Rehman et al., Cogent Mathematics 016, 3: 1798 PURE MATHEMATICS RESEARCH ARTICLE Petrović s inequality on coordinates related results Atiq Ur Rehman 1 *, Muhammad Mudessir 1, Hafiza Tahira Fazal Ghulam Farid 1 Received: 19 July 016 Accepted: 17 August 016 Published: 13 September 016 *Corresponding author: Atiq Ur Rehman, Department of Mathematics, COMSATS Institute of Information Technology, Attock, Pakistan E-mail: atiq@mathcity.org Reviewing editor: Lishan Liu, Qufu Normal University, China Additional information is available at the end of the article Abstract: In this paper, the authors extend Petrović s inequality to coordinates in the plane. The authors consider functionals due to Petrović s inequality in plane discuss its properties for certain class of coordinated log-convex functions. Also, the authors proved related mean value theorems. Subjects: Foundations & Theorems; Mathematics & Statistics; Science Keywords: Petrović s inequality; log-convexity; convex functions on coordinates 000 Mathematics subject classifications: Primary 6A51; Secondary 6D15 1. Introduction A function f : [a, b] R is called mid-convex or conven Jensen sense if for all x, y [a, b], the inequality x + y f is valid. f x+f y In 1905, J. Jensen was the first to define convex functions using above inequality see, Jensen, 1905; Robert & Varberg, 1974, p. 8 draw attention to their importance. ABOUT THE AUTHORS Atiq Ur Rehman Ghulam Farid are assistant professors in the Department of Mathematics at the COMSATS Institute of Information Technology CIIT, Attock, Pakistan. Their primary research interests include real functions, mathematical inequalities, difference equation. Muhammad Mudessir has successfully completed his MS degree in mathematics from CIIT in this year. He is a teacher in Government Pilot Secondary School, Attock, Pakistan. His area of research includes convex analysis inequalities in mathematics. Hafiza Tahira Fazal received her master of philosophy degree from National College of Business Administration Economics, Lahore, Pakistan. She is working as a lecturer in the Department of Mathematics at the University of Lahore, Sargodha, Pakistan from last two years. Her area of research includes inequalities in mathematics. PUBLIC INTEREST STATEMENT A real-valued function defined on an interval is called convef the line segment between any two points on the graph of the function lies above or on the graph. Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. One of the important subclass of convex functions is log-convex functions. Apparently, it would seem that logconvex functions would be unremarkable because they are simply related to convex functions. But they have some surprising properties. Recently, the concept of convex functions has been generalized by many mathematicians different functions related or close to convex functions are defined. In this work, the variant of Petrovic s inequality for convex functions on coordinates is given. Few generalization of the results related to it are given. 016 The Authors. This open access article is distributed under a Creative Commons Attribution CC-BY 4.0 license. Page 1 of 11
Rehman et al., Cogent Mathematics 016, 3: 1798 Definition 1 A function f : [a, b] R is said to be convef f tx +1 ty tf x+1 tf y 1.1 holds, for all x, y [a, b] t [0, 1]. A function f is said to be strictly convef strict inequality holds in 1.1. A mapping f : Δ R is said to be conven Δ if f tx +1 tz, ty +1 tw tf x, y+1 tf z, w for all x, y, z, w Δ, where Δ: =[a, b] [c, d] R t [0,1]. In Dragomir 001 gave the definition of convex functions on coordinates as follows. Definition Let Δ =[a, b] [c, d] R f : Δ R be a mapping. Define partial mappings f y : [a, b] R by f y u =f u, y 1. f x : [c, d] R by f x v =f x, v. 1.3 Then f is said to be convex on coordinates or coordinated conven Δ if f y f x are convex on [a, b] [c, d] respectively for all x [a, b] y [c, d]. A mapping f is said to be strictly convex on coordinates or strictly coordinated conven Δ if f y f x are strictly convex on [a, b] [c, d] respectively for all x [a, b] y [c, d]. One of the important subclass of convex functions is log-convex functions. Apparently, it would seem that log-convex functions would be unremarkable because they are simply related to convex functions. But they have some surprising properties. The Laplace transform of a non-negative function is a log-convex. The product of log-convex functions is log-convex. Due to their interesting properties, the log-convex functions appear frequently in many problems of classical analysis probability theory, e.g. see Farid, Marwan, & Rehman, 015; Niculescu, 01; Noor, Qi, & Awan, 013; Pečarić, & Rehman, 008a, 008b; Xi & Qi, 015; Zhang & Jiang, 01 the references therein. Definition 3 A function f : I R + is called log-convex on I if f αx + βy f α xf β y where α, β>0 with α + β = 1 x, y I. Definition 4 Alomari & Darus, 009 A function f : Δ R + is called log-convex on coordinates in Δ if partial mappings defined in 1. 1.3 are log-convex on [a, b] [c, d], respectively, for all x [a, b] y [c, d]. Remark 1 Every log-convex function is log convex on coordinates but the converse is not true in general. For example, f : [0, 1] [0, defined by f x, y =e xy is log-convex on coodinates but not log-convex. In Pečarić, Proschan, Tong 199, p. 154, Petrović s inequality for convex function is stated as follows. Theorem 1 Let [0, a R, x 1,, x n 0, a] n p 1,, p n be nonnegative n-tuples such that Page of 11
Rehman et al., Cogent Mathematics 016, 3: 1798 x j for j = 1,, 3,, n [0, a. If f is a convex function on [0, a, then the inequality f f + 1 f 0 1.4 is valid. Remark If f is strictly convex, then strict inequality holds in 1.4 unless x 1 = = x n n = 1. Remark 3 For = 1 i = 1,, n, the above inequality becomes f f + n 1f 0. 1.5 This was proved by Petrović in 193 see Petrović, 193. In this paper, we extend Petrović s inequality to coordinates in the plane. We consider functionals due to Petrović s inequality in plane discuss its properties for certain class of coordinated logconvex functions. Also we proved related mean value theorems.. Main results In the following theorem, we give our first result that is Petrović s inequality for coordinated convex functions. Theorem Let Δ=[0, a [0, b R, x 1,, x n 0, a] n, y 1,, y n 0, b] n, p 1,, p n q 1,, q n be non-negative n-tuples such that n p x i i x j n q y i i for j = 1,, n. Also let n p x i i [0, a, n p i 1 n q y i i [0, b. If f : Δ R is coordinated convex function, then q j f f, q j 1 f,0 i, Proof Let f x : [0, b R f y : [0, a R be mappings such that f x v =f x, v f y u =f u, y. Since f is coordinated convex on Δ, therefore f y is convex on [0, a. By Theorem 1, one has By setting y =, we have this gives [ ] + 1 f 0, q j 1 f 0, 0. f y f y + 1 f y 0. f f + 1 f 0,.1 q j f q j f + 1 q j f 0.. Again, using Theorem 1 on terms of right-h side for second coordinates, we have Page 3 of 11
Rehman et al., Cogent Mathematics 016, 3: 1798 q j f f, q j 1 f,0 [ ] 1 q j f 0 1 f 0, q j 1 f 0, 0. Using above inequities in., we get inequality.1. Remark 4 If f is strictly coordinated convex, then above inequality is strict unless all s y i s are not equal or n p i 1 n q j 1. Remark 5 If we take y i = 0 q j = 1, i, j = 1,...,n with f,0 f, then we get inequality 1.4. Let I R be an interval f : I R be a function. Then for distinct points u i I, i = 0, 1,. The divided differences of first second order are defined as follows: [u i, u i+1, f ]= f u i+1 f u i u i+1 u i, i = 0, 1,, f ]= [u 1, f ], f ] u u 0..3.4 The values of the divided differences are independent of the order of the points u 0 may be extended to include the cases when some or all points are equal, that is, f ]= lim u1 u 0, f ]=f u 0.5 provided that f exists. Now passing the limit u 1 u 0 replacing u by u in second-order divided difference, we have, u, f ]= lim u1 u 0, u, f ]= f u f u 0 u u 0 f u 0 u u 0, u u 0.6 provided that f exists. Also, passing to the limit u i u i = 0, 1, in second-order divided difference, we have [u, u, u, f ]=lim ui u, f ]=f u provided that f exists..7 One can note that, if for all u 0 I,, f ] 0, then f is increasing on I if for all u 0 I,, f ] 0, then f is convex on I. Now we define some families of parametric functions which we use in sequal. Let I =[0, a J =[0, b be intervals let for t c, d R, f t : I J R be a mapping. Then we define functions f t,y : I R by f t,y u =f t u, y Page 4 of 11
Rehman et al., Cogent Mathematics 016, 3: 1798 f t,x : J R by f t,x v =f t x, v, where x I y J. Suppose denotes the class of functions f t : I J R for t c, d such that t, f t,y ] u 0 I t [v 0, v 1, v, f t,x ] v 0, v 1, v J are log-convex functions in Jensen sense on c, d for all x I y J. We define linear functional Υ f as a non-negative difference of inequality.1 Υ f = f, q j 1 f,0 [ ] + 1 f 0, q j 1 f 0, 0 i, q j f..8 Remark 6 Under the assumptions of Theorem, if f is coordinated conven Δ, then Υ f 0. The following lemmas are given in Pečarić Rehman 008b. Lemma 1 Let I R be an interval. A function f : I 0, is log-conven Jensen sense on I, that is, for each r, t I f rf t f t + r if only if the relation m f t+mnf t + r + n f r 0 holds for each m, n R r, t I. Lemma If f is convex function on interval I then for all x 1, x, x 3 I for which x 1 < x < x 3, the following inequality is valid: x 3 x f x 1 +x 1 x 3 f x +x x 1 f x 3 0. Our next result comprises properties of functional defined in.8. Theorem 3 Let the functional Υ defined in.8 f t. Then the following are valid: a The function t Υ f t is log-conven Jensen sense on c, d. b If the function t Υ f t is continuous on c, d, then it is log-convex on c, d. c If Υ f t is positive, then for some r < s < t, where r, s, t c, d, one has [ Υ fs ] t r [ Υ fr ] t s[ Υ ft ] s r..9 Page 5 of 11
Rehman et al., Cogent Mathematics 016, 3: 1798 Proof a Let hu, v =m f t u, v+mnf t+r u, v + n f r u, v where m, n R t, r c, d. We can consider h y u =m f t,y u+mnf t+r f,yu+n r,y u h x v =m f t,x v+mnf t+r f,xv+n r,x v. Now we take, h y ]=m, f t,y ]+mn, f t+r [u,y]+n 0, f r,y ]. As, f t,y ] is log conven Jensen sense, so using Lemma 1, the right-h side of above expression is non-negative, so h y is convex on I. Similarly, one can show that h x is also convex on J, which concludes h is coordinated convex on Δ. By Remark 6, Υ h 0, that is, m Υ f t +mnυ f t+r +n Υ f r 0, so t Υ f t is log-conven Jensen sense on c, d. b Additionally, we have t Υ f t is continuous on c, d, hence we have t Υ f t is log-convex on c, d. c Since t Υ f t is log-convex on c, d, therefore for r, s, t c, d with r < s < t f t =logυ t in Lemma, we have t s log Υ f r +r t log Υ f s +s r log Υ f t 0, which is equivalent to.9. Example 1 Let t 0, φ t :[0, R be a function defined as φ t u, v = { u t v t, tt 1 t 1, uvlog u + log v, t = 1..10 Define partial mappings φ t,v :[0, R by φ t,v u =φ t u, v φ t,u :[0, R by φ t,u v =φ t u, v. As we have [u, u, u, φ t,v ]= φ t,v u = u t v t 0 t 0,. Page 6 of 11
Rehman et al., Cogent Mathematics 016, 3: 1798 This gives t, φ t,v ] is log-conven Jensen sense. Similarly, one can deduce that t [v 0, φ t,u ] is also log-conven Jensen sense. If we choose f t = φ t in Theorem 3, we get log convexity of the functional Υ φ t. In special case, if we choose φ t u, v =φ t u,1, then we get Butt, Pečarić, & Rehman, A. U. 011, Example 3. Example Let t [0, δ t :[0, R be a function defined as δ t u, v = { uve uvt t, t 0, u v, t = 0..11 Define partial mappings δ t,v :[0, R by δ t,v u =δ t u, v δ t,u :[0, R by δ t,u v =δ t u, v for all u, v [0,. As we have [u, u, u, δ t,v ]= δ t,v δu = e uvt v + uv 0 t 0,. This gives t, δ t,v ] is log conven Jensen sense. Similarly, one can deduce that t [v 0, δ t,u ] is also log-conven Jensen sense. If we choose f t = δ t in Theorem 3, we get log convexity of the functional Υ δ t. In special case, if we choose δ t u, v =δ t u,1, then we get Butt et al., 011, Example 8. Example 3 Let t [0, γ t :[0, R be a function defined as γ t u, v = { e uvt t, t 0, uv, t = 0..1 Define partial mappings γ t,v :[0, R by γ t,v u =γ t u, v γ t,u :[0, R by γ t,u v =γ t u, v. As we have [u, u, u, γ t,v ]= γ t,v u = tv e uvt 0 t 0,. This gives t, γ t,v ] is log-conven Jensen sense. Similarly one can deduce that t [v 0, γ t,u ] is also log-conven Jensen sense. If we choose f t = γ t in Theorem 3, we get log convexity of the functional Υ γ t. Page 7 of 11
Rehman et al., Cogent Mathematics 016, 3: 1798 In special case, if we choose γ t u, v =γ t u,1, then we get Butt et al., 011, Example 9. 3. Mean value theorems If a function is twice differentiable on an interval I, then it is convex on I if only if its second order derivative is non-negative. If a function f X: = f x, y has continuous second-order partial derivatives on interior of Δ, then it is convex on Δ if the Hessian matrix H f X = f X f X x y f X y x f X y is non-negative definite, that is, vh f Xv t is non-negative for all real non-negative vector v. It is easy to see that f : Δ R is coordinated convex on Δ iff f y f x, y x = f y x = f x, y y are non-negative for all interior points x, y in Δ. Lemma 3 Let f :Δ R be a function such that m 1 f x, y M 1 m f x, y y M for all interior points x, y in Δ. Consider the function ψ 1, ψ : Δ R defined as ψ 1 = 1 max{m 1, M }x + y f x, y ψ = f x, y 1 min{m 1, m }x + y then ψ 1, ψ are convex on coordinates in Δ. Proof Since ψ 1 x, y = max{m 1, M } f x, y 0 ψ 1 x, y y = max{m 1, M } f x, y y 0 for all interior points x, y in Δ, ψ 1 is convex on coordinates in Δ. Similarly, one can prove that ψ is also convex on coordinates in Δ. Theorem 4 Let f : Δ R be a mapping which has continuous partial derivatives of second order in Δ φx, y: = x + y. Then, there exist β 1, γ 1 β, γ in the interior of Δ such that Page 8 of 11
Rehman et al., Cogent Mathematics 016, 3: 1798 Υ f = 1 f β 1, γ 1 Υ φ Υ f = 1 f β, γ Υ φ y provided that Υ φ is non-zero. Proof Since f has continuous partial derivatives of second order in Δ, there exist real numbers m 1, m, M 1 M such that m 1 f x, y M 1 m f x, y M y, for all x, y Δ. Now consider functions ψ 1 ψ defined in Lemma 3. As ψ 1 is convex on coordinates in Δ, Υ ψ 1 0, that is 1 Υ max{m, M 1 }φx, y f x, y 0, this leads us to Υ f max{m 1, M }Υ φ. 3.1 On the other h, for function ψ, one has min{m 1, m }Υ φ Υ f. 3. As Υ φ 0, combining inequalities 3.1 3., we get min{m 1, m } Then there exist β 1, γ 1 β, γ in the interior of Δ such that Υ f Υ φ = f β 1, γ 1 Υ f Υ φ = f β, γ, y Υ f Υ φ max{m 1, M }. hence the required result follows. Theorem 5 Let ψ 1, ψ : Δ R be mappings which have continuous partial derivatives of second order in Δ. Then there exists η 1, ξ 1 η, ξ in Δ such that Υ ψ 1 Υ ψ = ψ 1 η 1,ξ 1 ψ η 1,ξ 1 3.3 Page 9 of 11
Rehman et al., Cogent Mathematics 016, 3: 1798 Υ ψ 1 Υ ψ = ψ 1 η,ξ y. ψ η,ξ y 3.4 Proof We define the mapping P: Δ R such that P = k 1 ψ 1 k ψ, where k 1 = Υ ψ k = Υ ψ 1. Using Theorem 4 with f = P, we have { } Υ P =0 = ψ k 1 1 x k ψ Υ φ Υ P =0 = Since Υ φ 0, we have k k 1 = k k 1 = ψ 1 η 1,ξ 1 ψ η 1,ξ 1 ψ 1 η,ξ y {, ψ η,ξ y ψ k 1 1 y k ψ y } Υ φ. which are equivalent to required results. Funding The authors received no direct funding for this research. Author details Atiq Ur Rehman 1 E-mail: atiq@mathcity.org ORCID ID: http://orcid.org/0000-000-7368-0700 Muhammad Mudessir 1 E-mail: mudessir001@gmail.com Hafiza Tahira Fazal E-mail: tahiramalik130@gmail.com Ghulam Farid 1 E-mails: faridphdsms@hotmail.com, ghlmfarid@ciit-attock. edu.pk 1 Department of Mathematics, COMSATS Institute of Information Technology, Attock, Pakistan. Department of Mathematics, University of Lahore, Sargodha Campus, Pakistan. Citation information Cite this article as: Petrović s inequality on coordinates related results, Atiq Ur Rehman, Muhammad Mudessir, Hafiza Tahira Fazal & Ghulam Farid, Cogent Mathematics 016, 3: 1798. References Alomari, M., & Darus, M. 009. On the Hadamard s inequality for log convex functions on coordinates. Journal of Inequalities Applications, Article ID 83147. 13. doi:10.1155/009/83147 Butt, S., Pečarić, J., & Rehman, A. U. 011. Exponential convexity of Petrović related functional. Journal of Inequalities Applications, 89, 16. doi:10.1186/109-4x-011-89 Dragomir, S. S. 001. On Hadamards inequality for convex functions on the co-ordinates in a rectangle from the plane. Taiwanese Journal of Mathematics, 4, 775 788. Farid, G., &, Marwan, M., & Rehman, A. U. 015. New mean value theorems generalization of Hadamard inequality via coordinated m-convex functions. Journal of Inequalities Applications, 11. Article ID 83. doi:10.1186/s13660-015-0808-z Jensen, J. 1905. Om konvekse Funktioner og Uligheder mellem Middelvaerdier [About convex functions inequalities between mean values]. Nyt tidsskrift for matematik, 16B, 49 69. Niculescu, C. P. 01. The Hermite-Hadamard inequality for log-convex functions. Nonlinear Analysis, 75, 66 669. doi:10.1016/j.na.011.08.066. Noor, M. A., Qi, F., & Awan, M. U. 013. Some Hermite- Hadamard type inequalities for log-h-convex functions. Analysis, 33, 367 375. doi:10.154/anly.013.13. Page 10 of 11
Rehman et al., Cogent Mathematics 016, 3: 1798 Pečarić, J., Proschan, F., & Tong, Y. L. 199. Convex functions, partial orderings statistical applications. New York, NY: Academic Press. Pečarić, J., & Rehman, A. U. 008a. On logarithmic convexity for power sums related results. Journal of Inequalities Applications, 1. Article ID 389410. doi:10.1155/008/389410 Pečarić, J., & Rehman, A. U. 008b. On logarithmic convexity for power sums related results II. Journal of Inequalities Applications, Article ID 30563. doi:10.1155/008/30563 Petrović, M. 193. Sur une fontionnelle. Publications de l Institut Mathématique, University of Belgrade, 149 146. Roberts, A. W., & Varberg, D. E. 1974. Convex functions Vol. 57. Academic Press. Xi, B. Y., & Qi, F. 015. Integral inequalities of Hermite- Hadamard type for α,m,log-convex functions on coordinates. Probl. Anal. Issues Anal., 4, 73 9. doi:10.15393/j3.art.015.89. Zhang, X., & Jiang, W. 01. Some properties of log-convex function applications for the exponential function. Computers & Mathematics with Applications, 63, 1111 1116. doi:10.1016/j.camwa.011.1.019. 016 The Authors. This open access article is distributed under a Creative Commons Attribution CC-BY 4.0 license. You are free to: Share copy redistribute the material in any medium or format Adapt remix, transform, build upon the material for any purpose, even commercially. The licensor cannot revoke these freedoms as long as you follow the license terms. Under the following terms: Attribution You must give appropriate credit, provide a link to the license, indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits. Cogent Mathematics ISSN: 331-1835 is published by Cogent OA, part of Taylor & Francis Group. Publishing with Cogent OA ensures: Immediate, universal access to your article on publication High visibility discoverability via the Cogent OA website as well as Taylor & Francis Online Download citation statistics for your article Rapid online publication Input from, dialog with, expert editors editorial boards Retention of full copyright of your article Guaranteed legacy preservation of your article Discounts waivers for authors in developing regions Submit your manuscript to a Cogent OA journal at www.cogentoa.com Page 11 of 11