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.“”“”..1———f(x)g(x)[a,b]f(x)≤g(x),x∈[a,b]ba乙f(x)dx≤ba乙g(x)dx.1f(x)[0,1]0＜α＜β＜1α0乙f(x)dxαββα乙f(x)dxf(x)0x≤α＜1f(x)≥f(α)α0乙f(x)dx≥βα乙f(α)dx=αf(α)1βα乙f(x)dx≤βα乙f(α)dx=(β-α)f(α)2121αα0乙f(x)dx≥f(α)≥1β-αβα乙f(x)dx33α(β-α)ββ-αβα0乙f(x)dx≥αββα乙f(α)dxβ-αβ1α0乙f(x)dxαββα乙f(x)dx210乙cosx1-x2姨dx≥10乙sinx1-x2姨dx..t=arccosxt=arcsinxπ20乙cos(sint)dt≥π20乙sin(cost)dtcos(sint)dt≥sin(cost)dtt∈0π2cos(sint)=sin(π2-sint)≥sin(c0st).t∈(0,π2)0cistπ20π2-sintπ2y=sinx(0,π2)t∈(0,π2)cost≤π2-sintcost+sint≤π2.t∈(0,π2)cost+sint=2姨sin(t+π2)≤2姨π2.2———f(x)[a,b]Φ(x)=xa乙f(t)dt,x∈[a,b]Φ[a,b]Φ'(x)=f(x).Φf(x)[a,b].f(x)[a,b](a,b)(a,b)f'(x)≥0f'(x)≤0f(x)[a,b].1f(x)[a,b]2f(x)f'(x)3423.3f(x)[a,b]ba乙xf(x)dx≥a+b2ba乙xf(x)dx.bx317000.O172.2A1673-260X201009-0014-02Vol.26No.9Sep.201026920109JournalofChifengUniversityNaturalScienceEdition14--F(x)=xa乙tf(t)dt-a+x2xa乙f(t)dtF(b)≥F(a)=0.F(x)=xa乙tf(t)dt-a+x2xa乙f(t)dtF(x)[a,b](a,b)F'(x)=12[(x-a)f(x)-xa乙f(t)dt]=12xa乙[f(x)-f(t)]dt∵f(x)[a,b]∴g'(x)≥0g(x)[a,b]∵F(a)=0∴F(b)0∴ba乙xf(x)dx-a+b2ba乙f(x)dx≥0∴ba乙xf(x)dx≥a+b2ba乙f(x)dx4f(x)g(x)[0,1]f(0)=0f'(x)≥0g'(x)≥0a∈[0,1]a0乙g(x)f'(x)dx-10乙f(x)g'(x)dx≥f(x)g(l)axF(x)=x0乙g(t)f'(t)dt-10乙f(t)g'(t)dt-f(x)g(l)F(a)≥0.F(x)=x0乙g(t)f'(t)dt-10乙f(t)g'(t)dt-f(x)g(l)x∈[0,1]F(x)[0,1](0,1)F'(x)=g(x)f'(x)-f'(x)g(l)=f'(x)[g(x)-g(l)]∵g'(x)≥0f'(x)≥0∴g(x)≤g(l)F'(x)≤0∴F(x)≥F(l)=10乙g(t)f'(t)dt-10乙f(t)g'(t)dt-f(l)g(l)=10乙d[g(x)f(x)]-f(l)g(l)=[f(l)g(l)-f(0)g(0)]-f(l)g(l)=0F(x)≥0x∈[0,1].∴a∈[0,1]a0乙g(x)f'(x)dx-10乙f(x)g'(x)dx≥f(a)g(l).5f(x)[0,1]x∈0,1)0f'x1f(0)=0[10乙f(x)dx]210乙f3(x)dx1x.t∈[0,1]F(t)=[10乙f(x)dx]2-10乙f3(x)dx,F(0)=0F'(t)=2f(t)[2[10乙f(t)dx-f2(t)]H(t)=210乙f(t)dx-f2(t)]H(0)=0H'(t)=2f(t)-2f(t)f'(t)=2f(t)[1-f'(t)]∵0f'(t)1∴f(t)[0,1].∴f(t)f(0)=0∴H'(t)≥0∴t∈[0,1]H(t)≥H(0)=0F'(t)0t∈[0,1]∴F(t)[0,1]F(l)F(0)=0[10乙f(x)dx]210乙f3(x)dxx0乙f(x)dx.———————————————————：〔1〕.()[M].:,2001.〔2〕.[J].,2007(10).〔3〕，.[M].:,2009.〔4〕.[M].:,2009.15--