Properties and inequalities for the (h 1. )- and (h 1

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1 X & Q, Coget Mthetcs 6, 3: 766 htt://dxdoorg/8/ PURE MATHEMATICS RESEARCH ARTICLE Proertes d equltes or the h - d h, -GA-covex uctos Bo-Y X * d Feg Q,3 Receved: 8 Mrch 6 Acceted: 7 Mrch 6 Frst Pulshed: 8 Arl 6 *Corresodg uthor: Bo-Y X, College o Mthetcs, Ier Mogol Uversty or Ntoltes, Toglo Cty, Ier Mogol Autooous Rego 843, Ch E-l: oytu78@qqco Revewg edtor: Lsh Lu, Quu Norl Uversty, Ch Addtol orto s vlle t the ed o the rtcle Astrct: I the er, the uthors troduce detos o the h -GA-covex uctos d the h, -GA-covex uctos, dscuss soe roertes o these ds o uctos, estlsh soe tegrl equltes or these uctos, d ly these equltes to costruct severl ore equltes Sujects: Advced Mthetcs; Alyss-Mthetcs; Mthetcl Alyss; Mthetcs & Sttstcs; Pure Mthetcs; Rel Fuctos; Scece Keywords: covex ucto; h-ga-covex ucto; h, -GA-covex ucto; roerty; equlty Mthetcs suject clssctos: 6A5; 6D5 Itroducto The ollowg detos re well ow the lterture ABOUT THE AUTHORS The Ier Mogol Uversty or Ntoltes loctes t the Horq Khorch, Horch grssld Ch d ws set u 958 The College o Mthetcs s oe o the eldest secltes d sujects the uversty Curretly, there re eers the reserch grou o the Theory o Covexty d Alctos The eers hve troduced severl ew otos o covex uctos d cotruted uch to the theory o covexty d lctos Feg Q receved hs PhD degree o Scece thetcs ro Uversty o Scece d Techology o Ch He s eg ull roessor t He Polytechc Uversty d Tj Polytechc Uversty Ch He ws the ouder o the School o Mthetcs d Iortcs t He Polytechc Uversty Ch He ws vstg roessors t Vctor Uversty Austrl d t Uversty o Hog Kog Ch He ws rt-te roessor t He Uversty, He Norl Uversty, d Ier Mogol Uversty or Ntoltes Ch He vsted Coehge Uversty Der, Kyugoo Ntol Uversty d other sx uverstes South Kore, d Atly Turey to tted cdec coerece held y the Ağrı İrh ÇeÇe Uversty Arl 6 He s or ws edtors o over tertol resected jourls Fro 993 to 6, he ulshed over 46 cdec rtcles reuted tertol jourls PUBLIC INTEREST STATEMENT The theory o covex uctos hs ortt lctos y thetcl sceces The oto o h-covex uctos c e used to derve lety o covex uctos lr to coo thetcs I curret er, the uthors exted the oto o h-covex uctos, troduce ore geerl oto o h-covex uctos, rove exsted otos or covex uctos, estlsh severl equltes or the geerl h-covex uctos, d d ther lctos 6 The Authors Ths oe ccess rtcle s dstruted uder Cretve Coos Attruto CC-BY 4 lcese Pge o 8

2 X & Q, Coget Mthetcs 6, 3: 766 htt://dxdoorg/8/ Deto A ucto :I R + =, R s sd to e GA-covex x t y t t x+ t y holds or ll x, y I d t, ] Deto Toder, 985 For :, ] R, d > d, ], tx + ty t x+ t y s vld or ll x, y, ] d t, ], the we sy tht s -covex ucto o, ] Deto 3 Hudz & Mlgrd, 994 Let s, ] A ucto :R =, R s sd to e s-covex the secod sese λx + λy λ s x+ λ s y holds or ll x, y I d λ, ] Deto 4 Shug, Y, & Q, 3 Let :I R + R d s, ] The s sd to e s-geoetrc-rthetclly covex ucto x λ y λ λ s x+ λ s y or x, y I d λ, ] Deto 5 X & Q, 5 For soe s, ], ucto :I R R s sd to e exteded s-covex λx + λy λ s x+ λ s y s vld or ll x, y I d λ, Deto 6 Pr, For s,, ] d >, ucto :, ] R s sd to e s, - covex λx + λy λ s x+ λ s y holds or ll x, y I d λ, Deto 7 Vrošec, 7 Let I, J R e tervls,, J, d h:j R e o-egtve ucto such tht h A ucto :I R s clled h-covex, or sy, SXh, I, s o-egtve d tx + ty ht x+h t y or ll x, y I d t, I the equlty Equto s reversed, the s sd to e h-cocve, or sy, SVh, I Deto 8 Özder, Ader, & Set, Let J R e tervl,, J d >, h:j R e o-egtve ucto such tht h We sy tht :, ] R s h, -covex ucto, or sy, SMXh,,, ], s o-egtve d tx + ty ht x+h t y or ll x, y, ] d λ, ] d, ] I the equlty Equto s reversed, the s sd to e h, -cocve, or sy, SMVh,,, ] Pge o 8

3 X & Q, Coget Mthetcs 6, 3: 766 htt://dxdoorg/8/ The ollowg equltes o Herte Hdrd tye were estlshed or soe o the ove covex uctos Theore Drgor & Toder, 993, Theore Let :R R e -covex d, ] I L, ] or < <, the { + x dx, } + Theore Drgor,, Theore Let :R R e -covex ucto wth, ] I L, ] or < <, the x+ x dx + ] + Theore 3 Sry, Sgl, & Yldr, 8, Theore 6 Let SXh, I d L, ] or, I wth < The + h x dx +] ht dt Theore 4 Phero, 8, Theore 4 I s s-covex the secod sese d o-egtve o I d x, x,, x I or 3 d soe s, ], the x + + x x s s x + x j = <j 3 Theore 5 Lt,, Theore Let h e o-egtve suer-ultlctve ucto I SXh, I d x,, x I, the = x = x h h x + x +, = 4 where x + = x Ths equlty s reversed SVh, I For ore orto o otos o vrous covex uctos d ther equltes o Herte Hdrd tye, lese reer to recetly ulshed rtcles 3, Bougo, 6, Hudz & Mlgrd, 994, Lt,, Shug et l, 3, X & Q, 3, X & Q, 5, 3, X, Wg, & Q, 4, 4 d closely relted reereces there Detos We ow troduce cocets o h -GA-covex uctos d h, -GA-covex uctos Deto Let h :, ] R such tht h or =, d :I R + R I x t y t h t x+h t y or x, y I d t, ], the s sd to e h -geoetrc-rthetclly covex ucto or, sly seg, h -GA-covex ucto I Equto s reversed, the s sd to e h -geoetrc-rthetclly cocve ucto or, sly seg, h -GA-cocve ucto Rer I s decresg d h -GA-covex ucto o R + d h t =h t =ht or t, ], the s h covex ucto o R + Pge 3 o 8

4 X & Q, Coget Mthetcs 6, 3: 766 htt://dxdoorg/8/ Rer By Deto, h t =h t =ht, the s h-ga-covex ucto; whe ht =t s or t, d s, ], h-ga-covex ucto s reduced to exteded s-ga-covex ucto; 3 whe ht =t or t, ], h-ga-covex ucto ecoes GA-covex ucto Deto Let h :, ] R d :, ], ] such tht h or =, A ucto :, ] R s sd to e h, -GA-covex, x t y tt h t x+th t y or ll x, y, ] d t, ] I the equlty Equto s reversed, we sy tht s h, -GA-cocve Rer 3 By the ove detos, we hve the ollowg ssertos I h t =h t =ht or ll t, ], the :, ] R s h, -GA-covex ucto; Let :, ] R e h, -GA-covex ucto d, ] Whe ht =t or t, ], the ucto s sd to e -GA-covex; 3 I :, ] R s h, -GA-covex ucto, ht =t s or t, d s, ], d, ], the s exteded s, -GA-covex; 4 I :, ] R s h, -GA-covex ucto d, ], the t s h-ga-covex o, ] Exle l R Let x = l x or x, ] d t =c t l or t, d < c, d soe Let h t =t l d h t =t l or t, d l, l R I l, l, the s decresg d h, -GA-covex ucto o, ]; l, l, the s decresg d h, -GAcocve ucto o, ] I Detos 8, lettg = 6, ht =t or ll t, ], x = 5, y = 9, d t = leds to t x + t y ht x h t y > Ths les tht x = l x s ot h, -covex ucto o, ] 3 Proertes Now we dscuss soe roertes o h, -GA-covex uctos Theore 3 Let h :, ] R such tht h or =, d let :I R + R + I :I R + s h -GA-covex ucto o I, the h t+h t or t, ]; I :I R + s h -GA-cocve ucto o I, the h t+h t or t, ] Proo I :I R + s h -GA-covex ucto o I, usg the h -GA-covexty o o I, we ot x = x t x t h t x+h t x =h t+h t] x or ll x I d t, ] Pge 4 o 8

5 X & Q, Coget Mthetcs 6, 3: 766 htt://dxdoorg/8/ The rest y e roved slrly Theore 3 s thus roved Theore 3 Let h :, ] R or =,, 3, 4, :, ] R +, d :, ], ] I s h, -GA-covex ucto o, ], h t h 3 t, d h t h 4 t or t, ], the s h 3, h 4, -GA-covex ucto o, ]; I s h, -GA-cocve ucto o, ], h t h 3 t, d h t h 4 t or t, ], the s h 3, h 4, -GA-cocve ucto o, ] Proo Sce s h, -GA-covex ucto o, ], h t h 3 t, d h t h 4 t or t, ], we hve x t y tt h t x+th t x h 3 t x+th 4 t x or ll x, ] d t, ] Sce x s h, -GA-cocve ucto o, ] d h t h t or t, ], we hve x t y tt h t x+th t x h 3 t x+th 4 t x or ll x, ] d t, ] The roo o Theore 3 s colete Corollry 3 Let h :, ] R d :, ] R or d :, ], ] I ht =x {h t} or t, ] d re h, -GA-covex o, ] or =,,,, the = s h, -GA-covex ucto o, ]; I ht = {h t} or t, ] d re h, -GA-cocve o, ] or =,,,, the = s h, -GA-cocve ucto o, ] Proo Ths ollows ro Theore 3 d ducto o Theore 33 d, ] Let h :, ] R such tht h or =,, :, ] R, g:, d] g, d], ], I s cresg or decresg, resectvely d h, -GA-covex ucto o, ] d u = gx s -geoetrclly covex or cocve, resectvely ucto o, d], the g s h, -GA-covex ucto o, d]; I s cresg or decresg, resectvely d h, -GA-cocve o, ] d u = gx s -geoetrclly covex or cocve, resectvely ucto o, d], the g s h, -GA-cocve ucto o, d] Proo Whe s decresg d h, -GA-covex ucto o, ], u = gx s -geoetrclly cocve o, d], the g x t y t gx] t gy] t or ll x, y, d] d t, ]; Pge 5 o 8

6 X & Q, Coget Mthetcs 6, 3: 766 htt://dxdoorg/8/ y = u s decresg d h, -GA-covex o, ], the g x t y t gx] t gy] t h t gx + h t gy or ll x, y, d] d t, ] Thereore, or ll cses etoed ove, the cooste g s h, -GA-covex ucto o, d] The rest y e roved slrly Theore 33 s thus roved 4 Jese tye equltes Now we re osto to estlsh equltes o Jese tye or h, -GA-covex uctos Theore 4 Let h :, ] R e uctos such tht h or =,, let h t h t h t t or ll t, t, ], let h e suer-ultlctve ucto d :, ], ], d let :, ] R e h, -GA-covex ucto o, ] The x w j= w j h w x + = = j= w j h w x 4 holds or ll x, ], w > such tht = w = d w = I h t h t h t t or ll t, t, ], h s su-ultlctve, d s h, -GA-cocve o, ], the the equlty Equto 4 s reversed Proo Whe =, tg t = w d t = w Deto es tht the equlty Equto 4 holds Suose tht the equlty Equto 4 holds or =, tht s, x w j= w j h w x + = = j= w j h w x 4 Whe = +, lettg Δ = + w =, y Deto d the hyothess Equto 4, we hve + = x w j= w j + = x w = x w Δ w j=,j w j Δ h w x +w h Δ x w Δ w Sce h s suer-ultlctve ucto, we ot h Δ h Δ h w or =,,, Ths les tht, whe = +, the equlty Equto 4 holds By ducto, Theore 4 s roved Rer 4 Uder the codtos o Theore 4, s decresg ucto o, ] d t = or t, ], the + =3 w h w x +w h Δ h x Δ + ] w + w j h x Δ =3 j= x w Δ j= w j Pge 6 o 8

7 X & Q, Coget Mthetcs 6, 3: 766 htt://dxdoorg/8/ = w x = x w h w x + h w x holds or ll x, ] d w > such tht = w = Corollry 4 Uder the codtos o Theore 4, w = = w =, the = ] x ] h x +h x = = 43 holds or ll x, ] or =,,, I h t h t h t t or ll t, t, ], h s su-ultlctve, d s h, -GA-cocve o, ], the the equlty Equto 43 s reversed Corollry 4 Let h:, ] R e suer-ultlctve ucto such tht h,, ], d :, ] R e h, -GA-covex ucto o, ] The the equlty = x w hw x = 44 holds or ll x, ] d w > such tht = w = I h s su-ultlctve d s h, -GAcocve o, ], the the equlty Equto 44 s reversed Proo Ths ollows ro Theore 4 y uttg h t =h t =ht d t = or ll t, ] d, ] Corollry 43 Let ht =t s or t, d s, ], :, ] R, d, ] The s s, -GA-covex ucto o, ] d oly = x w w s x = or ll x, ] d w > such tht = w = 5 Herte Hdrd tye equltes Now we re osto to estlsh soe ew Herte Hdrd tye equltes or h, - GA-covex uctos Theore 5 Let h :, ] R, h or =,, :, ], ], d :R + R e h, -GAcovex ucto o ], such tht L, ] d h L, ] or < < The h x dx + h l l l l Proo Sce = t t t t or t, ro the h, -GA covexty o o, ], we ot h t t t t + h I relcg t t d t t or t y x, the x dx Pge 7 o 8

8 X & Q, Coget Mthetcs 6, 3: 766 htt://dxdoorg/8/ t t dt = x dx l l d t t dt = The roo o Theore 5 s colete l l x dx 5 5 Theore 5 Let h :, ] R, h or =,,, ], d :R + R e h, -GA-covex ucto o ], such tht L, ] d h L, ] or < < The l l h t dt + x dx { h t dt + } h t dt I h t =h t =ht or ll t, ], we hve { } x dx +, + ht dt l l Proo Lettg x = t t or t, y the h, -GA-covexty o d Equto 5, we ot h t dt, x dx = l l The roo o Theore 5 s colete + t t dt { h t dt, h t dt h t dt + } h t dt Corollry 5 Let h t =t s d h t =t s or ll t,, let s, s, ] d, ], d let :R + R e h, -GA-covex ucto o ], such tht L, ] or < < The l l { x dx s + + s +, Proo Fro the h, -GA covexty o o, s + + s + Theore 53 Let h :, ] R, h or =,,, ], :R + R e h, -GA-covex ucto o ], such tht L, ], d h L, ] or < < The h x dx + h x dx l l l l { ] h +h h t dt ] + h + h h t dt, h +h ] + h + h h t dt ] } } h t dt ], we ot Pge 8 o 8

9 X & Q, Coget Mthetcs 6, 3: 766 htt://dxdoorg/8/ h t t + h Susttutg t t d t t or t y x d tegrtg o oth sdes o the ove equlty wth resect to t, ] led to Theore 53 s roved t t ] { h h t +h t + h h + h h t + h t h t +h t ] h t + h t Corollry 53 Let h:, ] R, h d, ], d :R + R e h, -GA-covex ucto o ], such tht L, ] d h L, ] or < < The ] x x+ dx h l l { + + +, + +, + +, } ht dt Proo Ths c e derved ro lettg h t =h t =ht or ll t, ] d cosderg the syetry etwee d Theore 4 Corollry 53 Uder the codtos o Corollry 53, ht =t s or t, d s, ], the s ] x x+ dx l l { s , + +, + } +, ], ]} h x dx + h x dx l l l l { ] h +h h t dt ] + h + h h t dt, h +h ] + h + h h t dt ] } h t dt Pge 9 o 8

10 X & Q, Coget Mthetcs 6, 3: 766 htt://dxdoorg/8/ Theore 54 Let h :, ] R such tht h or =,, :, ], ], d, g:r + R re h, -GA-covex uctos o ], such tht g L, ] or < < The h ] g xgxdx l l + h h ] x x gx+xg l l dx h ] x + l l g x dx 53 Proo Usg the h, -GA covexty o d g o, g h t t + h ] t t h gt t + h ] g t t ], we ot Lettg x = t t d x = t t or t, ] d tegrtg the equlty Equto 54 o, ] wth resect to t, we rrve t the equlty Equto 53 Theore 54 s thus roved Theore 55 Let h :, ] R, h or =,,,, ],, g:r + R I s h, -GA-covex ucto o ],, g s h, -GA-covex ucto o ],, d g L, ] d h, h L, ] or < <, the 54 xgx dx l l g h t dt + g h t dt ] + g + g h th t dt 55 Proo Let x = t t or, ] By the h, -GA-covexty o d g, we hve xgx dx = t t g t t dt l l h t + h t = g h t dt + g h t dt ] + g + g h th t dt The roo o Theore 55 s colete ] h tg+ h tg ] dt Corollry 55 Uder the codtos o Theore 55, h t =h t =ht or ll t, ], the l l h t dt + xgx dx g+ g+ g I rtculr, ht =t s or t,, s,], d = =, the g ] ] hth t dt Pge o 8

11 X & Q, Coget Mthetcs 6, 3: 766 htt://dxdoorg/8/ xgx dx l l g+ s + + Bs +, s + g where B deotes the well-ow Bet ucto g 6 Alctos I wht ollows we wll ly theores d corollres the ove secto to estlsh equltes or h, -GA-covex uctos Theore 6 Uder the codtos o Theore 4,, N wth 3 d The, or ll x, ] d w > such tht w = =, w =, d x + = x,, x = x, + ] ] g, whe h, we hve ] + h = = h w ] x + = = { h h x + = = ] x w + j= w j ] h h w w j h w + j= + = = x ] x = = ] }; 6 whe h =, we hve + x w + j= w j = = ] h w + w j h w + x + = j= { ] h w h x = = = + = ] = x ] }, 6 I h t h t h t t or ll t, t, ] s su-ultlctve, d s h, -GA-cocve o, ], the the equltes Equtos 6 d 6 re reversed Proo Usg the equlty Equto 4, t ollows tht = + = x w + j= w j Whe h, we hve + = + = x w + h j= w j ] h w h w ] h w h w ] = j= x + = x = = j= w j h w + ] w j h w + x h x = = x = Pge o 8

12 X & Q, Coget Mthetcs 6, 3: 766 htt://dxdoorg/8/ By the equlty Equto 43, t s esy to see tht ] h = h + = j= + = x w + Whe h =, we ot + x w + j= w j j= w j ] h w h w ] w j h w + { h h x + = = ] = = = j= The roo o Theore 6 s colete x = + = = x ] x = ] } ] h w + w j h w + x h w ] h w + w j h w + x + = j= = = = = { ] h w h x = + = x = ] = x ] } Corollry 6 Uder the codtos o Theore 6, w = = w =, whe h, we hve h + h ] ] h x h ] h = = h ] h h x + x + = = = = ] + = = x ] x = = ] ; 63 whe h =, we hve = + x ] = h + h { h h ] ] x + = = ] + x = = = ] = x ] } 64 Pge o 8

13 X & Q, Coget Mthetcs 6, 3: 766 htt://dxdoorg/8/ I h t h t h t t or ll t, t, ], h s su-ultlctve, d s h, -GA-cocve o, ], the the equltes Equtos 63 d 64 re reversed Corollry 6 Uder the codtos o Theore 6, w = w = = w = d t = or y t, ], whe h, we hve h + h ] ] h h x h = = h x + ] + h h x x ; = = = 65 whe h =, we hve + = = ] x h + h x = ] + h { h = x = } x 66 I h t h t h t t or ll t, t, ], h s su-ultlctve, d s h -GA-cocve o, ], the the equltes Equto 65 d 66 re reversed Corollry 63 Uder the codtos o Theore 6, h t =t s d h t =t s or ll t, d s, s, ], the whe s, ], s, d s s, we hve + ] ] s s s s s = = x ] s ] { x + x + s s = = = = ] + = = x ] x = = ] }; whe s s =, we hve = + = w s + x w + j= w j = j= ] w j x + = { s x = = + = ] = x ] } I s h, -GA-cocve o, ] d s s, the the equltes Equtos 67 d 68 re reversed Corollry 64 Uder the codtos o Theore 6, w = = w =, h t =h t =ht, d t = or t, ] d h d s decresg ucto o, ], the Pge 3 o 8

14 X & Q, Coget Mthetcs 6, 3: 766 htt://dxdoorg/8/ whe h <, we hve h h = + = x = = x h h x = = x + x = x ; = 69 whe h =, we hve + h x = = h = + x = = x + x = + x = x = Theore 6 Uder the codtos o Theore 6, l,, l N, the whe h, we hve h + x w + j= w j l l < <l = = ] h h w h w ] x = ]{ + w j h w + x = j= = h h + = = x ] + = ] x = = ] }; whe h =, we hve h w + l < <l = = j= + = where l + = l,, l = l x w + j= w j l ] w j h w + { ] h x = = = + j= x + x ]j j h w = ] }, 6 6 I h t h t h t t or ll t, t, ] s su-ultlctve, d s h, -GA-cocve o, ], the the equltes Equtos 6 d 6 re reversed Proo Usg the equlty Equto 4, t ollows tht Pge 4 o 8

15 X & Q, Coget Mthetcs 6, 3: 766 htt://dxdoorg/8/ l < <l = h w h w ] = + = x w + j= w j l x + w j h w + Whe h, usg the equlty Equto 43, we hve = j= x = h = l < <l = j= + = x w + j= w j l ] h h w h w ] x = ]{ + w j h w + x h h + = ] = + = = x ] x = ] } = Whe h =, we ot + l < <l = = h w + = = j= j= The roo o Theore 6 s colete x w + j= w j l ] w j h w + = = x h w ] h w + w j h w + x + { ] h w h x = = + j= x = ] = x ]j j } Corollry 6 Uder the codtos o Theore 6, w = = w =, whe h, we hve h h h l < <l = + x ] l = ] ] h h ] h x + = = { h h = + = ] x = + = x ] = ] }; Pge 5 o 8

16 X & Q, Coget Mthetcs 6, 3: 766 htt://dxdoorg/8/ whe h =, we hve I h t h t h t t or ll t, t, ], h s su-ultlctve, d s h, -GA-cocve l < <l = h + h = + x ] l o, ], the the ove equltes re reversed = ] ] x + = { ] h x = Corollry 6 Uder the codtos o Theore 6, w = = w = t, ], = + j= x ]j j } h = ] d t = or ll whe h, we hve h h l < <l = h h ] x + + = = whe h =, we hve x l h ] ] h h x = h x ; 6 I h t h t h t t or ll t, t, ], h s su-ultlctve, d s h, -GA-cocve o, ], the the equltes Equtos 6 d 63 re reversed + h l < <l ] x h l + h x = = { ] } h x x = j= 63 Corollry 63 Uder the codtos o Theore 6, w = = w = d h t =h t =ht or ll t, ], whe h, we hve h h l < <l = + = x ] l ] + = j= x ]j j x = = = ] ; 64 Pge 6 o 8

17 X & Q, Coget Mthetcs 6, 3: 766 htt://dxdoorg/8/ whe h =, we hve + x ] h l l < <l = = ] ] + = j= x ]j j + x = = = ] 65 I h s su-ultlctve d s h, -GA-cocve o, ], the the equltes Equtos 6 d 63 re reversed Corollry 64 Uder the codtos o Theore 6, w = = w =, t = or y t, ], h, d s decresg ucto o, ], the whe h <, we hve h x h l l < <l = h x l h l < <l = ] h h ] h h x = h h ] + + x x = = ] h h ] h h x = h h ] + + x x ; = whe h =, we hve = I h t h t h t t or ll t, t, ], h s su-ultlctve, d s h -GA-cocve o x l x l l < <l = l < <l = ] { ] h + h x +h h = } ] = + h x = { h x ] h + h, ], the the ove equltes re reversed = x = x } = x Pge 7 o 8

18 X & Q, Coget Mthetcs 6, 3: 766 htt://dxdoorg/8/ More rers Rer 5 Lettg = the equlty Equto 69 yelds the equlty Equto 4 Rer 6 Lettg h t =h t =ht the rst equlty Corollry 63 or ll t, ] yelds the equlty 39 X et l 4 Fudg Ths wor ws rtlly suorted y NNSF grt uer 3638] o Ch d y the Ier Mogol Autooous Rego Nturl Scece Foudto Project grt uer 5MS3], Ch Author detls Bo-Y X E-l: oytu78@qqco ORCID ID: htt://orcdorg/ Feg Q,3 E-ls: qeg68@glco, qeg68@hotlco ORCID ID: htt://orcdorg/ College o Mthetcs, Ier Mogol Uversty or Ntoltes, Toglo Cty, Ier Mogol Autooous Rego 843, Ch Dertet o Mthetcs, College o Scece, Tj Polytechc Uversty, Tj Cty 3387, Ch 3 Isttute o Mthetcs, He Polytechc Uversty, Jozuo Cty, He Provce 454, Ch Ctto orto Cte ths rtcle s: Proertes d equltes or the h - d h, -GA-covex uctos, Bo-Y X & Feg Q, Coget Mthetcs 6, 3: 766 Reereces B, R-F, Q, F, & X, B-Y 3 Herte--Hdrd tye equltes or the - d, -logrthclly covex uctos Flot, 7, 7 do:98/fil3b Bougo, L 6 New equltes out covex uctos Jourl o Iequltes Pure d Aled Mthetcs, 73, 3 Artcle 48 Retreved ro htt://wwwesde/ jourls/jipam/rtcle766htl Drgor, S S O soe ew equltes o Herte- Hdrd tye or -covex uctos Tg Jourl o Mthetcs, 33, Drgor, S S, & Toder, G 993 Soe equltes or -covex uctos Stud Uverstts Beş-Boly Mthetc, 38, 8 do:5556/jtj3334 Hudz, H, & Mlgrd, L 994 Soe rers o s-covex uctos Aequtoes Mthetce, 48, do: 7/BF83798 Lt, M A O soe equltes or h-covex uctos Itertol Jourl o Mthetcl Alyss, 4, Özder, M E, Ader, A O, & Set, E O h-- covexty d Hdrd-tye equltes Retreved ro htt://rxvorg/s/3663 Pr, J Soe Hdrd s tye equltes or coordted s, -covex gs the secod sese Fr Est Jourl o Mthetcl Sceces, 5, 5 6 Phero, M R 8 Lzhr Iequltes d s-covex heoeo New Zeld Jourl o Mthetcs, 38, 57 6 Sry, M Z, Sgl, A, & Yldr, H 8 O soe Hdrd-tye equltes or h-covex uctos Jourl o Mthetcl Iequltes,, do:753/j--3 Shug, Y, Y, H-P, & Q, F 3 Herte--Hdrd tye tegrl equltes or geoetrc-rthetclly s-covex uctos Alyss Much, 33, 97 8 do:54/ ly39 Toder, G 985 Soe geerlztos o the covexty I Proceedgs o the Colloquu o Aroxto d Otzto Cluj-Noc, Cluj: Uversty o Cluj-Noc Vrošec, S 7 O h-covexty Jourl o Mthetcl Alyss d Alctos, 36, 33 3 do:6/j j686 X, B-Y, & Q, F 3 Herte Hdrd tye equltes or uctos whose dervtves re o covextes Noler Fuctol Alyss d Alctos, 8, X, B-Y, & Q, F 3 Soe Herte--Hdrd tye equltes or deretle covex uctos d lctos Hcettee Jourl o Mthetcs d Sttstcs, 4, X, B-Y, & Q, F 5 Iequltes o Herte--Hdrd tye or exteded s-covex uctos d lctos to es Jourl o Noler d Covex Alyss, 6, X, B-Y, Wg, S-H, & Q, F 4 Proertes d equltes or the h- d h, -logrthclly covex uctos Cretve Mthetcs d Iortcs, 3, 3 3 X, B Y, Wg, S H, & Q, F 4 Soe equltes or h, -covex uctos Jourl o Iequltes d Alctos,, do:86/9-4x-4-6 The Authors Ths oe ccess rtcle s dstruted uder Cretve Coos Attruto CC-BY 4 lcese You re ree to: Shre coy d redstrute the terl y edu or ort Adt rex, trsor, d uld uo the terl or y urose, eve coerclly The lcesor cot revoe these reedos s log s you ollow the lcese ters Uder the ollowg ters: Attruto You ust gve rorte credt, rovde l to the lcese, d dcte chges were de You y do so y resole er, ut ot y wy tht suggests the lcesor edorses you or your use No ddtol restrctos You y ot ly legl ters or techologcl esures tht leglly restrct others ro dog ythg the lcese erts Pge 8 o 8