HERMITE-HADAMARD TYPE INEQUALITIES OF CONVEX FUNCTIONS WITH RESPECT TO A PAIR OF QUASI-ARITHMETIC MEANS FLAVIA CORINA MITROI nd CĂTĂLIN IRINEL SPIRIDON In this pper we estblish some integrl inequlities of Hermite-Hdmrd type in the frmework of Borel probbility mesures. AMS 2010 Subject Clssifiction: 26A51 26D15. Key words: convexity qusi-rithmetic men Borel probbility mesure. The Hermite-Hdmrd inequlity sserts tht for every continuous convex function f defined on n intervl [ b] nd every Borel probbility mesure µ on [ b] we hve HH where f b µ fxdµx b b µ b f + b µ b fb b µ = xdµx is the brycenter of µ. See [3] for detils. The im of this pper is to prove n nlogue of Hermite-Hdmrd inequlity in the frmework of qusi-rithmetic mens. Let I be n intervl nd ϕ : I R continuous incresing function. The weighted qusi-rithmetic men ssocited to ϕ is defined by the formul M [ϕ] b; 1 λ λ = ϕ 1 1 λ ϕ + λϕb for b I nd λ [0 1]. The weighted rithmetic men A b; 1 λ λ = 1 λ + λb corresponds to ϕx = x nd the weighted geometric men G b; 1 λ λ = 1 λ b λ MATH. REPORTS 1464 3 2012 291 295
292 Flvi Corin Mitroi nd Cătălin Irinel Spiridon 2 corresponds to ϕx = log x. Given pir of continuous incresing functions ϕ : [ b] R nd ψ : [c d] R function f : [ b] [c d] is clled M [ϕ] M [ψ] -convex if f M [ϕ] x y; 1 λ λ M [ψ] fx fy; 1 λ λ for every x y [ b] nd λ [0 1]. The theory of M [ϕ] M [ψ] -convex functions cn be deduced from the theory of usul convex functions. Indeed f is M [ϕ] M [ψ] -convex function if nd only if ψ f ϕ 1 is convex. This fct llows us to trnslte results known for convex functions into their counterprts for M [ϕ] M [ψ] -convex functions. We will next consider the cse of Hermite-Hdmrd inequlity. Our pproch is bsed on the concept of push-forwrd mesure. Given Borel probbility mesure µ on n intervl [ b] the pushforwrd of µ through continuous mp ϕ : [ b] R is defined by ϕ#µ A = µ ϕ 1 A for every Borel subset A of [ϕ ϕb]. This mesure llows the following chnge of vrible formul f ϕx dµ x = The brycenter of ϕ#µ is so if we put nd b ϕ#µ = Mξ = we obtin the identity 1 ϕb ϕ ϕb ϕ xdµ ϕ 1 x = ξ = ϕ 1 b ϕ#µ fxdµ ϕ 1 x. ϕxdµx b ϕb ϕξ ϕ ϕξ ϕb ϕ ϕξ ϕ ϕb ϕ = b Mξ. b Lemm 1. The brycenter of ϕ#µ verifies the formul 2 b ϕ#µ = Mξ b ϕ + b Mξ ϕb. b
3 Hermite-Hdmrd type inequlities of convex functions 293 Proof. In fct due to the identity 1. b ϕ#µ = ϕb b ϕ#µ ϕb ϕ ϕ + b ϕ#µ ϕ ϕb ϕ ϕb = Mξ b ϕ + b Mξ ϕb. b Theorem 1 [The Hermite-Hdmrd inequlity for M [ϕ] M [ψ] -convex functions]. Let f : [ b] [c d] be continuous M [ϕ] M [ψ] -convex function nd µ be Borel probbility mesure on [ b]. Then RHH fξ ψ 1 ψ fx dµx LHH where ξ = ϕ 1 b ϕ#µ. M [ψ] f fb; ϕb ϕξ ϕξ ϕ ϕb ϕ ϕb ϕ Proof. We pply the inequlity HH to ψ f ϕ 1. As we hve seen ψ f ξ = ψ f ϕ 1 b ϕ#µ ϕb ϕ nd the conclusion follows. ψ f ϕ 1 x dµ ϕ 1 x = ϕb b ϕ#µ ϕb ϕ ψ f + b ϕ#µ ϕ ϕb ϕ ψ fb ψ fx dµx Remrk 1. Theorem 1 ws proved for A M [ψ] -convex functions in [1 Theorem 3.3] under more restrictive conditions. The prticulr cse of G A- convex functions ws proved in [5] while the cse of G G-convex functions ppered in [2] nd [4]. We will cll the function Φ support of f if ψ Φ ϕ 1 = Ψ where Ψ is support line of the convex function ψ f ϕ 1. Theorem 2. Let f : [ b] [c d] be continuous M [ϕ] M [ψ] -convex function ψ concve nd µ be Borel probbility mesure on [ b]. Then fxdµx f ϕ 1 b ϕ#µ } = sup {ψ 1 ψ Φx dµx. Φ is support of f
294 Flvi Corin Mitroi nd Cătălin Irinel Spiridon 4 Proof. The proof is similr to [2 Theorem 3]. Detils re left to the reder. The Hermite-Hdmrd type inequlities proved in Theorem 1 re not just consequences of M [ϕ] M [ψ] -convexity but lso chrcterize it. The converse of Hermite-Hdmrd inequlity for M [ϕ] M [ψ] -convex functions reds s follows: Theorem 3. Let I J be two intervls nd f : I J continuous function. Assume tht ϕ : I R nd ψ : J R re continuous incresing functions. If for every compct subintervl [ b] of I nd for every tomless Borel probbility mesure µ on [ b] the function f stisfies either the inequlity RHH or LHH then f is M [ϕ] M [ψ] -convex. Proof. If RHH holds by Jensen s inequlity we conclude tht ψ f ϕ 1 is convex hence f is M [ϕ] M [ψ] -convex. It remins to consider tht LHH holds. We proceed by reductio d bsurdum. Assume tht f is not M [ϕ] M [ψ] -convex. Then there exists subintervl [x y] I nd number ε 0 1 such tht 3 f M [ϕ] x y; 1 ε ε > M [ψ] fx fy; 1 ε ε. Since f is continuous the inequlity 3 holds on n entire neighbourhood ε 1 ε 2 of ε. We choose ε 1 ε 2 the biggest neighbourhood with this property. Put = M [ϕ] x y; 1 ε 1 ε 1 nd b = M [ϕ] x y; 1 ε 2 ε 2 < b. The continuity of f ensures tht nd f = M [ψ] fx fy; 1 ε 1 ε 1 fb = M [ψ] fx fy; 1 ε 2 ε 2. Since we hve 1 t ε 1 + tε 2 ε 1 ε 2 for every t in 0 1 we infer from 3 tht f M [ϕ] b; 1 t t = f M [ϕ] M[ϕ] x y; 1 ε 1 ε 1 M [ϕ] x y; 1 ε 2 ε 2 ; 1 t t = f M [ϕ] x y; 1 1 t ε 1 tε 2 1 t ε 1 + tε 2 > M [ψ] fx fy; 1 1 t ε 1 tε 2 1 t ε 1 + tε 2 = M [ψ] M[ψ] fx fy; 1 ε 1 ε 1 M [ψ] fx fy; 1 ε 2 ε 2 ; 1 t t = M [ψ] f fb; 1 t t.
5 Hermite-Hdmrd type inequlities of convex functions 295 Thus it follows = > = ψ fx dµx ψ f M [ϕ] b; ψ M [ψ] f fb; ϕb ϕx ϕx ϕ dµx ϕ b ϕ ϕb ϕ ϕb ϕx ϕx ϕ ϕb ϕ ϕb ϕ ϕb ϕξ ϕξ ϕ ψ f + ϕb ϕ ϕb ϕ ψ fb. This is contrdiction completing the reductio d bsurdum. dµx Acknowledgements. The first uthor ws supported by CNCSIS Grnt 420/2008. The uthors re indebted to Professor Constntin P. Niculescu for his support nd useful discussions during the preprtion of this pper. REFERENCES [1] Ondrej Hutnik On Hdmrd type inequlities for generlized weighted qusi-rithmetic mens. J. Inequl. Pure Appl. Mth. 7 2006 3. [2] F.C. Mitroi nd C.I. Spiridon The Hermite-Hdmrd type inequlity for multiplictively convex functions. Mnuscript. [3] C.P. Niculescu nd L.-E. Persson Convex Functions nd their Applictions. A Contemporry Approch. CMS Books in Mthemtics vol. 23 Springer-Verlg New York 2006. [4] C.P. Niculescu The Hermite-Hdmrd inequlity for log-convex functions. Nonliner Anl. 75 2012 662 669. [5] X.-M. Zhng Y.-M. Chu nd X.-H. Zhng The Hermite-Hdmrd type inequlity of GA-convex functions nd its ppliction. J. Inequl. Appl. vol. 2010 Article ID 507560. Received 13 mrch 2011 University of Criov Deprtment of Mthemtics Street A.I. Cuz 13 200585 Criov Romni fcmitroi@yhoo.com ctlin gsep@yhoo.com