Tap chi so 32. thang 10-2018, trUdng Dai hpc Tay Nguyen ISSN 1 8 ^ 1 1 MO RONG CLA DINH LI TROI CHO HAIM LOI SUY RONG VA iTNG DUNG
Duong Quflc Huyi Ngay nhan bai: 01/9/2018; Ngay phan bifln thdng qua: 15/9/2018; Ngay duyet dang: 30/9/2018
Trong bai bao nay, chung tdi chi ra dinh Ii trdi ndi tiflng ciia Hardy-Littlewood-Pdlya vln con diing cho mdt Ic^D hara loi suy rdng cu thfl. U'ng dung ciia ket qua nay, ehiing tdi md rgng mgt sd bit ding ihiic ndi tiflng cho cac ham ldi suy rgng thudc ldp nay
Tir khoa Bd trgi. dinh li troi, hdm ldi suy rpng, bdt ddng thirc Karamata.
I. MO DAU
Bg trgi la mot cdng cu manh va quan trpng cua toan hpc, chiia dung nhiflu dac trung tinh tfl vdi cac irng dung khac nhau trong nhiflu ITnh vuc ciia toan hpc, li thuyet thdng tin, vat H, kinh tfl,.., Khai niem nay la mdt quan he tien thir tu giiia cac vecto trong W, dugc gidi thieu lan dau tifln bdi Hardy, Littlewood va Pdlya (xem Hardy, G. H. & ct al., 1934; Hardy, G. H & et al., 1929). Vdi mdi vecto x = (jC|,...,,r„)eE",takihieu x' =(x[,...,xl) la vecto nhan dirge tir ^ bang each sap xep lai cac thanh phan tpa dp ciia nd theo thii thu giam dan.
Khi do, vdi hai vecto x, y G M", ta ndi y trpi ban
•^, ki hieu-^"^-^, neu
k k
2 - ^ < * - X - ^ ' ' A = l>---,«,vdi ding thirc xay ra khi k = n. Quan he nay dugc dac trung bdi dinh li trgi noi tiflng ciia Hardy-Littlewood-Pdlya nhu sau.
Dinhli 1.1 (xem Day, P. W., 191 ^). Cdc khdng dinh sau ddy Id tirang duang ddi v&i x,y &W.
(1} x<y
(2) Y.M)^lLf^y.) vdi moi hdm ldi lien tuc f xac dinh trfln E .
{3} ^ thuge bao ldi eua tdp {z:z' =y').
(4) Ton tgi ma trdn ngdu nhiin kep A sao cho x = Ay_
Nhac lai ring mdt ma tran ^ = (a„) vudng cip
" dugc gpi la mdt ma tran ngiu nhifln kep nflu cac phan ni ciia nd khdng am va tdng eac phin tir trong mdi ddng va mdi cot dflu bing 1. Vi vay, dac
trung thii 4 trong Dinh li 1.1 chi ra ring mfli thanh phan tga do ciia vecto x la trung binh cd trpng ciia cae thanh phan tpa do eiia vecta V . Vfl mat hinh hgc, cd thfl ndi day la dac trung quan trong nhat ciia bd trdi.
Bdi vai trd va tim quan trgng cua bp trpi, dinh li nay da dugc xem xet md rdng tdi cac bii canh khac nhau ciia tinh ldi suy rdng dua trfln cac dac trung hinh hpc va giai tich ciia nd. Mdt trong cac kflt qua gan day ma ehiing tdi quan tam la cong trinh Clia Niculescu va Roventa (Niculescu, C.
P and Persson, L., 2006) nam 2017. Trong cong trinh dd, hg da sii dung d5c tnmg thii 4 trong Dinh Ii 1.1 de md rdng khai niera bd trdi den tinh loi suy rgng rit tdng quat. Tir dd, hp da thu dupe bSt dang thii:c cho trong dac tnmg 2 ciia Dinh li 1.1, That khdng may, do dac diflm cua ldp ham 16i suy rgng duoc xet, cdng trinh chi thu dugc diflu kien dii ma khdng phai dieu kifln can cho dac tnmg thii 2 Clia Dinh Ii 1.1.
Muc tiflu chinh ciia chiing tdi trong bai bao nay la md rdng dinh li trgi ndi tiflng cua Hardy- Littlewood-Pdlya (Dinh li 1.1) cho ldp ham s-15i (theo nghia thir hai) duoc gidi thiflu bdi Breckner (Breckner, W W., 1978). Mdt diflm quan trong trong ddng gdp nay la chiing tdi chi ra dac trung 2 la diflu kien cin va.dii cho bp trdi ddi vcri lop ham nay. Ndi riflng, ket qua nay ciing chinh ia mot md rdng cua bat dang thiic Karamata cho ldp ham s-ldi. Dua trfln dinh Ii radi nhan dupe, chiing toi chi ra mdt sd bit dang thiic ndi tieng khac doi voi ham ldi van cdn diing eho ldp ham s-ldi.
Vfl mpt sd kiflu ham ldi va eac van dfl lien quan, ban dpc cd thfl thara khao cac tai lieu (Anderson, G. D. & et al., 2007; Dragorair, S. S. and Pearce, C. E. M., 2000; Godunova, E. K and Levin, V. l, 1985; Niculescu, C. R and Persson, L., 2006).
2. NOl DUNG VA PHUONG PHAP NGHIEN ClTU
/ ThS, Khoa KHTN&CN Truong Dgi hoc Tdy Nguyen •
Tac gia hen he: Duang Qudc Huy: DT 0905616183; Email- [email protected].
Tap chi so 32, thang l(J-20i«, truong uai ngc Tay Nguyfln ISSN 1859-4611 Chung minh. Nflu x<y thi theo Dmh \\ I.l, 2.1. Npi dung nghien cuu
Ndi dung nghien cim cua bai bao thudc vfl ITnh _ A — i 's ' vuc giai tich ldi va bit ding thirc. Cu thfl hon, vin tdn tai raa tran ngau nhifln kep ^ - ( o , ) cap dfl nghifln ciiu ciia bai bao tap trung ehu yeu vao
li thuyet eac bd trgi, gan lifln vdi ham ldi suy rdng va bat dang thiic.
2.2, Phuang phdp nghien cuu
Ngoai viec sii dung phuong phap dac biflt hda va khai quat hda de cd dugc eai nhin tdng quan ve van dfl nghien ciiu, chung tdi sir dung cac dac trung hinh hoc va giai tich cua bd trdi. Bfln canh
dd, chiing tdi kflt hgp cac dac tnmg nay vdi cac tren, ta dugc dac trung ciia hinh hpc va giai tich ciia ldp hara
s-ldi dfl nhan dugc kflt qua mdi cho dinh H trdi.
3. KET QUA VA THAO LUAN 3.1. Dinh litrpi cho l&p hdm s-ldi
Trudc het, ta nhac lai khai niem ham s-l6i theo nghia thii hai dugc gidi thiflu bdi Breckner nam 1978.
nxn sao cho ^''L^^V^J vdi moi i = l,...,n.
Sir dung tinh s-ldi cua / vaBdde 3,1.3,tasuyra
/(^,)=/(Z«.>',)^ !;</(-»'.)
j=t / = i
vdi mpi i = l,...,n. Cgng n bit ding thirc I, ta dugc
tf(x,)<tp:f(y,) = tta;f(y^)
Mat khac, vi A la ma tran nglu nhifln kep va j e ( 0 , l ] nfln tacd
Dinh nghia 3.1.1 (Breckner, W. W., 1978).
Cho s e (0,1]. Ham sd / : [0, co) -> R dugc ggi la s-ldi (theo nghTa thir hai) nflu
/ ( f t + (1 - t)y) < I'fix) + (1 - ty f(y) V 6 i mpi x,y€[0,co) va moi r e [ 0 , l ] -
1
^^P'"'^"'-
Kflt hgp danh gia nay vdi bat dang thiic d trfln ta dugc
|;/(;c,)<«'-X/(3',). (*)
Ngugc Iai, nflu (*) diing vdi moi ham s-ldi lifln Kflt qua chinh ciia bai bao trong muc nay dugc ^^^ t^fln [0, oo) va mpi s e (0,1] thi bit ding thiic cho trong dinh li sau. (*) cung diing vdi mgi ham Idi lifln tuc xac dinh Dinh li 3.1.2. Cdc khdng dinh sau ddy tuang trfln [0,co). Hon nira, bit dang thiic (*) trd thanh
^ ddi vdl cdc vecta x,_yeM" cd thdnh phdn bit dang thiic trong (2) ciia Dinh li I.l. Do dd, khang dinh nguoc Iai la hifln nhifln.
3.2. Mpt so ung dung cua dinh litrpi Muc tiflu trude tifln ciia ehiing tdi d day la md rdng bit dang thirc Popoviciu (xem Bougoffa, L., 2006) sau day
khdng dm.
(1} x-<y
(2) ^fM^"'''HfM vai mgi hdm s-ldi liin tue trin [0, oo) va mgi s e (0,1].
Dfl chirng minh dinh li nay, ta can kflt qua bd trg sau day.
Bo dfl 3.1.3 (Varosanee, S., 2007; Theorem 19). Neu f la hdm s-ldi thi vai mot x, va mgi a, khdng dm'thoa mdn ^ i^, = 1 la ed bat ddng thuc loai Jensen
1=1 ' = 1
Bay gid ta se dua ra chiing minh ciia dinh li
Z/U,)^
« - 2/(- ^)
n-2tr 2
(3.1) vdi / la ham ldi xae dinh trfln / va cac J^, e / Dinh li 3.2.1 (Bat dang thiic loai Popoviciu).Cho / la ham s-ldi va jc,,...,x„ thuge mien xae dinh ctia / . Khi do, bat dang thiic sau diing.
5 1
Tap chi so 32. thang 10-2018, trU6ng Bai hoc Tay Nguyen ijji^TrajjJ-JWf
(,r-«)'- X / ( x , ) + - — / ( J — ) j;^\f( 2 ) + /( 2 '+-^(-7-)
-2- ^ Z / ( ^ T ^ ) - - 3 ^ ' 3~"'-
{"-2)S^ 2C/jimg i77/>i/i. Vi (3.1) dung voi moi ham loi nen no cung diing voi moi ham loi lien tuc xac dinh tren K . Do do, theo Dinh li 1.1, ta CO bo JV = (X|,...,JCp...,x^,.-.,x„,u,...,a) trong do
Muc tieu tiep theo cua chiing toi la mo rong bSt dang thtrc sau day ciia Via Titu Andreescu (xem BougoiTa, L , 2006): Voi moi x^,X2,XJ thupe mien xac dinh ciia ham loi / ta co
-troi hon bo x^+x,+x, f(Xt) + f(x,) + Ax,) + f { - ^ '-)
JC|+Xj X i + i , x„_,+x„ x„_^+x 3
4/(^)./(^)^/(^) I
Chu y rang trong bg y CO n-2 lan x^,...,x„ y^ (3,2) n lan " . Ap dung true tiep Dinh li 3.1.2 cho hai Ma rong bat dang thire (3.2) cho ham s-16i la
ho X va y gom n' -n thanh phan toa dp ta duoe ^uoc ket qua sau.
bit ddng thirc can chiing minh. pin^ |( 32.5. Cho x,,jj,Xj thuoc miin xac Mot he qua true tiep ciia dinh li tren khi K = 3 ^inh cua ham s-ldi ^ , tacd bat ddng thuc sau la nhu sau.
He qua 3.2.2. Dudi cdc gid thiet vd lei hieu ctia Dmh li 3.2.1 vdi n = 3 ta cd
12'-|/M + M)4-/(x,) + /(^'^y^')
X,+X,+ X,^ > - f(-^) + / ( ^ ^ ) + / ( - ^ ) 1.
nx,)+nx,)+f(x,)+3fc r ') ~i['' 2 ' '' 2 ' •" 2
Sii dung cimg phuong phap nhu trfln, ta nhan
Chirng minh. Ta xet bd y gdra ed 12 thanh phan X, +x, +x.
tpa do vdi 3 lan -'^P -^2' ^3 va 3 lan —L_
3
dugc ket qua sau. Tuong tu, ta xet bg z gdm 12 thanh phan tpa dp Dinh li 3.2.3 (Bat dang thirc Popoviciu suy vdi 4 lin ^^ ^^^ •'•2 "'"•^3 -^3 "*" •^i ^hj (J5_ bat
2 2 2 dang thiic (3.2) tucmg duong vdi rdng). Cho cdc sd khdng dm -^i ^ • • • > -^n thudc mien
xdc djnh ciia hdm J .Ki hiiu X =(x -) f-x )/n
vd y.='{hx~x,)/(n-\) v&i moi / = ! , . . . , „ . ^ / ( x ) > ^/(z,).
Khi do, niu f Id hdm s-ldi thi bdt ddng thue sau
diing. Vi bat dang thiic nay diing vdi mpi ham ldi nen nd cung diing vdi mgi ham ldi lien tuc xac dinh (n'-n)'- Z / ( > ' - ) ^ Z / ( - ^ > ) + " ( " - 2 ) / { ^ ) - trfln M.Dodd.theoDinhH 1.1,tacd z-<>.Kflt
hgp dieu nay voi Dinh li 3.1.2, ta nhan duge Mdt he qua true tiep ciia Dinh H 3.2.3 khi
n = 3 lanhusau. 1 2 ' - ^ £ / ( x ) > ^ / ( ^ , ) - He qua 3.2.4. Duai cdc gid thiet vd ki hieu cua
Dinh li 3.2.3 la cd Ro rang, bat ding thiic nay tuong duong vdi bat dang thirc can chiing minh.
Chil y ring cac kflt qua dupe chi ra trong bai bao
52
Tap chi Su j z . mang lu-zuio. uuuiig L/di HUL .t'ay Nguyfln ISSN 1859-4611 nay diing ddi vdi cac ham s-ldi, trong dd s e (0,1] s-ldi, chiing tdi da md rgng dmh H trdi ndi tiflng , De thay rang khi s =\ ham 1-ldi chinh la ham ciia Hardy-Littiewood-Pdlya. Ung dung quan 16i theo nghTa thdng thudng. Vi vay, dfl nhan duoc ^^°"8 '^"^ ^^^ ^"^ ™^ ""^^ '^^^'^ ^^ '^'f'^ 'l ^''*^' '^"^
cac kflt qua cua ham ldi theo nghTa thdng'thudng Hardy-Littlewood-Pdlya la mdt sd bat dang thuc chi cin cho J = 1 trong cac dinh li va hfl qua tren. "«' t'^"g ^^9"= ^^'et lap eho ham loi cung dugc k^^T I ITAIV ^^ ^^°^ ^^° ' ° ^ ^^^ ^'^°'' ^^^ ^^^ ^^^ *^""^ "^^
phucmg phap din dfln cac ket qua trong bai bao !a Trong bai bao nay, bang each su dung dac tmng ^ d i va cd thfl dugc ap dung dfl thu dupe cac ket hinh hgc va dac tnmg giai tich cua bg trgi va ham qua tuong tu cho idp ham ldi suy rgng khac.
.4N EXTENSION OF MAJORIZATION THEOREM FOR GENERALIZED CONVEXITY FUNCTIONS AND APPLICATIONS
Duong Quoc Huy^
Received Date: 01/9/2018; Revised Date: 15/9/2018; Accepted for Publication: 30/9/2018 SUMMARY
In this paper, we show that the famous majorization theorem of Hardy-Littlewood-Pdlya still holds tme for a certain class of generalized convexity flinctions. Using this result, we generalize some well- known inequalities for functions which belongs to this class.
Keywords- Majorization, majorization theorem, generalized convexity function, Karamata's inequality.
TAI LIEU THAM KHAO
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Breckner, W. W. (1978). Stetigkeitsaussagen fur eine Klasse verallgemeinerter konvexer fiinktionen in topologischen linearen Rauraen. Publ. Inst. Math. I'i, 13-20.
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Dragomir, S. S. and Pearce, C. E. M. (2000). Selected Topics on Hermite Hadamard Inequalities and Applications. RGMIA Monographs, Victoria University.
Godunova, E. K. and Levin, V I. (1985). Neravenstva dija funkcn sirokogo klassa, soderzascego vypuklye, monotonnye i nekotorye drugie vidy flinkii. Vyeislitel. Mat. i. Fiz. Mezvuzov. Sb. Nauc.
Trudov, MGPl, Moskva, 138-]42.
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58, 145-152.
Niculescu, C. P. and Persson, L. (2006). Convex functions and their applications. Springer Science + Business Media.
Niculescu, C. P. and Roventa, I. (2017). Hardy-Littlewood-Pdlya theorem of majorization in the framework of generalized convexity. Carpathian J. Math. 33, no. 1, 87-95.
Varosanee, S. (2007). On h-convexity / Math. Anal. Appl. 326, 303-311.
2 Master, Faculty of Natural Science and Technology. Tay Nguyen University;
Corresponding author. Duong Quoc Huy. Tel. 0905616183, [email protected] 53
