Here is the question 6.4 in the book Quantum optics an introduction Fox M.

The purpose of this exercise is to establish eqn 6.12 by a method similar to the one used in the previous exercise.

a, Use the Cauchy's inequality to show that ...

PLEASE SOLVE IT QUICK, I NEED IT TOMORROW.

(6.4) The purpose of this exercise is to establish eqn 6.12 by a method similar to the one used in the previous exercise. (a) Use Cauchy's inequality to show that: i=1NI(ti)I(ti+)21i=1N(I(ti)2+I(ti+)2). (b) Explain why, in any stationary light source (i.e. one in which the averages do not vary with time), we expect: iI(ti)2=iI(ti+)2. (c) The argument of part (b) allows us to rewrite the result of part (a) in the form: i=1NI(ti)I(ti+)i=1NI(ti)2. Following the definition of time averages given in part (c) of the previous exercise, we can write: I(t)I(t+)=N1i=1NI(ti)I(ti+). Use this definition to show that: I(t)I(t+)I(t)2. Hence derive eqn 6.12. (6.4) The purpose of this exercise is to establish eqn 6.12 by a method similar to the one used in the previous exercise. (a) Use Cauchy's inequality to show that: i=1NI(ti)I(ti+)21i=1N(I(ti)2+I(ti+)2). (b) Explain why, in any stationary light source (i.e. one in which the averages do not vary with time), we expect: iI(ti)2=iI(ti+)2. (c) The argument of part (b) allows us to rewrite the result of part (a) in the form: i=1NI(ti)I(ti+)i=1NI(ti)2. Following the definition of time averages given in part (c) of the previous exercise, we can write: I(t)I(t+)=N1i=1NI(ti)I(ti+). Use this definition to show that: I(t)I(t+)I(t)2. Hence derive eqn 6.12