Some estimates on the Hermite-Hadamard inequality through quasi-convex functions

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1 Annls of University of Criov, Mth. Comp. Sci. Ser. Volume 3, 7, Pges 8 87 ISSN: Some estimtes on the Hermite-Hdmrd inequlity through qusi-convex functions Dniel Alexndru Ion Abstrct. In this pper we estblish some estimtes of the right hnd side of Hermite- Hdmrd type inequlity in which some qusi-convex functions re involved. We lso point out some pplictions of our results to give estimtes for the pproximtion error of the integrl f(x) dx in the trpezoidl formul. Next, we extend our initil results to functions of severl vribles. Mthemtics Subject Clssifiction. 6D1, 39B6, 6A51, 6B5. Key words nd phrses. Hermite-Hdmrd inequlity, qusi-convex function, Iyengr s type inequlity, trpezoidl formul. 1. Introduction nd min results The clssicl Hermite-Hdmrd inequlity provides estimtes of the men vlue of continuous convex function f : [, b] R, ( ) + b f 1 f() + f(b) f(t) dt. (1) b The history of this inequlity begins with the ppers of Ch. Hermite [7] nd J. Hdmrd [6] in the yers (see, e.g. C. P. Niculescu nd L.-E. Persson [1] nd the references therein for some historicl notes on the Hermite-Hdmrd inequlity). Inequlity (1) hs triggered huge mount of interest over the yers. We just remember the recent studies in [1,, 3,, 5, 8, 9] on tht topic. An interesting problem relted with the Hermite-Hdmrd inequlity is the precision in the Hermite-Hdmrd inequlity. In tht context, we remember tht the left Hermite-Hdmrd inequlity cn be estimted by the inequlity of Ostrowski, f(x) 1 b f(t) dt ( 1 x +b + b ) M (b ), () while the right Hermite-Hdmrd inequlity cn be estimted by the inequlity of Iyengr, f() + f(b) 1 M (b ) 1 f(t) dt b M (b ) (f(b) f()), (3) where by M we denoted the Lipschitz constnt, i.e. M = sup{ f(x) f(y) x y ; x y}. The gol of this pper is to estblish n inequlity of type (3) in the cse when f : [, b] R is differentible mpping on (, b) for which either f is qusiconvex function or f p/(p 1) is qusi-convex function for rel number p > 1. We Received: My, 7. 8

2 SOME ESTIMATES ON THE HERMITE-HADAMARD INEQUALITY 83 remember tht the notion of qusi-convex function generlizes the notion of convex function. More exctly, function f : [, b] R is sid qusi-convex on [, b] if f((1 λ) x + λ y) sup{f(x), f(y)}, x, y [, b], λ [, 1]. Clerly, ny convex function is qusi-convex function. Furthermore, there exist qusi-convex functions which re not convex. In tht context, we point out n elementry exmple. The function g : [, ] R, { 1, for t [, 1] g(t) = t, for t ( 1, ], is not convex function on [, ] but it is qusi-convex function on [, ]. On the other hnd, we will show tht our results cn be used in order to give estimtes for the pproximtion error of the integrl f(x) dx in the trpezoidl formul. Finlly, we will remrk tht our estimtes of type (3) cn be extended to functions of severl vribles. The min results of this pper re given by the following theorems. Theorem 1. Assume, b R with < b nd f : [, b] R is differentible function on (, b). If f is qusi-convex on [, b] then the following inequlity holds true f() + f(b) 1 f(x)dx b (b ) sup{ f (), f (b) }. () Theorem. Assume, b R with < b nd f : [, b] R is differentible function on (, b). Assume p R with p > 1. If f p/(p 1) is qusi-convex on [, b] then the following inequlity holds true f() + f(b) 1 b f(x)dx (b ) (p + 1) 1/p [sup{ f () p/(p 1), f (b) p/(p 1) }] (p 1)/p. Remrk. The results of Theorems 1 nd generlize the results of Theorems. nd.3 in [].. Proof of the min results In order to prove Theorems 1 nd we point out n useful uxiliry result. Lemm.1. Assume, b R with < b nd f : [, b] R is differentible function on (, b). If f L 1 (, b) then the following equlity holds true I = f() + f(b) 1 b Proof. Integrting by prts we hve (1 t)f (t+(1 t)b) dt= f(x) dx = b f(t + (1 t)b) b (5) (1 t)f (t + (1 t)b) dt. (6) (1 t) 1 + Next, by the chnge of vrible x = t + (1 t)b we obtin I = f() + f(b) b (b ) f(t + (1 t)b) b f(x)dx. (7) Thus, we deduce tht equlity (6) holds true. The proof of Lemm.1 is complete. dt.

3 8 DANIEL ALEXANDRU ION Proof of Theorem 1. By Lemm.1, the fct tht f is qusi-convex on (, b) nd some elementry integrtion clculus we hve f() + f(b) 1 f(x)dx = b 1 b (1 t)f (t + (1 t)b)dt b b (1 t) f (t + (1 t)b) dt (1 t) sup{ f (), f (b) }dt = (b ) sup{ f (), f (b) } = (b ) sup{ f (), f (b) }. The proof of Theorem 1 is complete. Proof of Theorem. First, we point out tht 1 t p dt = / (1 t) p dt + 1/ (t 1) p dt = / (1 t) dt (1 t) p dt = 1 p + 1. Next, using the bove informtion, Lemm.1 nd the Hölder s inequlity we obtin f() + f(b) 1 f(x) dx b (1 t) f (t + (1 t)b) dt b b ( ) 1/p ( 1 t p dt where 1/p + 1/q = 1 (or, q = p/(p 1)). The proof of Theorem is complete. = b ( b (p + 1) [sup{ f () q, f (b) q }] 1/q, 1/p 3. Applictions of Theorems 1 nd to the trpezoidl formul Assume is division of the intervl [, b] such tht : = x < x 1 <... < x n 1 < x n = b. For given function f : [, b] R we consider the trpezoidl formul T (f, ) = n 1 i= ) 1/q f (t + (1 t)b) q dt ) 1/p ( 1 t p dt sup{ f () q, f (b) q } dt f(x i ) + f(x i+1 ) (x i+1 x i ). (8) It is well known tht if f is twice differentible on (, b) nd M = sup x (,b) f (x) < then f(x)dx = T (f, ) + E(f, ), (9) ) 1/q

4 SOME ESTIMATES ON THE HERMITE-HADAMARD INEQUALITY 85 where E(f, ) is the pproximtion error of the integrl f(x)dx by the trpezoidl formul nd stisfies, E(f, ) M n 1 (x i+1 x i ) 3. (1) 1 i= Clerly, if the function f is not twice differentible or the second derivtive is not bounded on (, b), then (1) does not hold true. In tht context, the following results re importnt in order to obtin some estimtes of E(f, ). Corollry 1. Assume, b R with < b nd f : [, b] R is differentible function on (, b). If f is qusi-convex on [, b] then for ech division of the intervl [, b] we hve, E(f, ) sup{ f (), f n 1 (b) } (x i+1 x i ). (11) Proof. We pply Theorem 1 on the sub-intervls [x i, x i+1 ], i =, n 1 given by the division. Adding from i = to i = n 1 we deduce T (f, ) f(x)dx 1 n 1 (x i+1 x i ) sup{ f (x i ), f (x i+1 ) }. (1) i= On the other hnd, for ech x i [, b] there exists α i [, 1] such tht x i = α i + (1 α i )b. Since f is qusi-convex we deduce Thus, i= f (x i ) sup{ f (), f (b) }, i =, n. sup{ f (x i ), f (x i+1 ) } sup{ f (), f (b) }, i =, n 1. (13) Reltions (1) nd (13) imply tht reltion (11) holds true. completely proved. Thus, Corollry 1 is A similr method s tht used in the proof of Corollry 1 but bsed on Theorem shows tht the following result is vlid. Corollry. Assume, b R with < b nd f : [, b] R is differentible function on (, b). Assume p R with p > 1. If f p/(p 1) is qusi-convex on [, b] then for ech division of the intervl [, b] we hve, E(f, ) sup{ f (), f (b) } (p + 1) 1/p. An extension of Theorem 1 n 1 (x i+1 x i ). The im of this section is to extend the result of Theorem 1 to functions of severl vribles. In this section we will denote by U n open nd convex set of R N (N 1). We sy tht function f : U R is qusi-convex on U if f((1 λ) x + λ y) sup{f(x), f(y)}, x, y U, λ [, 1]. The following uxiliry result holds true. i=

5 86 DANIEL ALEXANDRU ION Lemm.1. Assume f : U R is function. Then f is qusi-convex on U if nd only if for every x, y U the function ϕ : [, 1] R, ϕ(t) = f((1 t)x + ty) is qusi-convex on [, 1]. Proof. " ". Let x, y U be fixed. Assume tht ϕ : [, 1] R, ϕ(t) = f((1 t)x + ty) is qusi-convex on [, 1]. Let λ [, 1] be rbitrry, but fixed. Clerly, λ = (1 λ) + λ 1 nd thus, f((1 λ)x + λy) = ϕ(λ) = ϕ((1 λ) + λ 1) sup{ϕ(), ϕ(1)} = sup{f(x), f(y)}. It follows tht f is qusi-convex on U. " ".Assume tht f is qusi-convex on U. Let x, y U be fixed nd define ϕ : [, 1] R, ϕ(t) = f((1 t)x + ty). We show tht ϕ is qusi-convex on [, 1]. Let t 1, t [, 1] nd λ [, 1]. Then ϕ((1 λ)t 1 + λt ) = f([1 (1 λ)t 1 λt ]x + [(1 λ)t 1 λt ]y]) = f((1 λ)((1 t 1 )x + t 1 y) + λ((1 t )x + t y)) sup{f((1 t 1 )x + t 1 y), f((1 t )x + t y)} = sup{ϕ(t 1 ), ϕ(t )}. We deduce tht ϕ is qusi-convex on [, 1]. The proof of Lemm.1 is complete. Using the bove lemm we will prove n extension of Theorem 1 to functions of severl vribles. Proposition 1. Assume f : U R + is qusi-convex function on U. Then for ny x, y U nd ny, b (, 1) the following inequlity holds true 1 f((1 s)x + sy)ds + 1 f((1 s)x + sy)ds 1 b ( s ) b f((1 θ)x + θy)dθ ds b sup{f((1 )x + y), f((1 b)x + by)}. (1) Proof. We fix x, y U nd, b (, 1) with < b. Since f is qusi-convex, by Lemm.1 it follows tht the function is qusi-convex on [, 1]. Define Φ : [, 1] R, ϕ : [, 1] R, ϕ(t) = f((1 t)x + ty), Φ(t) = t ϕ(s) ds = t f((1 s)x + sy) ds. Obviously, Φ (t) = ϕ(t) for ll t (, 1). Since f(u) R + it results tht ϕ on [, 1] nd thus, Φ on [,1]. Applying Theorem 1 to the function Φ we obtin Φ() + Φ(b) 1 b nd we deduce tht reltion (1) holds true. Φ(s)ds b sup{φ (), Φ (b)},

6 SOME ESTIMATES ON THE HERMITE-HADAMARD INEQUALITY 87 Proposition 1 is completely proved. Remrk. We point out tht similr result s those of Proposition 1 cn be stted by using Theorem. References [1] R. P. Agrwl nd S. S. Drgomir, An ppliction of Hyshi s inequlity for differentible functions, Computers Mth. Applic. 3 (6) (1996), [] S. S. Drgomir nd R. P. Agrwl, Two Inequlities for Differentible Mppings nd Applictions to Specil Mens of Rel Numbers nd to Trpezoidl Formul, Appl. Mth. Lett. 11 (5) (1998), [3] S. S. Drgomir nd S. Wng, Applictions of Ostrowski s inequlity to the estimtion of error bounds for some specil mens nd for some numericl qudrture rule, Appl. Mth. Lett. 11 (1) (1998), [] S. S. Drgomir nd S. Wng, An inequlity of Ostrowski-Grüss type nd its pplictions to the estimtion of error bounds for some specil mens nd for some numericl qudrture rule, Computers Mth. Applic. 33 (11) (1997), 15-. [5] A. Flore nd C. P. Niculescu, A Hermite-Hdmrd inequlity for convex-concve symmetric functions, Bull. Soc. Sci. Mth. Roum., 5 (98) (7), No., [6] J. Hdmrd, Étude sur les propriétés des fonctions entières et en prticulier d une fonction considerée pr Riemnn, J. Mth. Pures et Appl. 58 (1893), [7] Ch. Hermite, Sur deux limites d une intégrle définie, Mthesis 3 (1883), 8. [8] M. Mihăilescu nd C. P. Niculescu, An extension of the Hermite-Hdmrd inequlity through subhrmonic functions, Glsgow Mthemticl Journl 9 (7), 1-6. [9] C. P. Niculescu nd L.-E. Persson, Old nd New on the Hermite-Hdmrd Inequlity, Rel Anl. Exchnge, 9 (3/), [1] C. P. Niculescu nd L.-E. Persson, Convex Functions nd their Applictions. A Contemporry Approch, CMS Books in Mthemtics vol. 3, Springer-Verlg, New York, 6. (Dniel Alexndru Ion) Deprtment of Mthemtics, University of Criov, 13, Al.I. Cuz st., 585, Criov, Romni E-mil ddress: dn

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