Another converse of Jensen s inequality

Transcription

1 Another converse of Jensen s nequalty Slavko Smc Abstract. We gve the best possble global bounds for a form of dscrete Jensen s nequalty. By some examples ts frutfulness s shown. 1. Introducton Throughout ths paper x = {x } represents a fnte sequence of real numbers belongng to a fxed closed nterval I = [a, b], a < b and p = {p }, p = 1 s a postve weght sequence assocated wth x. If f s a convex functon on I, then the well known Jensen s nequalty [1],[3]) asserts that 1.1) 0 p fx ) f p x ). There are many mportant nequaltes whch are partcular cases of Jensen s nequalty such as the weghted A G H nequalty, Cauchy s nequalty, the Ky Fan and Holder nequaltes etc. One can see that the lower bound zero s of global nature snce t does not depend on p, x but only on f and the nterval I whereupon f s convex. We gve n [1] an upper global bound that s, dependng on f and I only) whch happens to be better than already exstng ones. Namely, we proved that 1.) 0 ) p fx ) f p x ) T f a, b), wth T f a, b) := max[pfa) + 1 p)fb) fpa + 1 p)b)]. p Note that, for a strctly) postve convex functon f, Jensen s nequalty can be stated n the form 1.3) 1 f p x ). 000 Mathematcs Subject Classfcaton. Prmary 6B5, 6D0. Key words and phrases. Jensen s dscrete nequalty, Stolarsky means, best possble global bounds. 1

2 SIMIC It s not dffcult to prove that 1 s the best possble global lower bound for Jensen s nequalty wrtten n the above form. Our am here s to fnd the best possble global upper bound for ths nequalty. We shall show wth examples that n ths way one may obtan at once converses of some mportant nequaltes.. Results Our man result s contaned n the followng Theorem.1. Let f be a strctly) postve, twce contnuously dfferentable functon on I := [a, b], x I and 0 p, q 1, p + q = 1. We have. f f s strctly) convex functon on I, then.1) 1 f + qfb) max[pfa) ] := S f a, b); p x ) p fpa + qb). f f s strctly) concave functon on I, then.) 1 f p x ) max [ fpa + qb) p pfa) + qfb) ] := S f a, b). Both estmatons are ndependent of p. The next asserton shows that S f a, b)s f a, b)) exsts and s unque. Theorem.. There s unque p 0 0, 1) such that.3) S f a, b) = p 0fa) + 1 p 0 )fb) fp 0 a + 1 p 0 )b) Of partcular mportance s the followng Theorem.3. The expresson S f a, b) represents the best possble global upper bound for Jensen s nequalty wrtten n the form 1.3). 3. Proofs We shall gve proofs of the above assertons related to the frst part of Theorem.1. Proofs concernng concave functons go along the same lnes. Proof of Theorem.1. We apply the method already shown n [1]. Namely, snce a x b, there s a sequence t [0, 1] such that x = t a + 1 t )b. Hence, f p x ) = p ft a + 1 t )b) f p t a + 1 t )b)) fa) p t + fb)1 p t ) fa p t + b1. p t ) Denotng p t := p, 1 p t := q; p, q [0, 1], we get f pfa) + qfb) + qfb) = max[pfa) ] := S f a, b) p x ) fpa + qb) p fpa + qb)

3 ANOTHER CONVERSE 3 and Proof of Theorem.. Denote pfa) + qfb) F p) :=. fpa + qb) We get F p) = gp)/f pa + qb) wth Also, gp) := fa) fb))fpa + qb) pfa) + qfb))f pa + qb)a b). g p) = a b) pfa) + qfb))f pa + qb), g0) = fb)fa) fb) f b)a b)); g1) = fa)fb) fa) f a)b a)). Snce f s strctly convex on I and pa+qb I, we conclude that gp) s monotone decreasng on [0, 1] wth g0) > 0, g1) < 0. Therefore there exsts unque p 0 0, 1) such that gp 0 ) = F p 0 ) = 0. Also F p 0 ) = g p 0 )/f p 0 a + q 0 b) < 0 and the proof s done. Proof of Theorem.3. Let R f a, b) be an arbtrary global upper bound. By defnton, the nequalty f p x ) R f a, b) holds for arbtrary p and x [a, b]. In partcular, for #x =, x 1 = a, x = b, p 1 = p 0 we obtan that S f a, b) R f a, b) as requred. 4. Applcatons In the sequel we shall gve some examples to demonstrate the frutfulness of the assertons from Theorem.1. Snce all bounds wll be gven as a combnaton of means from the Stolarsky class, here s ts defnton. Stolarsky or extended) two-parametrc mean values are defned for postve values of x, y as rx s y s ) ) 1/s r), 4.1) E r,s x, y) := rsr s)x y) 0. sx r y r ) E means can be contnuously extended on the doman by the followng rx s y s ) {r, s; x, y) r, s R; x, y R + } 1/s r), sx r y )) rsr s) 0 r exp 1 s + xs log x y s log y x s y ), r = s 0 s E r,s x, y) = x s y 1/s, s slog x log y)) s 0, r = 0 xy, r = s = 0, x, x = y > 0, and n ths form are ntroduced by Keneth Stolarsky n [].

4 4 SIMIC Most of the classcal two varable means are specal cases of the class E. For example, E 1, = x+y s the arthmetc mean Ax, y), E 0,0 = xy s the geometrc mean Gx, y), E 0,1 = x y log x log y s the logarthmc mean Lx, y), E 1,1 = x x /y y ) 1 x y /e s the dentrc mean Ix, y), etc. More generally, the r-th power ) 1/r mean x r +y r s equal to Er,r. Example 1 Takng fx) = 1/x, after an easy calculaton t follows that S 1/x a, b) = Aa, b)/ga, b)). Therefore we obtan at once Proposton 1 If 0 < a x b, then the nequalty 4.) 1 p x ) p x ) a + b), 4ab holds for an arbtrary weght sequence p. Ths s the extended form of Schwetzer nequalty. Example For fx) = x we get that the maxmum of F p) s attaned at b a+b. the pont p 0 = Hence Proposton If 0 < a x b, then the followng means nequalty p x Aa, b) 4.3) 1 p x Ga, b) holds for an arbtrary weght sequence p. Specalzng the above nequalty, that s, puttng p = u / u, x = v /u and notng that 0 < u u U, 0 < v v V mply a = v/u x V/u = b, we obtan a converse of the well-known Cauchy s nequalty. Proposton 3 If 0 < u u U, 0 < v v V, then u 4.4) 1 v UV/uv + uv/uv ). u v ) In ths form the Cauchy s nequalty was stated n [3, p.80]. Note that the same result can be obtaned from Schwetzer s nequalty 4.) takng p = u v / u v, x = u /v. Example 3 Let fx) = x α, 0 < α < 1. Snce n ths case f s a concave functon, applyng the second part of Theorem.1, we get Proposton 4 If 0 < a x b, then 4.5) 1 p x ) α p x α Eα,1 a, b)e 1 α,1 a, b) G a, b) ) α1 α), ndependently of p. In the lmtng cases we obtan two mportant converses. Namely, wrtng 4.5) as p x 4.6) 1 p x α Eα,1 a, b)e 1 α,1 a, b) ) 1 α, )1/α G a, b) and, lettng α 0 +, the converse of generalzed A G nequalty arses.

5 ANOTHER CONVERSE 5 Proposton 5 If 0 < a x b, then p x La, b)ia, b) 4.7) 1 x p G. a, b) Note that the rght-hand sde of 4.7) s exactly the Specht constant cf [1]). Analogously, wrtng 4.5) n the form p x ) α ) 1 1 α Eα,1 a, b)e 4.8) 1 1 α,1 a, b) ) α, p x α G a, b) and takng the lmt α 1, we get Proposton 6 If 0 < a x b, then 4.9) 0 p x log x p x log p x ) p x log La, b)ia, b) G. a, b) Fnally, puttng n 4.5) p = v / v, x = u /v, α = 1/p, 1 α = 1/q, we obtan the converse of dscrete Hölder s nequalty. Proposton 7 If p, q > 1, 1 p + 1 u q = 1; 0 < a v b, then 4.10) 1 u ) 1/p v ) 1/q E1/p,1 a, b)e 1/q,1 a, b) ) 1 pq 1/p u v 1/q G a, b) It s nterestng to compare 4.10) wth the converse of Hölder s nequalty for ntegral forms cf [4]). References [1] S. Smc, On an upper bound for Jensen s nequalty, J. Inequal. Pure Appl. Math. 10/ Artcle ), 5pp. [] K. B. Stolarsky, Generalzatons of the logarthmc mean, Math. Mag. 48/ 1975), pp [3] 3) G. Polya, G. Szego, Aufgaben und Lehrsatze aus der Analyss, Sprnger-Verlag 1964). [4] S. Satoh, V. K. Tuan, M. Yamamoto, Reverse weghted L p norm nequaltes n convolutons, J. Inequal. Pure Appl. Math. 1/1, Artcle 7 000), 7pp. Mathematcal Insttute SANU, Kneza Mhala 36, Belgrade, Serba E-mal address: ssmc@turng.m.sanu.ac.rs