Show that #|z+1|+|1+z+z^2|+|1+z^3|>=1# ?

If #z##in##CC# , show that #|z+1|+|1+z+z^2|+|1+z^3|>=1#

1 Answer
Jim S
May 11, 2018
  • For #|z|>=1#

#|z+1|+|z^2+z+1|>=|(z^2+z+1)-(z+1)|=|z^2|=|z|^2>=1#

  • For #|z|<1#

#|z+1|+|z^2+z+1|>=|z||z+1|+|z^2+z+1|=#

#|z(z+1)|+|z^2+z+1|=|z^2+z|+|z^2+z+1|>=|(z^2+z+1)-(z^2+z)|=1#

Hence, #|z+1|+|1+z+z^2|>=1# , #z##in##CC#

and

#|z+1|+|1+z+z^2|+|1+z^3|>=|1+z|+|1+z+z^2|>=1#

,
"#=#" , #z=-1vvz=e^((2k+1)iπ)# , #k##in##ZZ#

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