Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 948430, 17 pages doi:10.1155/2010/948430 Research Article Generalization of an Inequality for Integral Transforms with Kernel and Related Results Sajid Iqbal, 1 J. P e ˇ cari ´ c, 1, 2 and Yong Zhou 3 1 Abdus Salam School of Mathematical Sciences, GC University, Lahore 54000, Pakistan 2 Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia 3 School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China Correspondence should be addressed to Sajid Iqbal, sajid uos2000@yahoo.com Received 27 March 2010; Revised 2 August 2010; Accepted 27 October 2010 Academic Editor: Andr ´ as Ront ´ o Copyright q 2010 Sajid Iqbal 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. We establish a generalization of the inequality introduced by Mitrinovi ´ candPe ˇ cari ´ c in 1988. We prove mean value theorems of Cauchy type for that new inequality by taking its difference. Furthermore, we prove the positive semidefiniteness of the matrices generated by the difference of the inequality which implies the exponential convexity and logarithmic convexity. Finally, we define new means of Cauchy type and prove the monotonicity of these means. 1. Introduction Let Kx, t be a nonnegative kernel. Consider a function u : a, b → R, where u ∈ Uv, K, and the representation of u is u  x    b a K  x, t  v  t  dt 1.1 for any continuous function v on a, b. Throughout the paper, it is assumed that all integrals under consideration exist and that they are finite. The following theorem is given in 1see also 2, page 235. Theorem 1.1. Let u i ∈ Uv, Ki  1, 2 and rt ≥ 0 for all t ∈ a, b.Alsoletφ : R  → R be a function such that φx is convex and increasing for x>0.Then  b a r  x  φ      u 1  x  u 2  x       dx ≤  b a s  x  φ      v 1  x  v 2  x       dx, 1.2 2 Journal of Inequalities and Applications where s  x   v 2  x   b a r  t  K  t, x  u 2  t  dt, u 2  t  /  0. 1.3 The following definition is equivalent to the definition of convex functions. Definition 1.2 see 2.LetI ⊆ R be an interval, and let φ : I → R be convex on I. Then, for s 1 ,s 2 ,s 3 ∈ I such that s 1 <s 2 <s 3 , the following inequality holds: φ  s 1  s 3 − s 2   φ  s 2  s 1 − s 3   φ  s 3  s 2 − s 1  ≥ 0. 1.4 Let us recall the following definition. Definition 1.3 see 3, p age 373.Afunctionh : a, b → R is exponentially convex if it is continuous and n  i,j1 ξ i ξ j h  x i  x j  ≥ 0 1.5 for all n ∈ N and all choices of ξ i ∈ R,x i  x j ∈ a, b,i,j 1, ,n. The following proposition is useful to prove the exponential convexity. Proposition 1.4 see 4. Let h : a, b → R. The following statements are equivalent. i h is exponentially convex. ii h is continuous, and n  i,j1 ξ i ξ j h  x i  x j 2  ≥ 0 1.6 for every n ∈ N,ξ i ∈ a, b, and x i ∈ a, b, 1 ≤ i ≤ n. Corollary 1.5. If h : a, b → R  is exponentially convex, then h is log-convex; that is, h  λx   1 − λ  y  ≤ h  x  λ h  y  1−λ ∀x, y ∈  a, b  ,λ∈  0, 1  . 1.7 This paper is organized in this manner. In Section 2, we give the generalization of Mitrinovi ´ c-Pe ˇ cari ´ c inequality and prove the mean value theorems of Cauchy type. We also introduce the new type of Cauchy means. In Section 3, we give the proof of positive semidefiniteness of matrices generated by the difference of that inequality obtained from the generalization of Mitrinovi ´ c-Pe ˇ cari ´ c inequality and also discuss the exponential convexity. At the end, we prove the monotonicity of the means. Journal of Inequalities and Applications 3 2. Main Results Theorem 2.1. Let u i ∈ Uv, Ki  1, 2, and rx ≥ 0 for all x ∈ a, b.AlsoletI ⊆ R be an interval, let φ : I → R be convex, and let u 1 x/u 2 x, v 1 x/v 2 x ∈ I.Then  b a r  x  φ  u 1  x  u 2  x   dx ≤  b a q  x  φ  v 1  x  v 2  x   dx, 2.1 where q  x   v 2  x   b a r  t  K  t, x  u 2  t  dt, u 2  t  /  0. 2.2 Proof. Since u 1   b a Kx, tv 1 tdt and v 2 t > 0, we have  b a r  x  φ  u 1  x  u 2  x   dx   b a r  x  φ  1 u 2  x   b a K  x, t  v 1  t  dt  dx   b a r  x  φ  1 u 2  x   b a K  x, t  v 2  t  v 1  t  v 2  t  dt  dx   b a r  x  φ   b a K  x, t  v 2  t  u 2  x  v 1  t  v 2  t  dt  dx. 2.3 By Jensen’s inequality, we get  b a r  x  φ  u 1  x  u 2  x   dx ≤  b a r  x    b a K  x, t  v 2  t  u 2  x  φ  v 1  t  v 2  t   dt  dx   b a   b a r  x  K  x, t  v 2  t  u 2  x  φ  v 1  t  v 2  t   dt  dx   b a φ  v 1  t  v 2  t   v 2  t    b a r  x  K  x, t  u 2  x  dx  dt   b a q  t  φ  v 1  t  v 2  t   dt. 2.4 Remark 2.2. If φ is strictly convex on I and v 1 x/v 2 x is nonconstant, then the inequality in 2.1 is strict. 4 Journal of Inequalities and Applications Remark 2.3. Let us note that Theorem 1.1 follows from Theorem 2.1. Indeed, let the condition of Theorem 1.1 be satisfied, and let u i ∈ U|v|,K;thatis, u 1  x    b a K  x, t  | v 1  t  | dt. 2.5 So, by Theorem 2.1, we have  b a q  x  φ      v 1  x  v 2  x       dx   b a q  x  φ  | v 1  x  | v 2  x   dx ≥  b a r  x  φ  u 1  x  u 2  x   dx. 2.6 On the other hand, φ is increasing function, we have φ  u 1  x  u 2  x    φ  1 u 2  x   b a K  x, t  | v 1  t  | dt  ≥ φ  1 u 2  x        b a K  x, t  v 1  t  dt        φ  | u 1  x  | u 2  x    φ      u 1  x  u 2  x       . 2.7 From 2.6 and 2.7,weget1.2. If f ∈ Ca, b and α>0, then the Riemann-Liouville fractional integral is defined by I α a f  x   1 Γ  α   x a f  t  x − t  α−1 dt. 2.8 We will use the following kernel in the upcoming corollary: K I  x, t   ⎧ ⎪ ⎨ ⎪ ⎩  x − t  α−1 Γ  α  ,a≤ t ≤ x, 0,x<t≤ b. 2.9 Corollary 2.4. Let u i ∈ Ca, b i  1, 2, and rx ≥ 0 for all x ∈ a, b.AlsoletI ⊆ R be an interval, let φ : I → R be convex, u 1 x/u 2 x, I α a u 1 x/I α a u 2 x ∈ I, and u 1 x,u 2 x have Riemann-Liouville fractional integral of order α>0.Then  b a r  x  φ  I α a u 1  x  I α a u 2  x   dx ≤  b a φ  u 1  t  u 2  t   Q I  t  dt, 2.10 where Q I  t   u 2  t  Γ  α   b t r  x  x − t  α−1 I α a u 2  x  dx, I α a u 2  x  /  0. 2.11 Journal of Inequalities and Applications 5 Let ACa, b be space of all absolutely continuous functions on a, b.ByAC n a, b, we denote the space of all functions g ∈ C n a, b with g n−1 ∈ ACa, b. Let α ∈ R  and g ∈ AC n a, b. Then the Caputo fractional derivative see 5, p. 270 of order α for a function g is defined by D α ∗a g  t   1 Γ  n − α   t a g n  s   t − s  α−n1 ds, 2.12 where n α1; the notation of α stands for the largest integer not greater than α. Here we use the following kernel in the upcoming corollary: K D  x, t   ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩  x − t  n−α−1 Γ  n − α  ,a≤ t ≤ x, 0,x<t≤ b. 2.13 Corollary 2.5. Let u i ∈ AC n a, b i  1, 2, and rx ≥ 0 for all x ∈ a, b.AlsoletI ⊆ R be an interval, let φ : I → R be convex, u n 1 t/u n 2 t, D α ∗a u 1 x/D α ∗a u 2 x ∈ I, and u 1 x,u 2 x have Caputo fractional derivative of order α>0.Then  b a r  x  φ  D α ∗a u 1  x  D α ∗a u 2  x   dx ≤  b a φ  u n 1  t  u n 2  t   Q D  t  dt, 2.14 where Q D  t   u n 2  t  Γ  n − α   b t r  x  x − t  n−α−1 D α ∗a u 2  x  dx, D α ∗a u 2  x  /  0. 2.15 Let L 1 a, b be the space of all functions integrable on a, b. For β ∈ R  , we say that f ∈ L 1 a, b has an L ∞ fractional derivative D β a f in a, b if and only if D β−k a f ∈ Ca, b for k  1, ,β1, D β−1 a f ∈ ACa, b,andD β a ∈ L ∞ a, b. The next lemma is very useful to give the upcoming corollary 6see also 5, p. 449. Lemma 2.6. Let β>α≥ 0,f∈ L 1 a, b has an L ∞ fractional derivative D β a f in a, b, and D β−k a f  a   0,k 1, ,  β   1. 2.16 Then D α a f  s   1 Γ  β − α   s a  s − t  β−α−1 D β a f  t  dt 2.17 for all a ≤ s ≤ b. 6 Journal of Inequalities and Applications Clearly D α a f is in AC  a, b  for β − α ≥ 1, D α a f is in C  a, b  for β − α ∈  0, 1  , 2.18 hence D α a f ∈ L ∞  a, b  , D α a f ∈ L 1  a, b  . 2.19 Now we use the following kernel in the upcoming corollary: K L  s, t   ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩  s − t  β−α−1 Γ  β − α  ,a≤ t ≤ s, 0,s<t≤ b. 2.20 Corollary 2.7. Let β>α≥ 0, u i ∈ L 1 a, bi  1, 2 has an L ∞ fractional derivative D β a u i in a, b, and rx ≥ 0 for all x ∈ a, b.AlsoletD β−k a u i a0 for k  1, ,β1 i  1, 2,letφ : I → R be convex, and D α a u 1 x/D α a u 2 x, D β a u 1 x/D β a u 2 x ∈ I.Then  b a r  x  φ  D α a u 1  x  D α a u 2  x   dx ≤  b a φ  D β a u 1  t  D β a u 2  t   Q L  t  dt, 2.21 where Q L  t   D β a u 2  t  Γ  β − α   b t r  x  x − t  β−α−1 D α a u 2  x  dx, D α a u 2  x  /  0. 2.22 Lemma 2.8. Let f ∈ C 2 I, and let I be a compact interval, such that m ≤ f   x  ≤ M, ∀x ∈ I. 2.23 Consider two functions φ 1 ,φ 2 defined as φ 1  x   Mx 2 2 − f  x  , φ 2  x   f  x  − mx 2 2 . 2.24 Then φ 1 and φ 2 are convex on I. Journal of Inequalities and Applications 7 Proof. We have φ  1  x   M − f   x  ≥ 0, φ  2  x   f   x  − m ≥ 0, 2.25 that is φ 1 ,φ 2 are convex on I. Theorem 2.9. Let f ∈ C 2 I,letI be a compact interval, u i ∈ Uv, Ki  1, 2, and rx ≥ 0 for all x ∈ a, b.Alsoletu 1 x/u 2 x, v 1 x/v 2 x ∈ I, v 1 x/v 2 x be nonconstant, and let qx be given in 2.2. Then there exists ξ ∈ I such that  b a  q  x  f  v 1  x  v 2  x   − r  x  f  u 1  x  u 2  x   dx  f   ξ  2  b a  q  x   v 1  x  v 2  x   2 − r  x   u 1  x  u 2  x   2  dx. 2.26 Proof. Since f ∈ C 2 I and I is a compact interval, therefore, suppose that m  min f  , M  max f  .UsingTheorem 2.1 for the function φ 1 defined in Lemma 2.8, we have  b a r  x   M 2  u 1  x  u 2  x   2 − f  u 1  x  u 2  x    dx ≤  b a q  x   M 2  v 1  x  v 2  x   2 − f  v 1  x  v 2  x    dx. 2.27 From Remark 2.2, we have  b a  q  x   v 1  x  v 2  x   2 − r  x   u 1  x  u 2  x   2  dx > 0. 2.28 Therefore, 2.27 can be written as 2  b a  q  x  f  v 1  x  /v 2  x  − r  x  f  u 1  x  /u 2  x   dx  b a  q  x  v 1  x  /v 2  x  2 − r  x  u 1  x  /u 2  x  2  dx ≤ M. 2.29 We have a similar result f or the function φ 2 defined in Lemma 2.8 as follows: 2  b a  q  x  f  v 1  x  /v 2  x  − r  x  f  u 1  x  /u 2  x   dx  b a  q  x  v 1  x  /v 2  x  2 − r  x  u 1  x  /u 2  x  2  dx ≥ m. 2.30 Using 2.29 and 2.30, we have m ≤ 2  b a  q  x  f  v 1  x  /v 2  x  − r  x  f  u 1  x  /u 2  x   dx  b a  q  x  v 1  x  /v 2  x  2 − r  x  u 1  x  /u 2  x  2  dx ≤ M. 2.31 8 Journal of Inequalities and Applications By Lemma 2.8, there exists ξ ∈ I such that  b a  q  x  f  v 1  x  /v 2  x  − r  x  f  u 1  x  /u 2  x   dx  b a  q  x  v 1  x  /v 2  x  2 − r  x  u 1  x  /u 2  x  2  dx  f   ξ  2 . 2.32 This is the claim of the theorem. Let us note that a generalized mean value Theorem 2.9 for fractional derivative was given in 7. Here we will give some related results as consequences of Theorem 2.9. Corollary 2.10. Let f ∈ C 2 I,letI be a compact interval, u i ∈ Ca, b i  1, 2, and rx ≥ 0 for all x ∈ a, b.Alsoletu 1 x/u 2 x, I α a u 1 x/I α a u 2 x ∈ I,letu 1 x/u 2 x be nonconstant, let Q I t be given in 2.11, and u 1 x, u 2 x have Riemann-Liouville fractional integral of order α>0. Then there exists ξ ∈ I such that  b a  Q I  x  f  u 1  x  u 2  x   − r  x  f  I α a u 1  x  I α a u 2  x   dx  f   ξ  2  b a  Q I  x   u 1 x u 2 x  2 − r  x   I α a u 1 x I α a u 2 x  2  dx. 2.33 Corollary 2.11. Let f ∈ C 2 I,letI be compact interval, u i ∈ AC n a, b i  1, 2, and rx ≥ 0 for all x ∈ a, b.Alsoletu n 1 t/u n 2 t, D α ∗a u 1 x/D α ∗a u 2 x ∈ I,letu n 1 x/u n 2 x be nonconstant, let Q D t be given in 2.15, and u 1 x,u 2 x have Caputo derivative of order α>0. Then there exists ξ ∈ I such that  b a  Q D  x  f  u n 1  x  u n 2  x   − r  x  f  D α ∗a u 1  x  D α ∗a u 2  x    dx  f   ξ  2  b a ⎛ ⎝ Q D  x   u n 1  x  u n 2  x   2 − r  x   D α ∗a u 1  x  D α ∗a u 2  x   2 ⎞ ⎠ dx. 2.34 Corollary 2.12. Let β>α≥ 0,f∈ C 2 I,letI be a compact interval, u i ∈ L 1 a, bi  1, 2 has an L ∞ fractional derivative, and rx ≥ 0 for all x ∈ a, b.LetD β−k a u i a0 for k  1, ,β1 i  1, 2, D α a u 1 x/D α a u 2 x, D β a u 1 x/D β a u 2 x ∈ I,letD β a u 1 x/D β a u 2 x be nonconstant, and let Q L t be given in 2.22. Then there exists ξ ∈ I such that  b a  Q L  x  f  D β a u 1  x  D β a u 2  x   − r  x  f  D α a u 1  x  D α a u 2  x    dx  f   ξ  2  b a ⎛ ⎝ Q L  x   D β a u 1  x  D β a u 2  x   2 − r  x   D α a u 1  x  D α a u 2  x   2 ⎞ ⎠ dx. 2.35 Journal of Inequalities and Applications 9 Theorem 2.13. Let f, g ∈ C 2 I,letI be a compact interval, u i ∈ Uv, Ki  1, 2, and rx ≥ 0 for all x ∈ a, b.Alsoletu 1 x/u 2 x,v 1 x/v 2 x ∈ I, v 1 x/v 2 x be nonconstant, and let qx be given in 2.2. Then there exists ξ ∈ I such that  b a q  x  f  v 1  x  /v 2  x  dx −  b a r  x  f  u 1  x  /u 2  x  dx  b a q  x  g  v 1  x  /v 2  x  dx −  b a r  x  g  u 1  x  /u 2  x  dx  f   ξ  g   ξ  . 2.36 It is provided that denominators are not equal to zero. Proof. Let us take a function h ∈ C 2 I defined as h  x   c 1 f  x  − c 2 g  x  , 2.37 where c 1   b a q  x  g  v 1  x  v 2  x   dx −  b a r  x  g  u 1  x  u 2  x   dx, c 2   b a q  x  f  v 1  x  v 2  x   dx −  b a r  x  f  u 1  x  u 2  x   dx. 2.38 By Theorem 2.9 with f  h, we have 0   c 1 2 f   ξ  − c 2 2 g   ξ     b a q  x   v 1  x  v 2  x   2 dx −  b a r  x   u 1  x  u 2  x   2 dx  . 2.39 Since  b a q  x   v 1  x  v 2  x   2 dx −  b a r  x   u 1  x  u 2  x   2 dx /  0, 2.40 so we have c 1 f   ξ  − c 2 g   ξ   0. 2.41 This implies that c 2 c 1  f   ξ  g   ξ  . 2.42 This is the claim of the theorem. Let us note that a generalized Cauchy mean-valued theorem for fractional derivative was given in 8. Here we will give some related results as consequences of Theorem 2.13. 10 Journal of Inequalities and Applications Corollary 2.14. Let f, g ∈ C 2 I,letI be a compact interval, u i ∈ Ca, b i  1, 2, and rx ≥ 0 for all x ∈ a, b.Alsoletu 1 x/u 2 x, I α a u 1 x/I α a u 2 x ∈ I,letu 1 x/u 2 x be nonconstant, let Q I t be given in 2.11, and u 1 x, u 2 x have Riemann-Liouville fractional derivative of order α>0. Then there exists ξ ∈ I such that  b a Q I  x  f  u 1  x  /u 2  x  dx −  b a r  x  f  I α a u 1  x  /I α a u 2  x  dx  b a Q I  x  g  u 1  x  /u 2  x  dx −  b a r  x  g  I α a u 1  x  /I α a u 2  x  dx  f   ξ  g   ξ  . 2.43 It is provided that denominators are not equal to zero. Corollary 2.15. Let f,g ∈ C 2 I,letI be a compact interval, u i ∈ AC n a, b i  1, 2, and rx ≥ 0 for all x ∈ a, b.Alsoletu n 1 t/u n 2 t,D α ∗a u 1 x/D α ∗a u 2 x ∈ I,letu n 1 x/u n 2 x be nonconstant, let Q D t be given in 2.15, and u 1 x, u 2 x have Caputo fractional derivative of order α>0. Then there exists ξ ∈ I such that  b a Q D  x  f  u n 1  x  /u n 2  x   dx −  b a r  x  f  D α ∗a u 1  x  /D α ∗a u 2  x  dx  b a Q D  x  g  u n 1  x  /u n 2  x   dx −  b a r  x  g  D α ∗a u 1  x  /D α ∗a u 2  x  dx  f   ξ  g   ξ  . 2.44 It is provided that denominators are not equal to zero. Corollary 2.16. Let β>α≥ 0,f,g∈ C 2 I,letI be a compact interval, u i ∈ L 1 a, bi  1, 2 has an L ∞ fractional derivative D β a u i in a, b, and rx ≥ 0 for all x ∈ a, b.AlsoletD β−k a u i a0 for k  1, ,β1 i  1, 2,D α a u 1 x/D α a u 2 x,D β a u 1 x/D β a u 2 x ∈ I,letD β a u 1 x/D β a u 2 x be nonconstant, and let Q L t be given in 2.22. Then there exists ξ ∈ I such that  b a Q L  x  f  D β a u 1  x  /D β a u 2  x   dx −  b a r  x  f  D α a u 1  x  /D α a u 2  x  dx  b a Q L  x  g  D β a u 1  x  /D β a u 2  x   dx −  b a r  x  g  D α a u 1  x  /D α a u 2  x  dx  f   ξ  g   ξ  . 2.45 It is provided that denominators are not equal to zero. Corollary 2.17. Let I ⊆ R  ,letI be a compact interval, u i ∈ Uv, Ki  1, 2, and rx ≥ 0 for all x ∈ a, b.Letu 1 x/u 2 x,v 1 x/v 2 x ∈ I,letv 1 x/v 2 x be nonconstant, and let qx be given in 2.2. Then, for s, t ∈ R \{0, 1} and s /  t,thereexistsξ ∈ I such that ξ  ⎛ ⎝ s  s − 1  t  t − 1   b a q  x  v 1  x  /v 2  x  t dx −  b a r  x  u 1  x  /u 2  x  t dx  b a q  x  v 1  x  /v 2  x  s dx −  b a r  x  u 1  x  /u 2  x  s dx ⎞ ⎠ 1/t−s . 2.46 [...]... Classical and New Inequalities in Analysis, vol 61 of c c c Mathematics and Its Applications (East European Series), Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993 4 M Anwar, N Latif, and J Peˇ ari´ , “Positive semidefinite matrices, exponential convexity for c c majorization, and related cauchy means,” Journal of Inequalities and Applications, vol 2010, Article ID 728251, 2010 5 G A Anastassiou,... α > 0, we well use the n α notation Ms,t instead of Ms,t and we replace vi x with ui x , ui x with D∗a ui x , and q x with QD x Remark 2.21 In the case of L∞ fractional derivative, we will use the notation Ms,t instead of β α Ms,t and we replace vi x with Da ui x , ui x with Da ui x , and q x with QL x 3 Exponential Convexity Lemma 3.1 Let s ∈ R, and let ϕs : R → R be a function defined as ⎧ s ⎪ x... 1 q x A x dx − v1 x /v2 x and B x b a s r x B x dx ⎞1/ 1−s ⎟ ⎠ , u1 x /u2 x Remark 2.19 In the case of Riemann-Liouville fractional integral of order α > 0, we well use α the notation Ms,t instead of Ms,t and we replace vi x with ui x , ui x with Ia ui x , and q x with QI x Journal of Inequalities and Applications 13 Remark 2.20 In the case of Caputo fractional derivative of order α > 0, we well use... Peˇ ari´ , “Generalizations of two inequalities of Godunova and Levin,” Bulletin c c c of the Polish Academy of Sciences, vol 36, no 9-10, pp 645–648, 1988 2 J E Peˇ ari´ , F Proschan, and Y L Tong, Convex Functions, Partial Orderings, and Statistical Applications, c c vol 187 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1992 3 D S Mitrinovi´ , J E Peˇ ari´ , and A M Fink,... Science-Businness Media, Dordrecht, The Netherlands, 2009 6 G D Handley, J J Koliha, and J Peˇ ari´ , “Hilbert-Pachpatte type integral inequalities for fractional c c derivatives,” Fractional Calculus & Applied Analysis, vol 4, no 1, pp 37–46, 2001 7 J J Trujillo, M Rivero, and B Bonilla, “On a Riemann-Liouville generalized Taylor’s formula,” Journal of Mathematical Analysis and Applications, vol 231, no 1, pp... continuous for s ∈ R and using Corollary 1.5, we get that s is logconvex Now by Definition 1.2 with f t log t and r, s, t ∈ R such that r < s < t, we get log which is equivalent to 3.4 t−r s ≤ log t−s r log s−r t , 3.10 Journal of Inequalities and Applications 15 i 1, 2 , and r x ≥ 0 for all x ∈ a, b Also let u1 x /u2 x , Corollary 3.3 Let ui ∈ C a, b α α Ia u1 x /Ia u2 x ∈ R , u1 x , u2 x have Riemann-Liouville... 3.12 For t, s, u, v ∈ R such that s ≤ v, t ≤ u, then the following inequality holds: Ms,t ≤ Mv,u 3.18 1, 2 and the assumption of Corollary 3.5 be Corollary 3.9 Let β > α ≥ 0, ui ∈ L1 a, b i satisfied, also let t be defined by 3.13 For t, s, u, v ∈ R such that s ≤ v, t ≤ u Then following inequality holds Ms,t ≤ Mv,u 3.19 Journal of Inequalities and Applications 17 References 1 D S Mitrinovi´ and J... Then ϕs is strictly convex on R for each s ∈ R Proof Since ϕs x s ∈ R xs−2 > 0 for all x ∈ R , s ∈ R, therefore, ϕ is strictly convex on R for each Theorem 3.2 Let ui ∈ U v, K let q x be given in 2.2 , and i 1, 2 , ui x , vi x > 0 i b t q x ϕt a v1 x v2 x dx − b r x ϕt a 1, 2 , r x ≥ 0 for all x ∈ a, b , u1 x u2 x dx 3.2 Then the following statements are valid a For n ∈ N and si ∈ R, i 1, , n, the... Riemann-Liouville fractional integral of order α > 0, let QI t be given in 2.11 , and b u1 x u2 x QI x ϕt t a Then the statement of Theorem 3.2 with t dx − b r x ϕt a instead of t α I a u1 x α I a u2 x dx 3.11 is valid Corollary 3.4 Let ui ∈ ACn a, b i 1, 2 , and r x ≥ 0 for all x ∈ a, b Also let n n α α u1 t /u2 t , D∗a u1 x /D∗a u2 x ∈ R , u1 x , u2 x have Caputo fractional derivative of order α > 0, let... given in 2.15 , and n b t QD x ϕt a u1 n u2 Then the statement of Theorem 3.2 with t x x b dx − r x ϕt a instead of t α D∗a u1 x α D∗a u2 x dx 3.12 is valid Corollary 3.5 Let β > α ≥ 0, ui ∈ L1 a, b i 1, 2 has L∞ fractional derivative, and r x ≥ 0 β−k α α 0 for k 1, , β 1 i 1, 2 , Da u1 x /Da u2 x , for all x ∈ a, b Also let Da ui a β β Da u1 x /Da u2 x ∈ R , let QL t be given in 2.22 , and β b t QL . Corporation Journal of Inequalities and Applications Volume 2010, Article ID 948430, 17 pages doi:10.1155/2010/948430 Research Article Generalization of an Inequality for Integral Transforms with Kernel and Related. organized in this manner. In Section 2, we give the generalization of Mitrinovi ´ c-Pe ˇ cari ´ c inequality and prove the mean value theorems of Cauchy type. We also introduce the new type of. that the right-hand side of 2.49 is mean, then for distinct s, t ∈ R it can be written as M s,t    t  s  1/t−s 2.52 12 Journal of Inequalities and Applications as mean in broader sense.