Two refinements of the classical Hermite-Hadamard integral inequality for log-convex functions and applications for special means are given

Dragomir, Sever S

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Refinements of the Hermite-Hadamard Integral Inequality for Log-Convex Functions

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REFINEMENTS OF THE HERMITE-HADAMARD INTEGRALINEQUALITY FOR LOG-CONVEX FUNCTIONSS.S. DRAGOMIRAbstract. Two refinements of the classical Hermite-Hadamard integral in-equality for log-convex functions and applications for special means are given.1. IntroductionLet I be an interval of real numbers.The function f : I → R is said to be convex on I if for all x, y ∈ I and t ∈ [0, 1],one has the inequality:f (tx+ (1− t) y) ≤ tf (x) + (1− t) f (y) .(1.1)A function f : I → (0,∞) is said to be log-convex or multiplicatively convexif log (f) is convex, or, equivalently, if for all x, y ∈ I and t ∈ [0, 1] one has theinequality:f (tx+ (1− t) y) ≤ [f (x)]t [f (y)]1−t .(1.2)We note that if f and g are convex functions and g is monotonic nondecreasing,then g ◦ f is convex. Moreover, since f = exp (log f), it follows that a log-convexfunction is convex, but the converse is not true [2, p. 7]. This fact is obvious from(1.2) as by the arithmetic-geometric mean inequality, we have[f (x)]t [f (y)]1−t ≤ tf (x) + (1− t) f (y)(1.3)for all x, y ∈ I and t ∈ [0, 1].The next inequality (see for example [2, p. 137]) is well known in the literatureas the Hermite-Hadamard inequalityf(a+ b2)≤ 1b− a∫ baf (x) dx ≤ f (a) + f (b)2,(1.4)where f : I → R is a convex function on the interval I and a, b ∈ I with a < b.For some recent results related to this classic result, see the papers [4] – [13] andthe books [1], [2] and [3] where further references are given.Date: June 20, 2000.1991 Mathematics Subject Classification. Primary 26D15; Secondary 26D99.Key words and phrases. Hermite-Hadamard Inequality, Log-Convex Functions.12 S.S. DRAGOMIRIn [13], S.S. Dragomir and B. Mond proved that the following inequalities ofHermite-Hadamard type hold for log-convex functions:f(a+ b2)≤ exp[1b− a∫ baln [f (x)] dx](1.5)≤ 1b− a∫ baG (f (x) , f (a+ b− x)) dx ≤ 1b− a∫ baf (x) dx≤ L (f (a) , f (b)) ≤ f (x) + f (b)2,where G (p, q) :=√pq is the geometric mean and L (p, q) := p−qln p−ln q (p 6= q) is thelogarithmic mean of the positive real numbers p, q (for p = q, we put L (p, p) = p).In this paper we prove another refinement of the Hermite-Hadamard Inequalityfor differentiable log-convex functions. Some applications for special means are alsogiven.2. The ResultsWe shall start with the following refinement of the Hermite-Hadamard inequalityfor log-convex functions.Theorem 1. Let f : I → (0,∞) be a differentiable log-convex function on theinterval of real numbers I˚ (the interior of I) and a, b ∈˚I with a < b. Then thefollowing inequalities hold:1b−a∫ ba f (x) dxf(a+b2)(2.1)≥ L(exp[f ′(a+b2)f(a+b2) (b− a2)], exp[−f′ (a+b2)f(a+b2) (b− a2)])≥ 1.Proof. Since f is differentiable and log-convex on I˚, we have thatlog f (x)− log f (y) ≥ ddt(log f) (y) (x− y)for all x, y ∈˚I, which gives thatlog[f (x)f (y)]≥ f′ (y)f (y)(x− y)for all x, y ∈˚I. That is,f (x) ≥ f (y) exp[f ′ (y)f (y)(x− y)]for all x, y ∈ I˚.(2.2)Now, if we choose y = a+b2 , we obtain:f (x)f(a+b2) ≥ exp[f ′ (a+b2 )f(a+b2) (x− a+ b2)], x ∈ [a, b] .(2.3)HERMITE-HADAMARD INTEGRAL INEQUALITY 3Integrating this inequality over x on [a, b] and using Jensen’s integral inequality, wededuce that:1b−a∫ ba f (x) dxf(a+b2) ≥ 1b− a∫ baexp[f ′(a+b2)f(a+b2) (x− a+ b2)]dx(2.4)≥ exp[1b− a∫ ba[f ′(a+b2)f(a+b2) (x− a+ b2)]dx]= 1.Now, as for α 6= 0 we have that1b− a∫ baexp (αx) dx =exp (αb)− exp (αa)α (b− a)= L [exp (αb) , exp (αa)] ,where L (·, ·) is the usual logarithmic mean, then1b− a∫ baexp[α(x− a+ b2)]dx=exp[α( b−a2)]− exp [−α ( b−a2 )]α[( b−a2)− (− ( b−a2 ))]= L(exp[α(b− a2)], exp[−α(b− a2)]).Using the above equality for α =f ′( a+b2 )f( a+b2 )the inequality (2.4) gives the desired result(2.1).The following corollary holds.Corollary 1. Let g : I → R be a differentiable convex function on I˚ and a, b ∈˚Iwith a < b. Then we have the inequality:1b−a∫ ba exp (g (x)) dxexp g(a+b2)(2.5)≥ L(exp[g′(a+ b2)(b− a2)], exp[−g′(a+ b2)(b− a2)])≥ 1.The following theorem also holds.Theorem 2. Let f : I → R be as in Theorem 1. Then we have the inequality:f(a)+f(b)21b−a∫ ba f (x) dx≥ 1 + log ∫ ba f (x) dx∫ ba f (x) exp[f ′(x)f(x)(a+b2 − x)]dx(2.6)≥ 1 + log[1b−a∫ ba f (x) dxf(a+b2) ] ≥ 1.4 S.S. DRAGOMIRProof. From the inequality (2.2) we havef(a+ b2)≥ f (y) exp[f ′ (y)f (y)(a+ b2− y)],for all y ∈ [a, b].Integrating over y and using Jensen’s integral inequality for exp (·) functions, wehave(b− a) f(a+ b2)≥∫ baf (y) exp[f ′ (y)f (y)(a+ b2− y)]dy≥∫ baf (y) dy · exp∫ ba f (y) [ f ′(y)f(y) (a+b2 − y)] dy∫ ba f (y) dy=∫ baf (y) dy · exp(∫ ba f′ (y)(a+b2 − y)dy∫ ba f (y) dy).A simple integration by parts gives∫ baf ′ (y)(a+ b2− y)dy =∫ baf (y) dy − f (a) + f (b)2(b− a) .Then we haveexp[1−f(a)+f(b)2 (b− a)∫ ba f (x) dx]≤∫ ba f (y) exp[f ′(y)f(y)(a+b2 − y)]dy∫ ba f (y) dy≤ (b− a) f(a+b2)∫ ba f (y) dy,which is equivalent to1−f(a)+f(b)2 (b− a)∫ ba f (x) dx≤ log∫ ba f (y) exp [ f ′(y)f(y) (a+b2 − y)] dy∫ ba f (y) dy≤ log[f(a+b2)1b−a∫ ba f (x) dx]from where we get the desired inequality.The following corollary is a natural consequence of the above theorem.Corollary 2. Let g : I → R be as in Corollary 1. Then we have the inequality:exp g(a)+exp g(b)21b−a∫ ba exp g (x) dx≥ 1 + log[ ∫ ba exp g (x) dx∫ ba exp[g (x)− (x− a+b2 ) g′ (x)] dx]≥ 1 + log[1b−a∫ ba exp g (x) dxexp g(a+b2) ] ≥ 1.HERMITE-HADAMARD INTEGRAL INEQUALITY 53. ApplicationsThe function f (x) = 1x , x ∈ (0,∞) is log-convex on (0,∞). Then we have1b− a∫ badxx= L−1 (a, b) ,f(a+ b2)= A−1 (a, b) ,f ′(a+b2)f(a+b2) = − 1A.Now, applying the inequality (2.1) for the function f (x) = 1x , we get the inequality:A (a, b)L (a, b)≥ L(exp(−b− a2A), exp(b− a2A))≥ 1,(3.1)which is a refinement of the well-known inequalityA (a, b) ≥ L (a, b) ,(3.2)where A (a, b) is the arithmetic mean and L (a, b) is the logarithmic mean of a, b,that is, A (a, b) = a+b2 , and L (a, b) =a−bln a−ln b .For f (x) = 1x , we also getf (a) + f (b)2= H−1 (a, b) ,where H (a, b) := 11a+1bis the harmonic mean of a, b. Now, using the inequality(2.6) we obtain another interesting inequality:L (a, b)H (a, b)≥ 1 + log[A (a, b)L (a, b)]≥ 1,(3.3)which is a refinement of the following well-known inequalityL (a, b) ≥ H (a, b) .(3.4)Similar inequalities may be stated for the log-convex functions f (x) = xx, x > 0or f (x) = ex + 1, x ∈ R, etc. We omit the details.References[1] S.S. DRAGOMIR and C.E.M. PEARCE, Selected Topics on Hermite-HadamardInequalities and Applications, RGMIA Monographs, Victoria University, 2000.http://rgmia.vu.edu.au/monographs.html[2] J.E. PECˇARIC´, F. PROSCHAN and Y.L. TONG, Convex Functions, Partial Orderings andStatistical Applications, Academic Press, 1991.[3] D.S. MITRINOVIC´, J.E. PECˇARIC´ and A.M. FINK, Classical and New Inequalities inAnalysis, Kluwer Academic Publishers, Dordrecht, 1993.[4] S.S. DRAGOMIR, J.E. PECˇARIC´ and J. SA´NDOR, A note on the Jensen-Hadamard in-equality, Anal. Num. Theor. Approx., 19 (1990), 29-34.[5] S.S. DRAGOMIR, Two functions in connection to Hadamard’s inequalities, J. Math. Anal.Appl., 167 (1992), 49-56.[6] S.S. DRAGOMIR, On Hadamard’s inequalities for convex functions, Mat. Balkanica, 6(1992), 215-222.[7] S.S. DRAGOMIR, Some integral inequalities for differentiable convex functions, Contribu-tions, Macedonian Acad. of Sci. and Arts, 13(1) (1992), 13-17.[8] S.S. DRAGOMIR, A refinements of Hadamard’s inequality for isotonic linear functional,Tamkang J. of Math. (Taiwan), 24 (1993), 101-106.6 S.S. DRAGOMIR[9] S.S. DRAGOMIR, D.S. MILOSEVIC´ and J. SANDOR, On some refinements of Hadamard’sinequalities and applications, Univ. Beograd, Publ. Elek. Fak., Ser. Math., 4 (1993), 21-24.[10] D. BARBU, S.S. DRAGOMIR and C. BUS¸E, A probabilistic argument for the convergenceof some sequences associated to Hadamard’s inequality, Studia Univ. “Babes-Bolyai”, Math.,38(1) (1993), 29-33.[11] S.S. DRAGOMIR and G. TOADER, Some inequalities for m-convex functions, Studia Univ.Babes¸-Bolyai Math, 38 (1993), 21-28.[12] S.S. DRAGOMIR, Some remarks on Hadamard’s inequalities for convex functions, ExtractaMath., 9(2) (1994), 88-94.[13] S.S. DRAGOMIR and B. MOND, Integral inequalities of Hadamard type for log-convexfunctions, Demonstratio Mathematica, 31 (2)(1998), 354-364.School of Communications and Informatics, Victoria University of Technology,Melbourne City MC, Victoria 8001, Australia.E-mail address: sever@matilda.vu.edu.auURL: http://rgmia.vu.edu.au/SSDragomirWeb.html

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Refinements of the Hermite-Hadamard Integral Inequality for Log-Convex Functions

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