Let K(r)be the complete elliptic integrals of the first kind for r∈(0,1)and f_(p)(x)=[(1−x)^(p)K(√x)].Using the recurrence method,we find the necessary and sufficient conditions for the functions−f′_(p),ln f_(p),−(...Let K(r)be the complete elliptic integrals of the first kind for r∈(0,1)and f_(p)(x)=[(1−x)^(p)K(√x)].Using the recurrence method,we find the necessary and sufficient conditions for the functions−f′_(p),ln f_(p),−(ln f_(p))^((i))(i=1,2,3)to be absolutely monotonic on(0,1).As applications,we establish some new bounds for the ratios and the product of two complete integrals of the first kind,including the double inequalities exp[r^(2)(1−r^(2))/^(64)]/(1+r)^(1/4)<K(r)/K(√r)<exp[−r(1−r)/4],π/2 exp[θ0(1−2r^(2))]<π/2 K(r′)/K(r)<π/2(r′/r)^(p)exp[θ_(p)(1−2r^(2))],K^(2)(1/√2)≤K(r)K(r′)≤1/√2rr′K^(2)(1/√2)for r∈2(0,1)and p≥13/32,where r′=√1−r^(2) and θ_(p)=2Γ(3/4)^(4)/π^(2)−p.展开更多
In the article,we prove that the inequalities H_(p)(K(r);E(r))>π/2;L_(q)(K(r);E(r))>π/2 hold for all r 2(0;1)if and only if p≥3=4 and q≥3=4,where Hp(a;b)and Lq(a;b)are respectively the p-th power-type Heroni...In the article,we prove that the inequalities H_(p)(K(r);E(r))>π/2;L_(q)(K(r);E(r))>π/2 hold for all r 2(0;1)if and only if p≥3=4 and q≥3=4,where Hp(a;b)and Lq(a;b)are respectively the p-th power-type Heronian mean and q-th Lehmer mean of a and b,and K(r)and E(r)are respectively the complete elliptic integrals of the first and second kinds.展开更多
In this paper,we present new bounds for the perimeter of an ellipse in terms of harmonic,geometric,arithmetic and quadratic means;these new bounds represent improvements upon some previously known results.
In the article, we present some refinements of three classes of transformation inequalities for zero-balanced hypergeometric functions by use of the updated monotonicity criterion for the quotient of power series.
In the article,we prove that the double inequalities Gp[λ1a+(1-λ1)b,λ1 b+(1-λ1)a]A1-p(a,b)<T[A(a,b),G(a,b)]<Gp[μ1 a+(1-μ1)b,μ1b+(1-μ1)a]A1-p(a,b),Cs[λ^(2) a+(1-λ2)b,λ2 b+(1-λ2)a]A1-s(a,b)<T[A(a,b)...In the article,we prove that the double inequalities Gp[λ1a+(1-λ1)b,λ1 b+(1-λ1)a]A1-p(a,b)<T[A(a,b),G(a,b)]<Gp[μ1 a+(1-μ1)b,μ1b+(1-μ1)a]A1-p(a,b),Cs[λ^(2) a+(1-λ2)b,λ2 b+(1-λ2)a]A1-s(a,b)<T[A(a,b),Q(a,b)]<Cs[μ2 a+(1-μ2)b,μ2 b+(1-μ2)a]A1-p(a,b)hold for all a,b>0 with a≠b if and only ifλ1≤1/2-(1-(2/π)2/p)1/2/2,μ1≥1/2-(2p)1/2/(4 p),λ2≤1/2+(2(3/(2 s)(E(21/2/2)/π)1/s)-1)1/2/2 andμ2≥1/2+s1/2/(4 s)ifλ1,μ1∈(0,1/2),λ2,μ2∈(1/2,1),p≥1 and s≥1/2,where G(a,b)=(ab)1/2,A(a,b)=(a+b)/2,T(a,b)=∫0π/2(a2 cos2 t+b2 sin2)1/2 tdt/π,Q(a,b)=((a2+b2)/2)1/2,C(a,b)=(a2+b2)/(a+b)and E(r)=∫0π/2(1-r^(2) sin^(2))1/2 tdt.展开更多
文摘Let K(r)be the complete elliptic integrals of the first kind for r∈(0,1)and f_(p)(x)=[(1−x)^(p)K(√x)].Using the recurrence method,we find the necessary and sufficient conditions for the functions−f′_(p),ln f_(p),−(ln f_(p))^((i))(i=1,2,3)to be absolutely monotonic on(0,1).As applications,we establish some new bounds for the ratios and the product of two complete integrals of the first kind,including the double inequalities exp[r^(2)(1−r^(2))/^(64)]/(1+r)^(1/4)<K(r)/K(√r)<exp[−r(1−r)/4],π/2 exp[θ0(1−2r^(2))]<π/2 K(r′)/K(r)<π/2(r′/r)^(p)exp[θ_(p)(1−2r^(2))],K^(2)(1/√2)≤K(r)K(r′)≤1/√2rr′K^(2)(1/√2)for r∈2(0,1)and p≥13/32,where r′=√1−r^(2) and θ_(p)=2Γ(3/4)^(4)/π^(2)−p.
基金Supported by the National Natural Science Foundation of China(11971142)the Natural Science Foundation of Zhejiang Province(LY19A010012)。
文摘In the article,we prove that the inequalities H_(p)(K(r);E(r))>π/2;L_(q)(K(r);E(r))>π/2 hold for all r 2(0;1)if and only if p≥3=4 and q≥3=4,where Hp(a;b)and Lq(a;b)are respectively the p-th power-type Heronian mean and q-th Lehmer mean of a and b,and K(r)and E(r)are respectively the complete elliptic integrals of the first and second kinds.
基金supported by the Natural Science Foundation of China(11971142)the Natural Science Foundation of Zhejiang Province(LY19A010012)。
文摘In this paper,we present new bounds for the perimeter of an ellipse in terms of harmonic,geometric,arithmetic and quadratic means;these new bounds represent improvements upon some previously known results.
基金supported by the Natural Science Foundation of China(61673169,11401191,11371125)the Tianyuan Special Funds of the Natural Science Foundation of China(11626101)the Natural Science Foundation of the Department of Education of Zhejiang Province(201635325)
文摘In the article, we present some refinements of three classes of transformation inequalities for zero-balanced hypergeometric functions by use of the updated monotonicity criterion for the quotient of power series.
基金supported by the Natural Science Foundation of China(61673169,11301127,11701176,11626101,11601485)。
文摘In the article,we prove that the double inequalities Gp[λ1a+(1-λ1)b,λ1 b+(1-λ1)a]A1-p(a,b)<T[A(a,b),G(a,b)]<Gp[μ1 a+(1-μ1)b,μ1b+(1-μ1)a]A1-p(a,b),Cs[λ^(2) a+(1-λ2)b,λ2 b+(1-λ2)a]A1-s(a,b)<T[A(a,b),Q(a,b)]<Cs[μ2 a+(1-μ2)b,μ2 b+(1-μ2)a]A1-p(a,b)hold for all a,b>0 with a≠b if and only ifλ1≤1/2-(1-(2/π)2/p)1/2/2,μ1≥1/2-(2p)1/2/(4 p),λ2≤1/2+(2(3/(2 s)(E(21/2/2)/π)1/s)-1)1/2/2 andμ2≥1/2+s1/2/(4 s)ifλ1,μ1∈(0,1/2),λ2,μ2∈(1/2,1),p≥1 and s≥1/2,where G(a,b)=(ab)1/2,A(a,b)=(a+b)/2,T(a,b)=∫0π/2(a2 cos2 t+b2 sin2)1/2 tdt/π,Q(a,b)=((a2+b2)/2)1/2,C(a,b)=(a2+b2)/(a+b)and E(r)=∫0π/2(1-r^(2) sin^(2))1/2 tdt.
