fi w a − r br dr 3.2 If f is an increasing convex function on a, b and f a 0, then the reverse inequality in 3.2 holds Let f be a positive decreasing concave function on a, b Assume that φ is a convex function on 0, ∞ , where fd b f a b−a b a t w t dt b − t w t dt 3.3 Then b b−a φ f t w t dt ≤ a φ 2r fd w ar b − r dr 3.4 If f is a decreasing convex function on a, b and f b 0, then the reverse inequality in 3.4 holds Theorem 3.2 Let f be a positive increasing concave function on a, b ; fi is defined in 3.1 , and Πs f : ϕs 2r fi w a − r br dr − b−a b ϕs f t w t dt 3.5 a Then Πs f is log-convex on 0, ∞ , and the following inequality holds for ≤ r < s < t < ∞: Πt−r f ≤ Πt−s f Πs−r f s r t 3.6 Let f be an increasing convex function on a, b , f a 0, Πs f : −Πs f Then Πs f is log-convex on 0, ∞ , and the following inequality holds for ≤ r < s < t < ∞: Πt−r f ≤ Πt−s f Πs−r f s r t Proof As in the proof of Theorem 2.1, we use Theorem 3.1 instead of Theorem 1.4 3.7 Journal of Inequalities and Applications Theorem 3.3 Let f and Πs f be defined as in Theorem 3.2(1), and let t, s, u, v ≥ be such that s ≤ u, t ≤ v, s / t, and u / v Then Πt f Πs f 1/ t−s ≤ Πv f Πu f 1/ v−u 3.8 Let f and Πs f be defined as in Theorem 3.2(2), and let t, s, u, v ≥ be such that s ≤ u, t ≤ v, s / t, and u / v Then, Πt f 1/ t−s ≤ Πs f Πv f 1/ v−u Πu f 3.9 Proof Similar to the proof of Theorem 2.2 Theorem 3.4 Let f be a positive decreasing concave function on a, b ; fd is defined as in 3.3 , and Γs f : ϕs 2r fd w ar b − r dr − b−a b ϕs f t w t dt 3.10 a Then Γs f is log-convex on 0, ∞ , and the following inequality holds for ≤ r < s < t < ∞: Γt−r f ≤ Γt−s f Γs−r f s r t 3.11 Let f be a decreasing convex function on a, b , f b 0, Γs f : −Γs f Then Γs is log-convex on 0, ∞ , and the following inequality holds for ≤ r < s < t < ∞: Γt−r f ≤ Γt−s f Γs−r f s r t 3.12 Proof As in the proof of Theorem 2.1, we use Theorem 3.1 instead of Theorem 1.4 Theorem 3.5 Let f and Γs f be defined as in Theorem 3.4(1), and let t, s, u, v ≥ be such that s ≤ u, t ≤ v, s / t, and u / v Then Γt f Γs f 1/ t−s ≤ Γv f Γu f 1/ v−u 3.13 Let f and Γs f be defined as in Theorem 3.4(2), and let t, s, u, v ≥ be such that s ≤ u, t ≤ v, s / t, and u / v Then Γt f Γs f 1/ t−s ≤ Γv f Γu f 1/ v−u 3.14 Journal of Inequalities and Applications Proof Similar to the proof of Theorem 2.2 Remark 3.6 Let w ≡ If f is a positive concave function on a, b , then the decreasing rearrangement f ∗ is concave on a, b By applying Theorem 3.4 to f ∗ , we obtain that Γs f ∗ is Γs f ∗ and we see that Theorem 3.4 log-convex Equimeasurability of f with f ∗ gives Γs f is equivalent to Theorem 2.1 Remark 3.7 Let w t tα with α > −1 Then Theorem 3.2 gives that if f is a positive increasing concave function on 0, , then Πα is log-convex, and s Πα Πα α 2s α s 1 s s−1 Πα s α α f t t dt f α f s t tα dt , s / 0, 1, log log f t t dt − α − α f t t dt α log s 1 f f t t dt − α t tα dt α t tα dt α − 1 3.15 , f t log f t tα dt, with zero for the function f t t If f is a positive decreasing concave function on 0, , then Theorem 3.4 gives that Γα s is log-convex, and Γα s s s−1 Γα α s α sB s 1, α s − f t tα dt log f t tα dt Γα 1 α 1−H α 1 − H α log α α log α 2 f s t tα dt , s / 0, 1, 1 α 1 α f t tα dt , 3.16 f t tα dt f t tα dt log f t tα dt − f t log f t tα dt, with zero for the function f t − t, where B ·, · is the beta function, and H α is the harmonic number defined for α > −1 with H α ψ α γ, where ψ is the digamma function and γ 0.577215 the Euler constant Cauchy Means Let us note that 2.12 , 2.17 , 3.8 , 3.9 , 3.13 , and 3.14 have the form of some known inequalities between means e.g., Stolarsky means, Gini means, etc Here we will prove that expressions on both sides of 3.8 are also means The proofs in the remaining cases are analogous 10 Journal of Inequalities and Applications Lemma 4.1 Let h ∈ C2 I , I interval in R, be such that h is bounded, that is, m ≤ h ≤ M Then the functions φ1 , φ2 defined by M t −h t , φ1 t h t − φ2 t m t , 4.1 are convex functions b Theorem 4.2 Let w be a nonnegative integrable function on a, b with a w x dx Let f be a positive increasing concave function on a, b , h ∈ C2 0, 2fi Then there exists ξ ∈ 0, 2fi , such that h 2r fi w a − r br dr − h ξ 2 2r fi w a − r Proof Set m minx∈ 0,2fi h x , M in Lemma 4.1, we have h f t w t dt a b br dr − b−a 4.2 f t w t dt a maxx∈ 0,2fi h x Applying 3.2 for φ1 and φ2 defined φ1 2r fi w a − r b−a b br dr ≥ b−a b b−a b br dr ≥ b b−a φ2 2r fi w a − r φ1 f t w t dt, a 4.3 φ2 f t w t dt, a that is, M 2 2r fi w a − r br dr − ≥ h 2r fi w a − r h 2r fi w a − r br dr − m ≥ 2r fi w a 1−r f t w t dt a br dr − b−a b−a 4.4 b h f t w t dt, a b h f t w t dt a br dr − b−a b 4.5 f t w t dt a By combining 4.4 and 4.5 , 4.2 follows from continuity of h Journal of Inequalities and Applications 11 Theorem 4.3 Let f be a positive increasing concave nonlinear function on a, b , and let w be a b nonnegative integrable function on a, b with a w x dx If h1 , h2 ∈ C2 0, 2fi , then there exists ξ ∈ 0, 2fi such that h1 ξ h2 ξ h 1 h 2r fi w a − r 2r fi w a − r b h a b h a br dr − 1/ b − a br dr − 1/ b − a f t w t dt f t w t dt , 4.6 provided that h2 x / for every x ∈ 0, 2fi Proof Define the functional Φ : C2 0, 2fi Φh → R with h 2r fi w a − r br dr − and set h0 Φ h2 h1 − Φ h1 h2 Obviously, Φ h0 0, 2fi such that Φ h0 h0 ξ 2 2r fi w a − r b−a b h f t w t dt, 4.7 a Using Theorem 4.2 , there exists ξ ∈ br dr − b−a b f t w t dt 4.8 a We give a proof that the expression in square brackets in 4.8 is nonzero actually strictly positive by inequality 3.2 for nonlinear function f Suppose that the expression in square brackets in 4.8 is equal to zero, which is by simple rearrangements equivalent to equality b t − a w t dt a b g t w t dt, b a where g t t − a w t dt b f a a t w t dt f t 4.9 Since g is positive concave function, it is easy to see that g t / t − a is decreasing function on a, b see , thus so x a b b a t − a w t dt t − a w t dt ≤ x g a g t w t dt ≤ a x a t − a w t dt x g t w t dt, x ∈ a, b , 4.10 a t w t dt for every x ∈ a, b Set x F x a t − a − g t w t dt 4.11 12 Journal of Inequalities and Applications Obviously, F x ≤ 0, F a we have b F b t−a 0 By 4.9 , obvious estimations and integration by parts, − g t w t dt ≥ a − 2g t dF t a b a b a t−a 2g t t − a − g t w t dt a b This implies b b F t d 2g t 4.12 ≥ a b 2g a − g t w t dt t t − a − g t w t dt, which is equivalent to t − a − g t w t dt This gives that g is a linear function, which obviously implies that f is a linear function Since the function f is nonlinear, the expression in square brackets in 4.8 is strictly 0, and this gives 4.6 Notice that Theorem 4.2 for h h2 positive which implies that h0 ξ implies that the denominator of the right-hand side of 4.6 is nonzero b Corollary 4.4 Let w be a nonnegative integrable function with a w x dx If f is a positive increasing concave nonlinear function on a, b , then for < s / t / / s there exists ξ ∈ 0, 2fi such that s s−1 t t−1 ξt−s t br dr − 1/ b − a 2r fi w a − r 2r fi w a − r xt and h2 x Proof Set h1 x br dr − 1/ b − a s 1 b t f a b s f a r w r dr r w r dr 4.13 xs , t / s / 0, in 4.6 , then we get 4.13 Remark 4.5 Since the function ξ → ξt−s is invertible, then from 4.13 we have ⎛ 1 s s−1 0