【摘 要】
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In the article,we prove that the double inequalities Gp[λ1a+(1-λ1)b,λ1b+(1-λ1)a]A1-p(a,b)＜T[A(a,b),G(a,b)]＜Gp[μ1a+(1-μ1)b,μ1b+(1-μ1)a]A 1-p(a,b),Cs[λ2a+(
【机 构】
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School of Mechanical and Electrical Engineering,Huzhou Vocational&Technical College,Huzhou 313000,Ch
论文部分内容阅读
In the article,we prove that the double inequalities Gp[λ1a+(1-λ1)b,λ1b+(1-λ1)a]A1-p(a,b)＜T[A(a,b),G(a,b)]＜Gp[μ1a+(1-μ1)b,μ1b+(1-μ1)a]A 1-p(a,b),Cs[λ2a+(1-λ2)b,λ2b+(1-λ2)a]A1-s(a,b)＜T[A(a,b),Q(a,b)]＜Cs[μ2a+(1-μ2)b,μ2b+(1-μ2)a]A1-p(a,b)hold for all a,b＞0 with a ≠ b if and only if λ1≤1/2-√1-(2/π)2/p/2,μ1≥1/2-√2p/(4p),λ2≤1/2+√23/(2s)(ε(√2/2)/π)1/s-1/2 and μ2≥1/2+√s/(4s)if λ1,μ1 ∈(0,1/2),λ2,μ2 ∈(1/2,1),p≥1 and s≥1/2,where G(a,b)= √ab,A(a,b)=(a+b)/2,T(a,b)=2 ∫π/20 √a2 cos2t+b2 sin2 tdt/π,Q(a,b)=√(a2+b2)/2,C(a,b)=(a2+b2)/(a+b)andε(r)= ∫0π/2 √1-r2 sin2 tdt.
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