Sharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means

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1 Qian t al. Journal of Inqualitis and Applications (015) 015:1 DOI /s R E S E A R C H Opn Accss Sharp bounds for Sándor man in trms of arithmtic, gomtric and harmonic mans Wi-Mao Qian 1, Yu-Ming Chu * and Xiao-Hui Zhang * Corrspondnc: School of Mathmatics and Computation Scincs, Hunan City Univrsity, Yiyang, 41000, China Full list of author information is availabl at th nd of th articl Abstract In th articl, w prsnt th bst possibl paramtrs α 1, α, β 1, β (0, 1) and α, α 4, β, β 4 (0, 1/) such that th doubl inqualitis α 1 A(a, b)+(1 α 1 )H(a, b)<x(a, b)<β 1 A(a, b)+(1 β 1 )H(a, b), α A(a, b)+(1 α )G(a, b)<x(a, b)<β A(a, b)+(1 β )G(a, b), H [ α a +(1 α )b, α b +(1 α )a ] < X(a, b)<h [ β a +(1 β )b, β b +(1 β )a ], G [ α 4 a +(1 α 4 )b, α 4 b +(1 α 4 )a ] < X(a, b)<g [ β 4 a +(1 β 4 )b, β 4 b +(1 β 4 )a ] hold for all a, b >0witha b.hr,x(a, b), A(a, b), G(a, b)andh(a, b)arthsándor, arithmtic, gomtric and harmonic mans of a andb, rspctivly. MSC: 6E60 Kywords: Sándor man; arithmtic man; gomtric man; harmonic man 1 Introduction Lt r R and a, b >0witha b. Thn th harmonic man H(a, b), gomtric man G(a, b), logarithmic man L(a, b), Siffrt man P(a, b), arithmtic man A(a, b), Sándor man X(a, b)[1]andrth powr man M r (a, b)ofa and b ar, rspctivly, dfind by and H(a, b)= ab a + b, G(a, b)= a b ab, L(a, b)= log a log b, (1.1) P(a, b)= a b arcsin( a b a+b ), + b G(a,b) A(a, b)=a, X(a, b)=a(a, b) P(a,b) 1 (1.) ( a r + b r ) 1/r M r (a, b)= (r 0), M 0 (a, b)= ab. (1.) It is wll known that M r (a, b) is continuous and strictly incrasing with rspct to r R for fid a, b >0witha b, and th inqualitis H(a, b)<g(a, b)<l(a, b)<p(a, b)<a(a, b) (1.4) hold for all a, b >0witha b. 015 Qian t al. This articl is distributd undr th trms of th Crativ Commons Attribution 4.0 Intrnational Licns (http://crativcommons.org/licnss/by/4.0/), which prmits unrstrictd us, distribution, and rproduction in any mdium, providd you giv appropriat crdit to th original author(s) and th sourc, provid a link to th Crativ Commons licns, and indicat if changs wr mad.

2 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag of 1 Rcntly, th Sándor man has attractd th attntion of svral rsarchrs. In [], Sándor stablishd th inqualitis X(a, b)< P (a, b) A(a, b), A(a, b)g(a, b) A(a, b)p(a, b) < X(a, b)< P(a, b) P(a, b) G(a, b), A(a, b)l(a, b) X(a, b)> G(a,b) L(a,b) 1 A(a, b)[p(a, b)+g(a, b)], X(a, b)>, P(a, b) P(a, b) G(a, b) A [ ( (a, b)g(a, b) P(a, b)l(a, b) L(a,b) A(a,b) 1 1 < X(a, b)<a(a, b) ) ] G(a, b), P(a, b) [ ] A(a, b)+g(a, b) 4/ A(a, b)+g(a, b) P(a, b)<x(a, b)<a 1/ (a, b), P 1/(log π log ) (a, b)a 1 1/(log π log ) (a, b) [ ] A(a, b)+g(a, b) < X(a, b)<p 1 (a, b) for all a, b >0witha b. Yang t al. [] provd that th doubl inquality M p (a, b)<x(a, b)<m q (a, b) (1.5) holds for all a, b >0witha b if and only if p 1/ and q log /(1 + log )= In [4], Zhou t al. provd that th doubl inquality H α (a, b)<x(a, b)<h β (a, b) (1.6) holds for all a, b >0witha b if and only if α 1/ and β log /(1 + log )= , whr H p (a, b)=[(a p +(ab) p/ + b p )/] 1/p (p 0)andH 0 (p)= ab is th pth powr-typ Hronian man of a and b. Inqualitis (1.4) and(1.5) togthr with th idntitis H(a, b) =M 1 (a, b), G(a, b) = M 0 (a, b)anda(a, b)=m 1 (a, b)ladtothinqualitis H(a, b)<g(a, b)<x(a, b)<a(a, b) (1.7) for all a, b >0witha b. Lt a, b >0witha b, [0, 1/], f ()=H[a +(1 )b, b +(1 )a]andg()=g[a + (1 )b, b +(1 )a]. Thn both functions f and g ar continuous and strictly incrasing on [0, 1/]. Not that f (0) = H(a, b)<x(a, b)<f (1/) = A(a, b) (1.8) and g(0) = G(a, b) <X(a, b) <g(1/) = A(a, b). (1.9)

3 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag of 1 Motivatd by inqualitis (1.7)-(1.9), w naturally ask: what ar th bst possibl paramtrs α 1, α, β 1, β (0, 1) and α, α 4, β, β 4 (0, 1/) such that th doubl inqualitis α 1 A(a, b)+(1 α 1 )H(a, b)<x(a, b)<β 1 A(a, b)+(1 β 1 )H(a, b), α A(a, b)+(1 α )G(a, b)<x(a, b)<β A(a, b)+(1 β )G(a, b), H [ α a +(1 α )b, α b +(1 α )a ] < X(a, b)<h [ β a +(1 β )b, β b +(1 β )a ], G [ α 4 a +(1 α 4 )b, α 4 b +(1 α 4 )a ] < X(a, b)<g [ β 4 a +(1 β 4 )b, β 4 b +(1 β 4 )a ] hold for all a, b >0witha b? Th purpos of this papr is to answr this qustion. Lmmas In ordr to prov our main rsults, w nd four lmmas, which w prsnt in this sction. Lmma.1 Lt p (0, 1) and f ()= 1 [(1 p) +1] p +(1 p)(1 ) arcsin(). (.1) Thn th following statmnts ar tru: (1) If p =/, thn f ()<0for all (0, 1). () If p =1/, thn thr ists λ 1 (0, 1) such that f ()>0for (0, λ 1 ) and f ()<0for (λ 1,1). Proof Simpl computations lad to f (0) = 0, f (1) = π, (.) f ()= 1 [p +(1 p)(1 )] f 1(), (.) whr f 1 ()=(1 p) 4 (1 p)( p) + p. (.4) (1) If p =/,thn(.4)ladsto f 1 ()= ( 7 ) <0 (.5) 9 for (0, 1). Thrfor, f ()<0for (0, 1) follows asily from (.), (.)and(.5). () If p =1/,thn(.4)ladsto f 1 (0) = >0, f 1 (1) = 1 <0, (.6) f 1 ()=(1 p)[ (1 p) ( p) ] < ( 1 p ) <0 (.7) for (0, 1).

4 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag 4 of 1 From (.6) and(.7) w clarly s that thr ists λ 0 (0, 1) such that f 1 () >0for (0, λ 0 )andf 1 ()<0for (λ 0,1). W divid th proof into two cass. Cas 1. (0, λ 0 ]. Thn f () > 0 follows asily from (.)and(.)togthrwithf 1 ()>0 on th intrval (0, λ 0 ). Cas. (λ 0,1).Thn(.) andf 1 ()<0onthintrval(λ 0,1) ladtothconclusion that f () is strictly dcrasing on [λ 0,1). From (.) andf (λ 0 ) > 0 togthr with th monotonicity of f () on[λ 0, 1) w clarly s that thr ists λ 1 (λ 0,1) (0, 1) such that f ()>0for (λ 0, λ 1 )andf ()<0for (λ 1,1). Lmma. Lt p (0, 1) and g()= p 1 +(1 p) (1 p) 1 + p arcsin(). (.8) Thn th following statmnts ar tru: (1) If p = 1/, thn g()>0 for all (0, 1). () If p =1/, thn thr ists μ 1 (0, 1) such that g()<0for (0, μ 1 ) and g()>0for (μ 1,1). Proof Simpl computations lad to g(0) = 0, g(1) = 1 p 1 π, (.9) g ()= 1 [p +(1 p) 1 ] g 1(), (.10) whr g 1 ()=p(p 1) 1 +1 p p. (.11) (1) If p = 1/, thn (.11)ladsto g 1 ()= 9( 1 1 ) >0 (.1) for (0, 1). Thrfor, g()>0 for all (0,1) follows asily from (.9), (.10)and(.1). () If p =1/,thn(.11)ladsto g 1 (0) = <0, g 1 (1) = 1 > 0, (.1) g 1 p(1 p) ()= >0 (.14) 1 for all (0, 1). From (.1) and(.14) w clarly s that thr ists μ 0 (0, 1) such that g 1 ()<0for (0, μ 0 )andg 1 ()>0for (μ 0,1).

5 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag 5 of 1 W divid th proof into two cass. Cas 1. (0, μ 0 ]. Thn g()<0for (0, μ 0 ] follows asily from (.9) and(.10) togthr with g 1 ()<0onthintrval(0,μ 0 ). Cas. (μ 0,1).Thn(.10)andg 1 ()>0onthintrval(μ 0,1)ladtothconclusion that g() is strictly incrasing on [μ 0, 1). Not that g(μ 0 )<0, g(1) = 1 π > 0. (.15) From (.15) and th monotonicity of g()onthintrval[μ 0, 1) w clarly s that thr ists μ 1 (μ 0,1) (0, 1) such that g()<0for (μ 0, μ 1 )andg()>0for (μ 1,1). Lmma. Lt p (0, 1/) and h()=arcsin() 1 [1 + (1 p) ] 1 (1 p). (.16) Thn th following statmnts ar tru: (1) If p = 1/ /6= , thn h()>0for all (0, 1). () If p = 1/ 1 1//= , thn thr ists σ 1 (0, 1) such that h()<0for (0, σ 1 ) and h()>0for (σ 1,1). Proof Simpl computations lad to h(0) = 0, h(1) = π, (.17) h ()= 1 [1 (1 p) ] h 1(), (.18) whr h 1 ()= ( 16p 4 p +4p 8p +1 ) 4 + ( 16p 4 +p p +16p ) + ( 6p 6p +1 ). (.19) (1) If p = 1/ /6, thn (.19)ladsto h 1 ()= 4 9 ( 7 ) <0 (.0) for (0, 1). Thrfor, h() > 0 for all (0, 1) follows asily from (.17) and(.18) togthrwith (.10). () If p = 1/ 1 1//, thn h 1 (0) = ( 6p 6p +1 ) >0, h 1 (1) = 4p(1 p)<0, (.1) h 1 ()=4( 16p 4 p +4p 8p +1 ) + ( 16p 4 +p p +16p ). (.)

6 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag 6 of 1 Not that 16p 4 p +4p 8p +1= >0, (.) 16p 4 p +16p 1 = < 0. (.4) It follows from (.)-(.4)that h 1 () <4( 16p 4 p +4p 8p +1 ) + ( 16p 4 +p p +16p ) = ( 16p 4 p +16p 1 ) <0 (.5) for (0, 1). From (.1) and(.5) w clarly s that thr ists σ 0 (0, 1) such that h 1 ()>0for (0, σ 0 )andh 1 ()<0for (σ 0,1). W divid th proof into two cass. Cas 1. (0, σ 0 ]. Thn h()<0for (0, σ 0 ] follows asily from (.17) and(.18) togthr with h 1 ()>0onthintrval(0,σ 0 ). Cas. (σ 0,1).Thn(.18)andh 1 ()<0onthintrval(σ 0,1)ladtothconclusion that h() is strictly incrasing on (σ 0, 1). Thrfor, thr ists σ 1 (σ 0,1) (0, 1) such that h() <0for (σ 0, σ 1 )andh() >0for (σ 1, 1) follows from (.17) andh(σ 0 )<0 togthr with th monotonicity of h()onthintrval(σ 0,1). Lmma.4 Lt p (0, 1/) and J()=arcsin() 1 1 (1 p). (.6) Thn th following statmnts ar tru: (1) If p = 1/ 6/6= , thn J()>0for all (0, 1). () If p = 1/ 1 1/ /= , thn thr ists τ 1 (0, 1) such that J()<0for (0, τ 1 ) and h()>0for (τ 1,1). Proof Simpl computations lad to J(0) = 0, J(1) = π, (.7) J ()= 1 [1 (1 p) ] J 1(), (.8) whr J 1 ()= ( 16p 4 p +4p 8p +1 ) ( 1p 1p +1 ). (.9) (1) If p = 1/ 6/6, thn (.9)ladsto J 1 ()= 4 9 >0 (.0) for (0, 1).

7 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag 7 of 1 Thrfor, J() > 0 for all (0, 1) follows asily from (.7) and(.8) togthrwith (.0). () If p = 1/ 1 1/ /, thn (.9)ladsto J 1 (0) = ( 1p 1p +1 ) <0, J 1 (1) = 4p ( 4p 8p +p +1 ) >0, (.1) J 1 ()=( 16p 4 p +4p 8p +1 ) >0 (.) for (0, 1). It follows from (.1) and(.) that thr ists τ 0 (0, 1) such that J 1 () <0for (0, τ 0 )andj 1 ()>0for (τ 0,1). W divid th proof into two cass. Cas 1. (0, τ 0 ]. Thn J()<0for (0, τ 0 ] follows asily from (.7) and(.8) togthr with J 1 ()<0onthintrval(0,τ 0 ). Cas. (τ 0,1).Thn(.8)andJ 1 ()>0onthintrval(τ 0,1)ladtothconclusion that J() is strictly incrasing on (τ 0,1). Thrfor, thr ists τ 1 (τ 0,1) (0, 1) such that J()<0for (τ 0, τ 1 )andj()>0 for (τ 1, 1) follows from (.7) andj(τ 0 ) < 0 togthr with th monotonicity of J() on th intrval (τ 0,1). Main rsults Thorm.1 Th doubl inquality α 1 A(a, b)+(1 α 1 )H(a, b)<x(a, b)<β 1 A(a, b)+(1 β 1 )H(a, b) holds for all a, b >0with a bifandonlyifα 1 1/ = and β 1 /. Proof Sinc H(a, b), X(a, b)anda(a, b) ar symmtric and homognous of dgr on, w assum that a > b >0.Lt =(a b)/(a + b) (0, 1) and p (0, 1). Thn (1.1)and(1.)lad to X(a, b) H(a, b) A(a, b) H(a, b) = 1 arcsin() 1 (1 ), (.1) Lt log X(a, b) pa(a, b)+(1 p)h(a, b) = 1 arcsin() 1 log [ p +(1 p) ( 1 )]. (.) 1 arcsin() F()= 1 log [ p +(1 p) ( 1 )]. (.) Thn simpl computations lad to F ( 0 +) =0, (.4) F(1) = log p 1, (.5) F ()= 1 f (), (.6) 1 whr f ()isdfindby(.1).

8 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag 8 of 1 W divid th proof into two cass. Cas 1. p =/.Thn(.)-(.4) and(.6) togthr with Lmma.1(1) lad to th conclusion that X(a, b)< A(a, b)+ 1 H(a, b). (.7) Cas. p =1/. Thn(.6) and Lmma.1() lad to th conclusion that thr ists λ 1 (0, 1) such that F() is strictly incrasing on (0, λ 1 ] and strictly dcrasing on [λ 1,1). Not that (.5)bcoms F(1) = 0. (.8) It follows from (.)-(.4) and(.8) togthr with th picwis monotonicity of F() that X(a, b)> 1 ( A(a, b)+ 1 1 ) H(a, b). (.9) Not that 1 arcsin() 1 (1 ) lim = 0 +, (.10) lim 1 1 arcsin() 1 (1 ) = 1. (.11) Thrfor, Thorm.1 follows from (.7) and(.9) in conjunction with th following statmnts. If α 1 >/,thnquations(.1) and(.10) lad to th conclusion that thr ists δ 1 (0, 1) such that X(a, b)<α 1 A(a, b)+(1 α 1 )H(a, b) for all a > b >0with (a b)/(a + b) (0, δ 1 ). If β 1 <1/, thnquations(.1) and(.11) lad to th conclusion that thr ists δ (0, 1) such that X(a, b)>β 1 A(a, b)+(1 β 1 )H(a, b) for all a > b >0with (a b)/(a + b) (1 δ,1). Thorm. Th doubl inquality α A(a, b)+(1 α )G(a, b)<x(a, b)<β A(a, b)+(1 β )G(a, b) holds for all a, b >0with a bifandonlyifα 1/ and β 1/ = Proof Sinc A(a, b), G(a, b)andx(a, b) ar symmtric and homognous of dgr on, w assum that a > b >0.Lt =(a b)/(a + b) (0, 1) and p (0, 1). Thn (1.1)and(1.)lad to X(a, b) G(a, b) A(a, b) G(a, b) = 1 arcsin() , (.1)

9 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag 9 of 1 Lt log X(a, b) pa(a, b)+(1 p)g(a, b) = 1 arcsin() 1 log [ p +(1 p) 1 ]. (.1) 1 arcsin() G()= 1 log [ p +(1 p) 1 ]. (.14) Thn simpl computations lad to G ( 0 +) = 0, (.15) G(1) = log p 1, (.16) G ()= 1 g(), (.17) 1 whr g()isdfindby(.8). W divid th proof into two cass. Cas 1. p = 1/. Thn (.1)-(.15) and(.17) togthr with Lmma.(1) lad to th conclusion that X(a, b)> 1 A(a, b)+ G(a, b). (.18) Cas. p =1/. Thn from Lmma.() and (.17) w know that thr ists μ 1 (0, 1) such that G() is strictly dcrasing on (0, μ 1 ] and strictly incrasing on [μ 1, 1). Not that (.16)bcoms G(1) = 0. (.19) It follows from (.1)-(.15)and(.19) togthr with th picwis monotonicity of G() that X(a, b)< 1 ( A(a, b)+ 1 1 ) G(a, b). (.0) Not that 1 arcsin() 1 1 lim = 1 1, (.1) 1 arcsin() 1 1 lim 1 1 = 1 1. (.) Thrfor, Thorm. follows asily from (.1) and(.18) togthrwith(.0)- (.). Thorm. Lt α, β (0, 1/). Thn th doubl inquality H [ α a +(1 α )b, α b +(1 α )a ] < X(a, b)<h [ β a +(1 β )b, β b +(1 β )a ]

10 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag 10 of 1 holds for all a, b >0with a bifandonlyifα 1/ 1 1//= and β 1/ /6= Proof Sinc H(a, b)andx(a, b) ar symmtric and homognous of dgr on, w assum that a > b >0.Lt =(a b)/(a + b) (0, 1) and p (0, 1/). Thn (1.1)and(1.)ladto Lt log H[pa +(1 p)b, pb +(1 p)a] X(a, b) = log [ 1 (1 p) ] 1 arcsin() +1. (.) H()=log [ 1 (1 p) ] 1 arcsin() +1. (.4) Thn simpl computations lad to H ( 0 +) =0, (.5) H(1) = 1 + log +log ( p p ), (.6) H ()= 1 h(), (.7) 1 whr h()isdfindby(.16). W divid th proof into four cass. Cas 1. p = 1/ /6. Thn (.)-(.5) and(.7) togthr with Lmma.(1) lad to [( 1 X(a, b) <H 6 ) ( 1 a ) ( 1 b, 6 ) ( 1 b ) ] a. Cas. 0 < p < 1/ /6. Lt q =(1 p) and 0 +, thn 1/ < q < 1 and powr sris pansion lads to ( H()= q 1 ) + o ( ). (.8) Equations (.), (.4)and(.8) lad to th conclusion that thr ists 0 < δ <1such that X(a, b)>h [ pa +(1 p)b, pb +(1 p)a ] (.9) for all a > b >0with(a b)/(a + b) (0, δ). Cas. p = 1/ 1 1//. Thn (.7) and Lmma.() lad to th conclusion that thr ists σ 1 (0, 1) such that H() is strictly dcrasing on (0, σ 1 ] and strictly incrasing on [σ 1,1). Not that (.6)bcoms H(1) = 0. (.0)

11 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag 11 of 1 Thrfor, [( 1 X(a, b) >H 1 1 ) ( 1 a ) ( 1 b, 1 1 ) ( 1 b follows from (.)-(.5)and(.0) togthr with th picwis monotonicity of H(). Cas 4. 1/ 1 1// < p < 1/. Thn (.6)ladsto ) ] a H(1) > 0. (.1) Equations (.) and(.4) togthr with inquality (.1) imply that thr ists 0 < δ <1suchthat X(a, b)<h [ pa +(1 p)b, pb +(1 p)a ] for a > b >0with(a b)(a + b) (1 δ,1). Thorm.4 Lt α 4, β 4 (0, 1/). Thn th doubl inquality G [ α 4 a +(1 α 4 )b, α 4 b +(1 α 4 )a ] < X(a, b)<g [ β 4 a +(1 β 4 )b, β 4 b +(1 β 4 )a ] holds for all a, b >0with a bifandonlyifα 4 1/ 1 1/ /= and β 4 1/ 6/6= Proof Sinc G(a, b)andx(a, b) ar symmtric and homognous of dgr on, w assum that a > b >0.Lt =(a b)/(a + b) (0, 1) and p (0, 1/). Thn (1.1)and(1.)ladto G[pa +(1 p)b, pb +(1 p)a] log X(a, b) = 1 log[ 1 (1 p) ] 1 arcsin() +1. (.) Lt K()= 1 log[ 1 (1 p) ] 1 arcsin() +1. (.) Thn simpl computations lad to K ( 0 +) =0, (.4) K(1) = 1 + log + 1 log( p p ), (.5) K ()= 1 J(), (.6) 1 whr J()isdfindby(.6). W divid th proof into four cass.

12 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag 1 of 1 Cas 1. p = p 0 = 1/ 6/6. Thn X(a, b)<g [ p 0 a +(1 p 0 )b, p 0 b +(1 p 0 )a ] follows from (.)-(.4)and(.6) togthr with Lmma.4(1). Cas. 0 < p < 1/ 6/6. Lt q =(1 p) and 0 +, thn / < q < 1 and powr sris pansion lads to K()= 1 ( q ) + o ( ). (.7) From (.), (.)and(.7) w clarly s that thr ists 0 < δ <1suchthat X(a, b)>g [ pa +(1 p)b, pb +(1 p)a ] for a > b >0with(a b)/(a + b) (0, δ). Cas. p = p 1 = 1/ 1 1/ /. Thn (.6) and Lmma.4() lad to th conclusion that thr ists τ 1 (0, 1) such that K() isstrictlydcrasingon(0,τ 1 ]andstrictly incrasing on [τ 1,1). Not that (.5)bcoms K(1) = 0. (.8) Thrfor, X(a, b)>g [ p 1 a +(1 p 1 )b, p 1 b +(1 p 1 )a ] follows from (.)-(.4)and(.8) togthr with th picwis monotonicity of K(). Cas 4. 1/ 1 1/ / < p < 1/. Thn (.5)ladsto K(1) > 0. (.9) Equations (.) and(.) togthr with inquality (.9) imply that thr ists 0 < δ <1suchthat X(a, b)<g [ pa +(1 p)b, pb +(1 p)a ] (.40) for a > b >0with(a b)/(a + b) (1 δ,1). Compting intrsts Th authors dclar that thy hav no compting intrsts. Authors contributions All authors contributd qually to th writing of this papr. All authors rad and approvd th final manuscript. Author dtails 1 School of Distanc Education, Huzhou Broadcast and TV Univrsity, Huzhou, 1000, China. School of Mathmatics and Computation Scincs, Hunan City Univrsity, Yiyang, 41000, China.

13 Qian t al. Journal of Inqualitis and Applications (015) 015:1 Pag 1 of 1 Acknowldgmnts Th rsarch was supportd by th Natural Scinc Foundation of China undr Grants , and , and th Natural Scinc Foundation of Zhjiang Provinc undr Grant LY1A Rcivd: Fbruary 015 Accptd: Jun 015 Rfrncs 1. Sándor, J: Two sharp inqualitis for trigonomtric and hyprbolic functions. Math. Inqual. Appl. 15(), (01). Sándor, J: On two nw mans of two variabls. Nots Numbr Thory Discrt Math. 0(1), 1-9 (014). Yang, Z-H, Wu, L-M, Chu, Y-M: Sharp powr man bounds for Sándor man. Abstr. Appl. Anal. 014, Articl ID (014) 4. Zhou, S-S, Qian, W-M, Chu, Y-M, Zhang, X-H: Sharp powr-typ Hronian man bounds for th Sándor and Yang mans. J. Inqual. Appl. 015, Articl ID 159 (015)

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