On Several Inequalities Deduced Using a Power Series Approach

Transcription

1 It J Cotemp Mth Sceces, Vol 8, 203, o 8, HIKARI Ltd, wwwm-hrcom O Severl Iequltes Deduced Usg Power Seres Approch Lored Curdru Deprtmet of Mthemtcs Poltehc Uversty of Tmsor P-t Vctore, No2, Tmsor, Rom Copyrght c 203 Lored Curdru Ths s ope ccess rtcle dstruted uder the Cretve Commos Attruto Lcese, whch permts urestrcted use, dstruto, d reproducto y medum, provded the orgl wor s properly cted Astrct The m of ths pper s to study wht would ecome severl equltes usg the power seres method Also some pplctos wll e preseted Mthemtcs Suject Clssfcto: 26D5 Keywords: Rdo s equlty, power seres Itroducto It s ecessry to recll the equlty of J Rdo whch ws pulshed [8] For every rel umers p>0, x 0, > 0 for, we hve the followg equlty: x p+ p x p+ p, p > 0 Theorem [4] For,x > 0, p, {, 2,, }, N d 2 the equlty tes plce, x p+ p x p+ {xp+ + mx p <j p + xp+ j p x + x j p+ j + j } p

2 856 Lored Curdru wth equlty f d oly f x = x 2 2 = = x Also we wll use the followg two equltes whch were eucted d proved [2] d [3] Theorem 2 [2] If N {}, R +,,c,d,x R +,X = x, cx >dmx x d m [,, p R +, the: 2 X + x m cx dx p + m c d p p m+ X m p Theorem 3 [3] If N {},,, x R +, {,, }, X = x d m, t, u [,, such tht X t >mx x t, the: 3 x m X t m+tu+ xt u t u X m tu It s lso ecessry to recll the exteso of equlty whch s stroger th the Rdo s equlty, d ws gve y [6] d lso cosequece Theorem 4 [6] For every 2, p > 0, > 0, x 0,, t holds: x p+ p + xp+ 2 p xp+ p x + x x p p + x + x j p x j j x 2 +p mx <j j + j p Deotg x = λ,, we hve the equvlet form: λ p+ + 2 λ p λ p+ λ + 2 λ λ p p j λ + j λ j p λ λ j 2 +p mx <j + j p Corollry [6] For every 2, p > 0, x 0,, wth s = x + x x, the followg exteso of Nestt s equlty holds: x s x + p p s + p x x j x + x j p x x j 2 +p mx <j s x s x j [x + x j s x 2 + x2 j ]p We use elow lso the ext result, whch s gve [9]

3 Iequltes deduced usg power seres pproch 857 Theorem 5 [9] For every 2, p, 0, > 0,, the followg equltes hold: 25, 0 Δ [p] ; p Δ [p] ; = = Δ [p ] ; d 0 Δ [p] ; p 4 M mm p m p where m M, for =,, 2 The results =, I ext result equlty oted usg power seres for equlty 2 from Theorem 2, see [2] s gve Theorem 6 If N {}, R +,,c,d,x R +,X = x, cx >dmx x,m [,, p R +, d ddto X + x <, {, 2,, } the: X + x cx dx p X x p+ X p + X + c d p Proof Usg equlty 2, X + x m + cx dx p X m p+ X c d p whe m N d replcg m y d summg the for {, 2,, m} we ot m X + x m + p+ cx = dx p X X = c d p Now tg to ccout the hypothess, 0 <X +x <, {, 2,, } d R +,,c,d,x R +, {, 2,, } we c otce tht, 0 + X = X + X <X + mx < Therefore whe m teds to fty we hve cx dx X + x p X + x p+ Xc p d + X p + X or cx dx p X + x X + x p+ X p + + X c d p

4 858 Lored Curdru Now we gve elow form of equlty from Theorem, see [4], oted usg power seres pproch, see [7] d [5] Theorem 7 For, x > 0, {, 2,, }, N, 2 f x <, {, 2,, } the equlty tes plce, x + x j 2 = x 2 x = x 2 = = x + mx <j { x 2 x2 j + x j x j + j x + x j } Proof We use the sme method le efore By equlty, we hve, x p+ {xp+ p <j p + xp+ j p j x p+ p + mx x p+ p+ p x + x j p+ + j p } + x + xp+ j p p x + x j p+ j + j, p <j Tg to ccout equlty whe p N s replced y l d summg for l {, 2,, p}, we ot: p p l= x x l+ l= x l+ + p l+ p l+ xj p l+ x + x j + + j + j, j + j l= l= l= <j Whe p teds to fty, ecuse x <, {, 2,, } d the seres re coverget d we hve, x x x + P x P x P < + x x + j x x j j + j j j x +x j x + x j, + + j j <j Therefore x 2 = x 2 x = = x + mx { x 2 + x2 j x + x j 2 <j x j x j + j x + x j } = We c lso see wht wll ecome the equlty from Theorem 2, see [6] y usg power seres method

5 Iequltes deduced usg power seres pproch 859 Theorem 8 For, x > 0, {, 2,, }, N, 2 f x <, {, 2,, } the equlty tes plce, = x 2 x = x 2 = = x + + mx { + j x j j x 2 <j j [ + j x + x j ] } 2 Proof We use the equlty from Theorem 4 d we hve x p+ p + xp+ 2 p xp+ p x + x x p p + x + x j p x j j x 2 +p mx x + x x p+ <j j + j p p +p x + x j p x j j x 2, j + j p <j Whe p N s replced y l d the summg for l {, 2,, p}, we ot: p x l+ p l+ + j x + x x l+ l= = l+ l= p l x + x j l x j j x 2 = x j j x 2 p l x + x j l, j + j l j + j + j l= l= <j By hypothess, x <, <j we see tht x +x j + j <, <j d therefore whe p teds to fty we ot, x 2 = x 2 = x = = x x j j x 2 j + j 2 = x +x j + j = + j x j j x 2 j [ + j x + x j ], 2 <j Ths mples the requred equlty, f we te mxmum for ll <j,, j N the rght sde of the lst equlty The well-ow equlty, x = x 2 where 0 <x<, see [5] ws lso used efore

6 860 Lored Curdru I ext result lso the power seres s used order to see wht wll ecome the equlty form Corollry 3, see [6] uder more restrctve codtos o the umers x,, x Corollry 2 For every 2, x 0,, wth s = x + x x, d s > mx <j, {x,, x2 +x2 j x +x j } the followg equlty tes plce: x s s x s s + + mx <j x x j x x j 2 s x s x j l= [x + x j s x 2 + x2 j ] [x + x j s x 2 + x2 j ]2 Proof Usg the hypothess we hve, s x >, s > d x + x j < x + x j s x 2 + x 2 jor s x <, < d x +x j < sofwe s x +x j s x 2 +x2 j chge p l, where l {, 2,, p} d p N d te the elow sum, we ot, p x p l s x l s + l + l= l= p x x j x + x j l x x j 2 l s x s x j [x + x j s x 2 + x2 j ]l d whe p teds to fty we hve, s x x s x s s + x x j x x j 2 [x + x j s x 2 + x2 j ] [x + x j s x 2 +, x2 j ]2 <j The usg the sme techque s prevous theorem we ot the desred mxmum d the requred equlty Cosderg Theorem 3 we c ot elow two dfferet equltes Frst cse s whe m teds to fty d the secod equlty s oted whe u teds to fty The we study the cse whe d m d u teds to fty Theorem 9 If N {},,, x R +, {,, }, X = x d t, u [,, such tht X t >mx x t, d x <, {,, } the: X t xt u x x tu+ t u X tu X

7 Iequltes deduced usg power seres pproch 86 Proof Le prevous demostrto summg whe m teds to fty we hve, tu+ X t xt u x t u X tu X Theorem 0 If N {},,, x R +, {,, }, X = x d t, m [,, t N such tht X t >x t +, {,, } the: x m X t xt t+ m X m t t t X Proof I equlty 3, we cosder l sted of u, l {,, u} d u turl umer, u d summg we ot: u x m u m+tl+ l= X t x t l l= t l Xm tl Becuse X t >xt +, {,, } d,, x R + results Xt P xt + or Xt xt + It s ow tht t x + + x xt + + xt f t N {0}, see [5] d therefore X t Xt > t or t X t t < If u teds to fty we hve x m m X m X t xt X t t Cosequece If N {},,, x R +, {,, }, X = x d t, m [,, t N such tht X t >xt +, {,, } d x <, {,, } the: X t xt x x t+ X t t t X X

8 862 Lored Curdru It would e lso terestg to see wht wll ecome some pplctos of Theorem d Theorem 2, see [3] Now we see wht wll ecome the equltes 25 d 26 from Theorem 23, see [9] f o umers, M d m we hve some restrctos Theorem For every 2, 0, > 0,, the followg equltes hold: 0 = = = = = 2 = = = 2 2 = 2 = 2 = 3 = = = 3 2 = = 2, d = 0 = = = = 4 M m M 2 m 2, = where m M<, {,, } Proof We deduce the equlty y the sme techque s efore, tg to ccout tht the codto m M<, {,, } result < P = d <, {,, } P = We shll eucte elow the tegrl form of ths lst equlty usg the sme techques s [] Theorem 2 Let fx 0, gx > 0 d f f,g :[, ] R + e two tegrle fuctos o [, ] wth m fx M, x [, ] d M < gx the 2 fxgx gxdx 0 gx fx dx gxdx fxdx fxdx gxdx fxg 2 xdx gx fx 2 dx g 3 x gx fx 2 dx fxdx gxdx 2 gxdx 2 2 fxdx 3 gxdx gxdx fxdx 2

9 Iequltes deduced usg power seres pproch 863 d 2 fxgx gxdx 0 gx fx dx gxdx fxdx 4 M m M gxdx 2 m 2 Proof Let N d x = +, {0,,, } We use prevous theorem, we put fx sted of d gx sted of, the we multply y d ot the correspodg Rem sums of the fuctos fg, f, g, fg 2 g f g f 2 d g 3 g f 2 elow our equlty: fg 0 σ g f, Δ,x σ f,δ,x σ g, Δ,x σ g, Δ,x σ f,δ,x fg 2 σ g f, Δ,x 2 σ f,δ,x σg, Δ,x 2 σ g, Δ,x 2 σ f,δ,x 2 σ f,δ,x g 3 σ σ g, Δ,x g f, Δ σ g, Δ,x 3,x 2 σ g, Δ,x σ f,δ,x 2 d fg 0 σ g f, Δ,x σ f,δ,x σ g, Δ,x σ g, Δ,x σ f,δ,x 4 M m M σ g, Δ 2 m 2,x, where Δ =x 0,x,, x s the dvso, x re the termedte pots d m fx gx M<, {,, } Whe teds to fty, we ot the equltes Refereces [] D M Btetu-Gurgu, D Mrghdu d O T Pop, A ew geerlzto of Rdo s equltes d pplctos, Cretve Mth Iform, 20 20, No, [2] D M Btetu-Gurgu, N Stcu, Oe equlty d some pplctos, Jourl of Scece d Arts, 203, 2, 23, 3-34 [3] D M Btetu-Gurgu, N Stcu, Some geerlztos of IMO equlty, Jourl of Scece d Arts, 203, 2, 23, [4] D Mrghdu, Geerlztos d refemets for Bergstrom d Rdo s equltes, Jourl of Scece d Arts, 2008, 8,, [5] J Mooj, Iequltes v power seres d Cuchy-Schwrz equlty, J Koree Soc Mth Educ Ser B: Pure Appl Mth, Vol 9, Numer 3 August 202, [6] C Mortc, A ew refemet of the Rdo equlty, Mth Commu, 6 20, [7] C Mortc, A Power Seres Approch to Some Iequltes, The Amerc Mthemtcl Mothly, Vol 9, No 2Ferury 202, pp 47-5

10 864 Lored Curdru [8] J Rdo, Uer de solut ddtve Megefutoe, Weer Stzugser 22 93, [9] A Rtu, N Mculete, Severl refemets d couterprts of Rdo s equlty, sumtted Receved: August 3, 203