HERMITE-HADAMARD TYPE INEQUALITIES OF CONVEX FUNCTIONS WITH RESPECT TO A PAIR OF QUASI-ARITHMETIC MEANS

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1 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

2 292 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 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

4 294 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 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 [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 [4] C.P. Niculescu The Hermite-Hdmrd inequlity for log-convex functions. Nonliner Anl [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 Article ID Received 13 mrch 2011 University of Criov Deprtment of Mthemtics Street A.I. Cuz Criov Romni fcmitroi@yhoo.com ctlin gsep@yhoo.com