JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2026, Vol. 61 ›› Issue (8): 99-115.doi: 10.6040/j.issn.1671-9352.0.2024.333
RONG Luqing, LIAO Zhongqi, PANG Chuanlong
CLC Number:
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Mathematics in Practice and Theory, 2019, 49(8):79-88. 附录 命题1证明 (1)(əw*1)/(əI)=(g0)/2>0;(əp*1)/(əI)=(3g0)/4>0;(əπ*s1)/(əI)=(g0(α+Ig0+λg0))/4>0; (əπ*m1)/(əI)=(g0(α+Ig0+λg0))/8>0;(əπ*sc1)/(əI)=(3g0(α+Ig0+λg0))/8>0; (2)(əw*1)/(əλ)=-(g0)/2<0;(əp*1)/(əλ)=-(g0)/4<0;(əπ*s1)/(əλ)=(g0(α+Ig0+λg0))/4>0; (əπ*m1)/(əλ)=(g0(α+Ig0+λg0))/8>0;(əπ*sc1)/(əλ)=(3g0(α+Ig0+λg0))/8>0,证明完毕。 命题2证明 (əw*2)/(əI)=(g0)/2>0;(əp*2)/(əI)=(3g0)/4>0;(əπ*s2)/(əI)=(g0(α+Ig0))/4>0; (əπ*m2)/(əI)=(g0(α+Ig0))/8>0;(əπ*sc2)/(əI)=(3g0(α+Ig0))/8>0,证明完毕。 命题3证明 (1)(əπ*s3)/(əλ)=(2η(α+Ig0+λg0)(Iα+λα+8ηg0))/((2Iλ+λ2-8η+I2)2)>0; (əπ*m3)/(əλ)=(4η2(α+Ig0+λg0)[2g0(2Iλ+λ2-8η+I2)-4(α+Ig0+λg0)(I+λ)] )/((2Iλ+λ2-8η+I2)3)>0; (əπ*sc3)/(əλ)=(-2η(αI+2αλ+2ηg0)[g0((I+λ)2-12η)((I+λ)2-8η)-(α+Ig0+λg0)(I+λ)((I+λ)2-16η)] )/((2Iλ+λ2-8η+I2)3)>0; (2)(əπ*s3)/(əI)=(2ηg0(α+Ig0+λg0)(8η-I2-2Iλ-λ2)-η(α+Ig0+λg0)2(-2I-2λ))/((8η-I2-2Iλ-λ2)2)>0; (əπ*m3)/(əI)=(8η2(α+Ig0+λg0)[g0(2Iλ+λ2-8η+I2)-2(λ+I)(α+Ig0+λg0)] )/((2Iλ+λ2-8η+I2)3)>0; (əπ*sc3)/(əI)=(2η(α+g0(I+λ))[-g0((I+λ)2-8η)((I+λ)2-12η)+(α+g0(I+λ))(λ+I)((I+λ)2-16η)] )/((2Iλ+λ2-8η+I2)3)>0; (3)(əπ*s3)/(əη)=(-(α+Ig0+λg0)2(I+λ)2)/((8η-(I+λ)2)2)<0; (əπ*m3)/(əη)=(8η(α+Ig0+λg0)2(2Iλ+λ2+I2))/((2Iλ+λ2-8η+I2)3)<0; (əπ*sc3)/(əη)=(-(α+Ig0+λg0)2(2Iλ+λ2+I2)((2Iλ+λ2+I2)-16η))/((2Iλ+λ2-8η+I2)3)<0,证明完毕。 命题4证明 (1)由π*s1-π*s2=((α+Ig0+λg0)2)/8-((α+Ig0)2)/8=((2α+2Ig0+λg0)λg0)/8>0得,π*s1>π*s2, 由π*s3-π*s1=(-(I+λ)2(α+Ig0+λg0)2)/(8(2Iλ+λ2-8η+I2))>0得,π*s3>π*s1,因此π*s3>π*s1>π*s2得证; (2)由π*m1-π*m2=((α+Ig0+λg0)2)/(16)-((α+Ig0)2)/(16)=((2α+2Ig0+λg0)λg0)/(16)>0得,π*m1>π*m2,由π*m3-π*m1=(-(2Iλ+λ2-16η+I2)(α+Ig0+λg0)2(I+λ)2)/(16(2Iλ+λ2-8η+I2)2)>0得;π*m3>π*m1, 因此π*m3>π*m1>π*m2得证; (3)由(1)、(2)且πsc=πm+πs易得π*sc3>π*sc1>π*sc2,证明完毕。 命题5证明 (1)(əw*4)/(ək)=(4η(-4η+Iλ+I2)[α+g0(I+λ)] )/([(4k-8)η+2Iλ+λ2+I2] 2),因此当0<η<(I(I+λ))/4时,(əw*4)/(ək)>0,当(I(I+λ))/4<η<1时,(əw*4)/(ək)<0;(əg*4)/(ək)=(4η(α+Ig0+λg0)(I+λ))/((4ηk+λ2-8η+I2+2Iλ)2)>0;(əp*4)/(ək)=(4η[α+g0(I+λ)] (-2η+I2+Iλ))/((4ηk+λ2-8η+I2+2Iλ)2), 因此当0<η<(I(I+λ))/2时,(əp*4)/(ək)>0,当(I(I+λ))/2<η<1时,(əp*4)/(ək)<0;(əU*s4)/(ək)=(4η2(α+Ig0+λg0)2)/((4ηk+λ2-8η+I2+2Iλ)2)>0;(əU*m4)/(ək)=-(32η3(α+Ig0+λg0)2)/((4ηk+λ2-8η+I2+2Iλ)3)>0;(əU*sc4)/(ək)=(4η2(a+Ig0+λg0)2[4ηk+(λ+I)2-16η] )/([4ηk+(λ+I)2-8η] 3)>0; (2)(əw*4)/(əη)=(4(I+λ)[(λg0+(k-1)(α+Ig0))(I+λ)-αλ(k-2)] )/([(4k-8)η+(I+λ)2] 2), 因此当0 |
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