
设f(z)在|z|<=1上解析,并且|f(z)|<=1,试证明|f'(0)|<=1
设f(z)在|z|<=1上解析,并且|f(z)|<=1,试证明|f'(0)|<=1详细一点的过程,谢谢...
设f(z)在|z|<=1上解析,并且|f(z)|<=1,试证明|f'(0)|<=1
详细一点的过程,谢谢 展开
详细一点的过程,谢谢 展开
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由Cauchy积分公式, f'(0) = 1/(2πi)·∫{|z| = 1} f(z)/z² dz.
故|f'(0)| = 1/(2π)·|∫{|z| = 1} f(z)/z² dz|
≤ 1/(2π)·∫{|z| = 1} |f(z)/z²| |dz|
= 1/(2π)·∫{|z| = 1} |f(z)| |dz|
≤ 1/(2π)·∫{|z| = 1} 1 |dz|
= 1.
故|f'(0)| = 1/(2π)·|∫{|z| = 1} f(z)/z² dz|
≤ 1/(2π)·∫{|z| = 1} |f(z)/z²| |dz|
= 1/(2π)·∫{|z| = 1} |f(z)| |dz|
≤ 1/(2π)·∫{|z| = 1} 1 |dz|
= 1.
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