ON HADAMARD S INEQUALITIES FOR THE PRODUCT OF TWO CONVEX MAPPINGS DEFINED IN TOPOLOGICAL GROUPS. f(a) + f(b) 2 b a. f(x)dx,

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1 ON HADAMARD S INEQUALITIES FOR THE PRODUCT OF TWO CONVE MAPPINS DEFINED IN TOPOLOICAL ROUPS M. A. LATIF, S. S. DRAOMIR 1,, AND E. MOMONIAT Abstract. In this paper we study Hermite-Hadamard type inequalities for the product of two midconvex and quasi-midconvex functions and give some applications of our results. 1. Introduction Let f : I R be a cnovex mapping defined on the interval I of real numbers and a, b I with a < b. The following double inequality: ) a + b 1.1) f 1 b fa) + fb) fx)dx b a a is known as Hadamard s inequality for convex mappings. The history of this inequality goes back to the papers of Ch. Hermite [9 and J. Hadamard [8 in 1883 and 1893 respectively. This inequality produces some classical inequalities of means for particular choice of the mapping f. The inequality 1.1) attracted a number of mathematicians and is generalized, extended, and refined it in a number of ways see e.g. [3, 4 and [7). Also some mappings naturally connected with 1.1) are defined and properties of these mappings are discussed by many mathematicians see e.g. [5, 6). We discuss only recent studies in this paper. A generalization of the left side of 1.1) for convex functions defined on a convex subset of R n is the following inequality from [13 1.) f0) 1 µ) fx)dx, where R n is a convex bounded symmetrical set that is, if x x, f is a lower semicontinuous convex function f : R and µ) is the volume of the set. In [1, Morassaei established Hadamard s inequality for midconvex and quasimidconvex functions in topological groups and discussed some of the properties of the mapping naturally connected with the Hadamard s inequality for globally midconvex function defined in a topological group. Some of the main results from [1 are stated in the following theorems. Date: March 9, Mathematics Subject Classification. Primary 6A51, A10; Secondary 6D07, 6D15. Key words and phrases. convex functions, Hadamard s inequality, midconvex function, quasimidconvex function, topological groups. This paper is in final form and no version of it will be submitted for publication elsewhere. 1

2 M. A. LATIF, S. S. DRAOMIR 1,, AND E. MOMONIAT Theorem 1. [1 Let be a locally compact group and an open symmetric set relative to a with 0 < µ) <. If f : R is measurable and locally midconvex in a and f L 1 ). If ω : R is non-negative symmetric to a and ω L 1 ) such that fω L 1 ), then 1.3) fa) ωaz)dµz) faz)gaz)ωaz)dµz), where µ is the Haar measure. Theorem. [1 Suppose that be a locally compact group and an open symmetric set relative to a with 0 < µ ) < and e. Let f be measurable and quasi-midconvex real-valued function on such that f L ). If ω : R is a non-negative and symmetric to a and ω L ), then 1.4) fa) ωaz)dµz) faz)ωaz)dµz) + Ia), where Ia) = 1 faz) faz 1 ) ωaz)dµz). Furthermore, I a) satisfies the following inequality 1.5) 0 I a) { 1 min f az)dµz) f az)faz 1 )dµz) ω az)dµz) ) } 1, faz) ωaz)dµz). Theorem 3. Let be a locally compact group and an open symmetric set relative to a with 0 < µ ) <. If f are measurable real valued P-function on such that f L 1 ). If ω : R is non-negative symmetric to a, ω L 1 ) and fω L 1 ), then 1.6) f a) ω az) dµ z) faz)gaz)ω az) dµ z). We give a result similar to 1.) for the product of two convex functions defined on a convex bounded symmetrical subset of R n. We will also give our results for the product of two midconvex and qusi-midconvex mappings defined in a topological groups in Section 3. Applications of the obtained results are given as well in Section 3.. A Secondary Result Theorem 4. Let f, g be two convex functions defined on a convex bounded symmetrical subset of R n, we have.1) f0)g0) 1 [fx)gx) + fx)g x) dx µ) = 1 [fx)gx) + f x)gx) dx. µ) Proof. Consider the transformation of the R n in itself defined by h : R n R n, h = h 1,..., h n )

3 HADAMARD S INEQUALITIES IN TOPOLOICAL ROUPS 3 and Then h) = and since D h 1,..., h n ) D x 1,..., x n ) = h i x 1,..., x n ) = x i, i = 1,,..., n. Thus we have, by the change of variables that = 1) n f x 1,..., x n ) g x 1,..., x n ) dx 1...dx n = f h 1 x 1,..., x n ),..., h n x 1,..., x n )) g h 1 x 1,..., x n ),..., h n x 1,..., x n )) D h 1,..., h n ) D x 1,..., x n ) dx 1...dx n = f x 1,..., x n ) g x 1,..., x n ) dx 1...dx n and f x 1,..., x n ) g x 1,..., x n ) dx 1...dx n = f h 1 x 1,..., x n ),..., h n x 1,..., x n )) g h 1 x 1,..., x n ),..., h n x 1,..., x n )) D h 1,..., h n ) D x 1,..., x n ) dx 1...dx n = f x 1,..., x n ) g x 1,..., x n ) dx 1...dx n. Now by the convexity of f and g on, we get that f0,..., 0)g0,..., 0) x1 x = f,..., x n x n x1,..., x n ) + x 1,..., x n ) = f ) x1 x g ) g,..., x ) n x n ) x1,..., x n ) + x 1,..., x n ) 1 4 [f x 1,..., x n ) + f x 1,..., x n ) [g x 1,..., x n ) + g x 1,..., x n ) = 1 4 [f x 1,..., x n ) g x 1,..., x n ) + f x 1,..., x n ) g x 1,..., x n ) +f x 1,..., x n ) g x 1,..., x n ) + f x 1,..., x n ) g x 1,..., x n )

4 4 M. A. LATIF, S. S. DRAOMIR 1,, AND E. MOMONIAT which gives by integration on that.) f0,..., 0)g0,..., 0)dx 1...dx n 1 [ f x 1,..., x n ) g x 1,..., x n ) dx 1...dx n 4 + f x 1,..., x n ) g x 1,..., x n ) dx 1...dx n + f x 1,..., x n ) g x 1,..., x n ) dx 1...dx n. Hence.1) follows from.). This completes the proof of the Theorem. 3. Main Results Now we prove Hadamard s type inequalities for product of midconvex and quasiconvex functions defined in a topological groups. Before we proceed to prove our results we give some definitions from [1, 11 and [1. We recall that for a group,, e), a topology on is compatible with the group structure when the maps : x, y) xy multiplication) and : x x 1 inverse) are continuous. A group together with a topology compatible with its group structure is a topological group. A Haar measure on is a measure µ : Σ [0, ), with a σ-algebra containing all Borel subsets of, such that µ ) = 1 and µ γs) = µ S) for all γ, S Σ, where γs = {γα : α S}. Definition 1. [1 Let be a topological group, a non-empty open subset of and f a real-valued function on. We say that f is globally right) midconvex if fa) faz) + faz 1 ) for all a, z such that a, az, az 1. We say that f is locally right) midconvex in a if there exists an open symmetric set V = V 1 from e such that for all z such that az, az 1. fa) faz) + faz 1 ) Definition. [11 Let be a topological group, a non-empty open subset of and f a real-valued function on. The mapping f is called quasi-right) midconvex, if faz) max{fa), faz )} for every a, z so that a, az, az. Note that a is midpoint of az 1 and az, and az is midpoint of a and az. Definition 3. [1, Definition 1, page 4 Let be an open subset of topological group, and a. is said to be symmetric relative to a, if a 1 is symmetric and e a 1. Definition 4. [1, Definition, Page 4 Let be a topological group and an open set. A function ω : R is called symmetric relative to a, if z ; az, az 1 and ωaz) = ωaz 1 ). We now give our main result.

5 HADAMARD S INEQUALITIES IN TOPOLOICAL ROUPS 5 Theorem 5. Suppose that be a locally compact group and an open symmetric set relative to a with 0 < µ) <. Let f, g : R + be measurable and locally midconvex in a and f, g L 1 ). If ω : R is nonnegative symmetric to a and ω L 1 ) such that fgω L 1 ), we have 3.1) fa)ga) 1 [ ωaz)dµz) faz)gaz)ωaz)dµz) + = 1 [ faz)gaz)ωaz)dµz) + faz)gaz 1 )ωaz)dµz) faz 1 )gaz)ωaz)dµz), where µ is the Haar measure. Proof. Since f and g are midconvex in a, therefore we have fa) faz) + faz 1 ) and ga) gaz) + gaz 1 ). for any z. From these inequalities we get that 4fa)ga) faz)gaz) + faz 1 )gaz) + faz)gaz 1 ) + faz 1 )gaz 1 ). Since ω is non-negative and symmetric relative to a, we have 4fa)ga)ωaz) faz)gaz)ωaz) + faz 1 )gaz)ωaz 1 ) + faz)gaz 1 )ωaz) + faz 1 )gaz 1 )ωaz 1 ).

6 6 M. A. LATIF, S. S. DRAOMIR 1,, AND E. MOMONIAT Integrating this inequality over, we get that 4fa)ga) ωaz)dµz) faz)gaz)ωaz)dµz) + faz 1 )gaz)ωaz 1 )dµz) + faz)gaz 1 )ωaz)dµz) + faz 1 )gaz 1 )ωaz 1 )dµz) = fz)gz)ωz)dµz) + fz 1 )gz)ωz 1 )dµz) a 1 a 1 + fz)gz 1 )ωz)dµz) + fz 1 )gz 1 )ωz 1 )dµz) a 1 a 1 = fz)gz)ωz)χ a 1 z) dµz) + fz 1 )gz)ωz 1 )χ a 1 z) dµz) + fz)gz 1 )ωz)χ a 1 z) dµz) + fz 1 )gz 1 )ωz 1 )χ a 1 z) dµz) = fz)gz)ωz)χ a 1 z) dµz) + fz 1 )gz)ωz 1 )χ a 1 z 1 ) dµz) + fz)gz 1 )ωz)χ a 1 z) dµz) + fz 1 )gz 1 )ωz 1 )χ a 1 z 1 ) dµz). That is 3.) fa)ga) ωaz)dµz) fz)gz)ωz)χ a 1 dµz) + fz)gz 1 )ωz)χ a 1 z) dµz) = fz)gz)ωz)dµz) + fz)gz 1 )ωz)dµz) a 1 a 1 = 1 [ faz)gaz)ωaz)dµz) + faz)gaz 1 )ωaz)dµz). Since faz)gaz 1 )ωaz)dµz) = faz)gaz 1 )ωaz 1 )dµz) = fz)gz 1 )ωz 1 )dµz) a 1 = fz)gz 1 )ωz 1 )χ a 1 z) dµz) = fz)gz 1 )ωz 1 )χ a 1 z 1 ) dµz) = fz 1 )gz)ωz)χ a 1 z) dµz) = fz 1 )gz)ω z) dµz) a 1 = faz 1 )gaz)ωaz)dµz).

7 HADAMARD S INEQUALITIES IN TOPOLOICAL ROUPS 7 Thus, we also have 3.3) fa)ga) ωaz)dµz) 1 [ faz)gaz)ωaz)dµz) + faz 1 )gaz)ωaz)dµz). Consequently the inequality 3.1) follows from 3.) and 3.3). This completes the proof of the theorem. Remark 1. If we take a = e and ω 1 on in Theorem 5, we have 3.4) fe)ge) 1 µ) [ fz)gz)ωz)dµz) + [ = 1 µ) which is similar to.1). fz)gz)ωz)dµz) + fz)gz 1 )ωz)dµz) Remark. If g 1 on in Theorem 5, we have 3.5) fa) ωaz)dµz) faz)gaz)ωaz)dµz) which is a similar result as proved in [1, Theorem 1. fz 1 )gz)ωz)dµz) Remark 3. If we take a = e, ω 1 and g 1 on in Theorem 5, we have 3.6) fe) 1 fz)dµz) µ) which is the same result as proved in [1, Remark1. Theorem 6. Suppose that be a locally compact group and an open symmetric set relative to a with 0 < µ ) < and e. Let f and g be measurable and quasi-midconvex non-negative real-valued functions on such that fg L ). If ω : R is a non-negative and symmetric to a and ω L ), we have 3.7) fa)ga) ωaz)dµz) 1 faz)gaz)ωaz)dµz) + 1 faz)gaz 1 )ωaz)dµz) + Ia) = 1 faz)gaz)ωaz)dµz) + 1 faz 1 )gaz)ωaz)dµz) + Ia), where Ia) = 1 faz) gaz) gaz 1 ) ωaz)dµz) + 1 faz) faz 1 ) gaz)ωaz)dµz) + 1 faz) faz 1 ) gaz) gaz 1 ) ωaz)dµz). 4

8 8 M. A. LATIF, S. S. DRAOMIR 1,, AND E. MOMONIAT Furthermore, I a) satisfies the following inequality 3.8) 0 I a) 1 faz 1 ) gaz) gaz 1 ) )) dµz) ω az)dµz) gaz 1 ) faz) faz 1 ) )) dµz) ω az)dµz) faz 1 ) faz)) gaz) gaz 1 ) )) dµz) 1 f az)dµz) g az)ω az)dµz) gaz)gaz 1 )ω az)dµz) + 1 g az)dµz) min f az)ω az)dµz) faz)faz 1 )ω az)dµz) + 1 f az)dµz) faz)f 1 az)dµz) g az)ω az)dµz) gaz)gaz 1 )ω az)dµz), 1 g az)dµz) g az)gaz 1 )dµz) ω az)dµz), f az)ω az)dµz) + 1 f az)dµz) f az)faz 1 )dµz) g az)ω az)dµz) + 1 g az)dµz) g az)gaz 1 )dµz) f az)ω az)dµz) f az)faz 1 )ω az)dµz), 3 faz)gaz)ωaz)dµz) + 3 faz)gaz 1 )ωaz)dµz). Proof. Since is symmetric set relative to a, thus for any z and by the quasi-midconvexity of f and g, we have and fa) = max{faz), faz 1 )} = faz) + faz 1 ) + faz) faz 1 ) ga) = max{gaz), gaz 1 )} = gaz) + gaz 1 ) + gaz) gaz 1 ). Now by the non-negativity of f and g, we get 3.9) fa)ga) 1 [ faz)gaz) + faz)gaz 1 ) + faz) gaz) gaz 1 ) 4 + faz 1 )gaz) + faz 1 )gaz 1 ) + faz 1 ) gaz) gaz 1 ) + faz) faz 1 ) gaz) + faz) faz 1 ) gaz 1 ) + faz) faz 1 ) gaz) gaz 1 ). Since ω is non-negative and symmetric relative to a, we have from 3.9) that fa)ga) ωaz)dµz) 1 faz)gaz)ωaz)dµz) + 1 faz)gaz 1 )ωaz)dµz) + Ia) = 1 faz)gaz)ωaz)dµz) + 1 faz 1 )gaz)ωaz)dµz) + Ia).

9 HADAMARD S INEQUALITIES IN TOPOLOICAL ROUPS 9 Hence 3.7) is proved, where 3.10) Ia) = 1 faz) gaz) gaz 1 ) ωaz)dµz) + 1 faz) faz 1 ) gaz)ωaz)dµz) + 1 faz) faz 1 ) gaz) gaz 1 ) ωaz)dµz). 4 Now by Cauchy-Schwartz inequality, we observe from 3.10) that 3.11) 0 Ia) faz) gaz) gaz 1 ) )) dµz) ω az)dµz) gaz) faz) faz 1 ) )) dµz) ω az)dµz) faz 1 ) faz)) gaz) gaz 1 ) )) dµz) ω az)dµz). Again by Cauchy-Schwartz inequality, we have from 3.10) the following inequality 0 Ia) f az)dµz) faz) faz 1 ) ) ω az)dµz) gaz) gaz 1 ) ) ω az)dµz) g az)dµz) faz) faz 1 ) ) dµz) gaz) gaz 1 ) ) ω az)dµz) which is equivalent to 3.1) 0 Ia) 1 f az)dµz) g az)ω az)dµz) 1 + g az)dµz) gaz)gaz 1 )ω az)dµz) f az)ω az)dµz) faz)faz 1 )ω az)dµz) + 1 f az)dµz) faz)f 1 az)dµz) g az)ω az)dµz) gaz)gaz 1 )ω az)dµz).

10 10 M. A. LATIF, S. S. DRAOMIR 1,, AND E. MOMONIAT Using Cauchy-Schwartz inequality again, we have from 3.10) the following inequality 3.13) 0 Ia) 1 f az)ω az)dµz) g az)ω az)dµz) g az)dµz) f az)ω az)dµz) g az)gaz 1 )dµz) f az)dµz) g az)dµz) f az)faz 1 )dµz) g az)gaz 1 )dµz) f az)faz 1 )ω az)dµz). Lastly, by using the properties of absolute value, we have 3.14) 0 Ia) 3 faz)gaz)ωaz)dµz) + 3 faz)gaz 1 )ωaz)dµz). The inequality 3.8) follows from 3.11)-3.14). theorem. This completes the proof of the Corollary 1. Suppose the assumptions of Theorem 6 are satisfied and if g 1 on in Theorem 6, we have 3.15) fa) ωaz)dµz) faz)ωaz)dµz) + Ia), where Ia) = 1 faz) faz 1 ) ωaz)dµz). Furthermore, I a) satisfies the following inequality 3.16) 0 I a) 1 µ) f az)ω az)dµz) faz)faz 1 )ω az)dµz) ) 1, min 1 f az)dµz) f az)faz 1 )dµz) ω az)dµz) ) 1, faz)ωaz)dµz). Definition 5. [1 The function f : R is said to be a P-function in, if fa) faz) + faz 1 ) for all a and z such that az, az 1. Theorem 7. Let be a locally compact group and an open symmetric set relative to a with 0 < µ ) <. If f, g are measurable non-negative real valued P -functions on such that fg L 1 ). If ω : R is non-negative

11 HADAMARD S INEQUALITIES IN TOPOLOICAL ROUPS 11 symmetric to a, ω L 1 ) and fgω L 1 ), we have 3.17) f a) g a) ω az) dµ z) faz)gaz)ω az) dµ z) + faz)gaz 1 )ω az) dµ z) = faz)gaz)ω az) dµ z) + faz 1 )gaz)ω az) dµ z). Proof. Since f and g are P -functions and ω is non-negative and symmetric to a, we have f a) g a) ω az) faz) + faz 1 ) ) gaz) + gaz 1 ) ) ω az) Integrating this inequality on, we get = faz)gaz)ω az) + faz 1 )gaz 1 )ω az) + faz)gaz 1 )ω az) + faz 1 )gaz)ω az). f a) g a) ω az) dµ z) faz)gaz)ω az) dµ z) + faz 1 )gaz 1 )ω az) dµ z) + faz)gaz 1 )ω az) dµ z) + faz 1 )gaz)ω az) dµ z) = faz)gaz)ω az) dµ z) + faz)gaz 1 )ω az) dµ z) = faz)gaz)ω az) dµ z) + faz 1 )gaz)ω az) dµ z). Hence the proof of the theorem is completed. Corollary. Suppose that the assumptions of Theorem 7 are satisfied and if g 1 on, we have 3.18) f a) ω az) dµ z) 4 faz)ω az) dµ z). Corollary 3. If we take a = e and ω 1 on in Corollary, we have 3.19) f e) 4 fz)dµ z). µ ) Some of the applications of our results are given in the following remarks. Remark 4. Set = R. Since R is an abelian additive group, thus, for all a, z R, a z and a + z are points for which a is the midpoint. Now, if a z = y and a + z = x, then a = x+y. If we take = [ b, b, we get a = 0 and y = x. Hence

12 1 M. A. LATIF, S. S. DRAOMIR 1,, AND E. MOMONIAT from Theorem 5, we have b 3.0) f0)g0) ωx)dx b [ 1 b b fx)gx)ωx)dx + fx)g x)ωx)dx b b [ = 1 b b fx)gx)ωx)dx + f x)gx)ωx)dx. b b If ω x) 1 for all x [ b, b in 3.0), we obtain [ 3.1) f0)g0) 1 b b fx)gx)dx + fx)g x)dx 4b b b [ = 1 b fx)gx)dx + 4b b b b f x)gx)dx Remark 5. If in the Theorem 5, = R n with an additive operation and = is an open bounded symmetric and convex subset of R n, then the result of Theorem 4 holds. References [1 A. Chademan and F. Mirzapour, Midconvex functions in locally compact groups, Proc. Amer. Math. Soc., 17, ), [ K. Chandrasekharan, A course on topological groups, Volume 9 of Texts and readings in mathematics, Hindustan Book Agency, New Delhi, 1996, 117 pp. [3 S. S. Dragomir and C. E. M. Pearce, Quasi-convex functions and Hadamard s inequalities, Bull. Austral. Math. Soc., 57, 1998), [4 S. S. Dragomir and C. E. M. Pearce, Select topics on Hermite-Hadamard inequalities and applications, Melbourne and Adelaide, 000. [5 S. S. Dragomir, A mapping in connection to Hadamard s inequality, An Ostro. Akad. Wiss. Math.-Natur Wien), ), [6 S. S. Dragomir, Two mappings in connection to Hadamard s inequality, J. Math. Anal. Appl., ), [7 S. S. Dragomir, On Hadamard s inequality for convex functions, Math. Balkanica, 6 199), 15-. [8 J. Hadamard, Étude sur les propriétés des fonctions entières et en particulier d une fonction considerée par Riemann, J. Math Pures Appl., ), [9 Ch. Hermite, Sur deux limites d une intégrale définie, Mathesis ), 8. [10 P. J. Higgins, An Introduction to Topological roups, London Mathematical Society Lecture Note Series, No. 15. Cambridge University Press, London-New York, 1974, 109 pp. [11 F. Mirzapour and A. Morassaei, Quasi-convex functions in topological groups, Int. J. Appl. Math., 16, 4 004), [1 A. Morassaei, On Hadamard s inequality for convex mappings defined in topological groups and connected results, J. Math, Inequal., Vol. 4, No. 3, 010, [13 A. M. Rubinov, Abstract convexity and global optimization, Kluwer Academic Publishers, Dordrecht, 000. School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 050, Johannesburg, South Africa address: m amer latif@hotmail.com 1 School of Engineering and Science, Victoria University, P. O. Box 1448, Melbourne City, MC8001, Australia.

13 HADAMARD S INEQUALITIES IN TOPOLOICAL ROUPS 13 School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 050, Johannesburg, South Africa address: sever.dragomir@vu.edu.au School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 050, Johannesburg, South Africa address: Ebrahim.Momoniat@wits.ac.za