ON A CONVEXITY PROPERTY. 1. Introduction Most general class of convex functions is defined by the inequality
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1 Krgujevc Journl of Mthemtics Volume 40( (016, Pges ON A CONVEXITY PROPERTY SLAVKO SIMIĆ Abstrct. In this rticle we proved n interesting property of the clss of continuous convex functions. This leds to the form of pre-hermite-hdmrd inequlity which in turn dmits generliztion of the fmous Hermite-Hdmrd inequlity. Some further discussion is lso given. 1. Introduction Most generl clss of convex functions is defined by the inequlity ( φ(x + φ(y x + y φ. A function which stisfies this inequlity in certin closed intervl I is clled convex in tht intervl. Geometriclly it mens tht the midpoint of ny chord of the curve y = φ(x lies bove or on the curve. Denote now by Q the fmily of weights i.e., non-negtive rel numbers summing to 1. If φ is continuous, then the inequlity (1.1 pφ(x + qφ(y φ(px + qy holds for ny p, q Q. Moreover, the equlity sign tkes plce only if x = y or φ is liner (cf. [1]. The sme is vlid for so-clled Jensen functionl, defined s J φ (p, x := ( p i φ(x i φ pi x i, where p = {p i } n 1 Q, x = {x i } n 1 I, n. Key words nd phrses. Continuous convex functions, Pre-Hermite-Hdmrd inequlities, Generliztion of Hermite-Hdmrd inequlity. 010 Mthemtics Subject Clssifiction. 39B6(6D15. Received: Februry 7, 016. Accepted: July 1,
2 ON A CONVEXITY PROPERTY 167 Geometriclly, the inequlity (1.1 sserts tht ech chord of the curve y = φ(x lies bove or on the curve.. Results nd Proofs Min contribution of this pper is the following Proposition.1. Let f( be continuous convex function defined on closed intervl [, b] := I. Denote ( s + t F (s, t := f(s + f(t f. Prove tht (.1 mx F (s, t = F (, b. s,t I Proof. It suffices to prove tht the inequlity F (s, t F (, b holds for < s < t < b. In the sequel we need the ssertion stted in Lemm.1 (which is of independent interest. Lemm.1. Let f( be continuous convex function on some intervl I R. If x 1, x, x 3 I nd x 1 < x < x 3, then (i f(x f(x 1 f ( ( x +x 3 f x1 +x 3 ; (ii f(x 3 f(x f ( x 1 +x 3 ( f x1 +x. We shll prove the first prt of the lemm; proof of the second prt goes long the sme lines. Since x 1 < x < (x + x 3 / < x 3, there exist p, q; 0 p, q 1, p + q = 1 such tht x = px 1 + q( x +x 3. Hence, f(x 1 f(x ( x + x 3 + f = 1 [ ( f(x 1 f px 1 + q x ] ( + x 3 x + x 3 + f 1 [ ( ( ] ( x + x 3 x + x 3 f(x 1 pf(x 1 + qf + f = q f(x 1 + q ( f x + x 3 ( q f x 1 + q ( x + x 3 ( ( q = f x x + x (x px 1 ( x1 + x 3 = f.
3 168 S. SIMIĆ For the proof of second prt we cn tke x = p ( x 1 +x + qx3 nd proceed s bove. Now, pplying the prt (i with x 1 =, x = s, x 3 = b nd the prt (ii with x 1 = s, x = t, x 3 = b, we get ( ( f(s f( s + b + b (. f f ; ( ( f(b f(t s + b s + t (.3 f f, respectively. Subtrcting (. from (.3, the desired inequlity follows. Remrk.1. A chllenging tsk is to find geometric proof of the property (.1. We shll quote now couple of importnt consequences. The first one is used in number of rticles lthough we never sw proof of it. Corollry.1. Let f be defined s bove. If x, y [, b] nd x + y = + b, then f(x + f(y f( + f(b. Proof. Obvious, s simple ppliction of Proposition.1. Corollry.. Under the conditions of Proposition.1, the double inequlity ( + b (.4 f f(p + qb + f(pb + q f( + f(b holds for rbitrry weights p, q Q. Proof. Applying Proposition.1 with s = p + qb, t = pb + q; s, t I we get the right-hnd side of (.4. The left-hnd side inequlity is obvious since, by definition, [ ] ( f(p + qb + f(pb + q (p + qb + (pb + q + b f = f. Remrk.. The reltion (.4 represents kind of pre-hermite-hdmrd inequlities. Indeed, integrting both sides of (.4 over p [0, 1], we obtin the form of Hermite- Hdmrd inequlity (cf. [], ( + b f 1 b f(tdt f( + f(b. Moreover, the inequlity (.4 dmits generliztion of the Hermite-Hdmrd inequlity. Proposition.. Let g be n rbitrry non-negtive nd integrble function on I. Then, with f defined s bove, we get ( + b b (.5 f g(tdt (g(t+g(+b tf(tdt (f(+f(b g(tdt.
4 ON A CONVEXITY PROPERTY 169 Proof. Multiplying both sides of (.4 with g(p + qb nd integrting over p [0, 1], we obtin ( + b f g(tdt (f(t + f( + b tg(tdt (f( + f(b f(tdt b b b, nd, becuse the inequlity (.5 follows. (f(t + f( + b tg(tdt = (g(t + g( + b tf(tdt, We shll give in the sequel some illustrtions of this proposition. Corollry.3. For ny f tht is convex nd continuous on I := [, b], 0 < < b nd α R/{0}, we hve ( + b α b [ f t α 1 + ( + b t α 1] f(tdt f( + f(b. b α α Also, for α 0, we get Corollry.4. ( + b log(b/ f + b Similrly, Corollry.5. ( π f ( π f 4 π 0 π/ 0 f(t log(b/ dt [f( + f(b] t( + b t + b. f(t sin tdt f(0 + f(π; [sin t + cos t]f(tdt f(0 + f ( π. Estimtions of the convolution of symmetric kernel on symmetric intervl re lso of interest. Corollry.6. Let f nd g be defined s bove on symmetric intervl [, ], > 0. Then we hve tht f(0 g(tdt [g( t + g(t]f(tdt [f( + f(] g(tdt. Remrk.3. There remins the question of possible extensions of the reltion (.1. In this sense one cn try to prove, long the lines of the proof of (.1, tht where mx F (p, q; x, y = F (p, q;, b, p,q Q;x,y [,b] F (p, q; x, y := pf(x + qf(y f(px + qy.
5 170 S. SIMIĆ Anywy the result will be wrong, s simple exmples show (prt from the cse f(x = x. On the other hnd, it ws proved in [3] tht for p i Q nd x i [, b] there exist p, q Q such tht (.6 J f (p, x = ( p i f(x i f pi x i pf( + qf(b f(p + qb, for ny continuous function f which is convex on [, b]. Therefore, n importnt conclusion follows. Corollry.7. For rbitrry p i Q nd x i [, b], we hve tht ( pi f(x i f pi x i mx[pf( + qf(b f(p + qb] := T f (, b, p where T f (, b is n optiml upper globl bound, depending only on nd b (cf. [3]. An nswer to the bove remrk is given by the next Proposition.3. If f is continuous nd convex on [, b], then Proof. We shll prove just tht mx F (p, q; x, y F (, b. p,q Q;x,y [,b] F (p, q; x, y F (x, y, for ll p, q Q nd x, y [, b]. Indeed, ( x + y F (x, y F (p, q; x, y = qf(x + pf(y + f(px + qy f ( x + y f(qx + py + f(px + qy f ( ( (qx + py + (px + qy x + y f f = 0. The rest of the proof is n ppliction of Proposition.1. Putting there x =, y = b nd combining with (.6, we obtin nother globl bound for Jensen functionl. Corollry.8. We hve tht ( + b J f (p, x f( + f(b f := T f(, b. The bound T f (, b is not so precise s T f(, b but is much esier to clculte.
6 ON A CONVEXITY PROPERTY 171 References [1] G. H. Hrdy, J. E. Littlewood nd G. Poly, Inequlities, Cmbridge University Press, Cmbridge, [] C. P. Nikulesku nd L. E. Persson, Old nd new on the Hermite-Hdmrd inequlity, Rel Anl. Exchnge 9( (003/4, [3] S. Simić, Best possible globl bounds for Jensen functionl, Proc. Amer. Mth. Soc. 138(7 (010, Mthemticl Institute SANU Knez Mihil 36, Belgrde, Serbi E-mil ddress: ssimic@turing.mi.snu.c.rs
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