Hcettepe Journl o Mthemtics nd Sttistics Volume 43 6 4 935 94 Hermite-Hdmrd type ineulities or hrmoniclly convex unctions İmdt İşcn Abstrct The uthor introduces the concept o hrmoniclly convex unctions nd estlishes some Hermite-Hdmrd type ineulities o these clsses o unctions AMS Clssiiction: Primry 6D5; Secondry 6A5 Keywords: Hrmoniclly convex unction Hermite-Hdmrd type ineulity Received 7 : 7 : 3 : Accepted 7 : : 3 Doi : 567/HJMS443759 Introduction Let : I R R be convex unction deined on the intervl I o rel numbers nd b I with < b The ollowing ineulity + b b b xdx + b holds This double ineulity is known in the literture s Hermite-Hdmrd integrl ineulity or convex unctions Note tht some o the clssicl ineulities or mens cn be derived rom or pproprite prticulr selections o the mpping Both ineulities hold in the reversed direction i is concve For some results which generlize improve nd extend the ineulities we reer the reder to the recent ppers see [ 3 4 6 5 7] The min purpose o this pper is to introduce the concept o hrmoniclly convex unctions nd estlish some results connected with the right-hnd side o new ineulities similr to the ineulity or these clsses o unctions Some pplictions to specil mens o positive rel numbers re lso given Deprtment o Mthemtics Fculty o Arts nd Sciences Giresun University 8 Giresun Turkey Emil: imdti@yhoocom imdtiscn@giresunedutr
Min Results Deinition Let I R\ {} be rel intervl A unction : I R is sid to be hrmoniclly convex i xy ty + tx tx + ty or ll x y I nd t [ ] I the ineulity in is reversed then is sid to be hrmoniclly concve Exmple Let : R x x nd g : R gx x then is hrmoniclly convex unction nd g is hrmoniclly concve unction The ollowing proposition is obvious rom this exmple: 3 Proposition Let I R\ {} be rel intervl nd : I R is unction then ; i I nd is convex nd nondecresing unction then is hrmoniclly convex i I nd is hrmoniclly convex nd nonincresing unction then is convex i I nd is hrmoniclly convex nd nondecresing unction then is convex i I nd is convex nd nonincresing unction then is hrmoniclly convex The ollowing result o the Hermite-Hdmrd type holds 4 Theorem Let : I R\ {} R be hrmoniclly convex unction nd b I with < b I L[ b] then the ollowing ineulities hold + b b b x x The ove ineulities re shrp dx + b Proo Since : I R is hrmoniclly convex unction we hve or ll x y I with t in the ineulity xy y + x x + y Choosing x y t+tb + b tb+t tb+t + we get t+tb Further integrting or t [ ] we hve 3 + b dt + tb + t Since ech o the integrls is eul to ineulity rom 3 b b dt t + tb x dx we obtin the let-hnd side o the x
The proo o the second ineulity ollows by using with x nd y b nd integrting with respect to t over [ ] Now consider the unction : R x thus xy tx + ty ty + tx or ll x y nd t [ ] Thereore is hrmoniclly convex on We lso hve nd + b + b b b x dx x which shows us the ineulities re shrp For inding some new ineulities o Hermite-Hdmrd type or unctions whose derivtives re hrmoniclly convex we need simple lemm below 5 Lemm Let : I R\ {} R be dierentile unction on I nd b I with < b I L[ b] then 4 Proo Let + b b b b x x dx t tb + t dt tb + t I b By integrting by prt we hve I Setting x I t tb + t t tb + t tb+t dx + b tb + t b dt x b tb+t b b x x dx which gives the desired representtion 4 dt tb + t dt we obtin dt
6 Theorem Let : I R be dierentile unction on I b I with < b nd L[ b] I is hrmoniclly convex on [ b] or then 5 where + b b λ λ b ln λ λ 3 b x b x dx [ λ + λ 3 b ] + b b b + 3 + b b 3 ln b 3b + λ λ 4 + b 4 + b b 3 ln 4 Proo From Lemm 5 nd using the Hölder ineulity we hve + b b b b b x x dx t tb + t dt tb + t t tb + t dt t tb + t dt tb + t Hence by hrmoniclly convexity o on [ b] we hve + b b b b x x dx t tb + t dt t [ t + t b ] tb + t dt b λ [ λ + λ 3 b ]
It is esily check tht t tb + t dt b ln + b 4 t t tb + t dt b 3b + b 3 ln + b 4 t t tb + t dt b b + 3 + b b 3 ln + b 4 7 Theorem Let : I R be dierentile unction on I b I with < b nd L[ b] I is hrmoniclly convex on [ b] or > then p + 6 + b b p + b b x x dx p µ + µ b where µ µ [ + b [b ] ] b [ b [b + b] ] b
Proo From Lemm 5 Hölder s ineulity nd the hrmoniclly convexity o on [ b]we hve + b b x b x dx b p t p dt where n esy clcultion gives 7 nd 8 tb + t dt tb + t b p p + t + t b tb + t dt t dt tb + t [ + b [b ] ] b t dt tb + t [ b [b + b] ] b Substituting eutions 7 nd 8 into the ove ineulity results in the ineulity 6 which completes the proo 3 Some pplictions or specil mens Let us recll the ollowing specil mens o two nonnegtive number b with b > : The rithmetic men A A b : + b The geometric men G G b : 3 The hrmonic men H H b : + b
4 The Logrithmic men L L b : b ln b ln 5 The p-logrithmic men b p+ p+ p L p L p b : p R\ { } p + b 6 the Identric men I I b e b b b These mens re oten used in numericl pproximtion nd in other res However the ollowing simple reltionships re known in the literture: H G L I A It is lso known tht L p is monotoniclly incresing over p R denoting L I nd L L 3 Proposition Let < < b Then we hve the ollowing ineulity H G L A Proo The ssertion ollows rom the ineulity in Theorem 4 or : R x x 3 Proposition Let < < b Then we hve the ollowing ineulity H G A b Proo The ssertion ollows rom the ineulity in Theorem 4 or : R x x 33 Proposition Let < < b nd p \ {} Then we hve the ollowing ineulity H p+ G L p p A p+ b p+ Proo The ssertion ollows rom the ineulity in Theorem 4 or : R x x p+ p \ {} 34 Proposition Let < < b Then we hve the ollowing ineulity H ln H G ln I A ln b ln b Proo The ssertion ollows rom the ineulity in Theorem 4 or : R x x ln x
Reerences [] Drgomir SS nd Perce CEM Selected Topics on Hermite-Hdmrd Ineulities nd Applictions RGMIA Monogrphs Victori University [] İşcn İ A new generliztion o some integrl ineulities or α m-convex unctions Mthemticl Sciences 7-8 3 doi:86/5-7456-7- [3] Kvurmcı H Özdemir ME nd Avcı M New Ostrowski type ineulities or m-convex unctions nd pplictions Hcettepe Journl o Mthemtics nd Sttistics 4 35 45 [4] Prk J New integrl ineulities or products o similr s-convex unctions in the irst sense Interntionl Journl o Pure nd Applied Mthemtics 8 4 585-596 [5] Set E Ozdemir ME nd Drgomir SS On Hdmrd-type ineulities involving severl kinds o convexity Journl o Ineulities nd Applictions Article ID 86845 pges doi:55// 86845 [6] Sulimn WT Reinements to Hdmrd s ineulity or log-convex unctions Applied Mthemtics 899-93 [7] Zhng T-Y Ji A-P nd Qi F On integrl ineulities o Hermite Hdmrd type or s-gometriclly convex unctions Abstrct nd Applied Anlysis Article ID 56586 4 pges doi:55//56586