now a term is a square so its an upward parabola and the expressions is less than zero so clearly it has 2 real roots and the expressions indicates those points where the curve is bellow x axis between the 2 roots!!!

here D < 0 since here the roots are not there

 

so 4(ab + bc) <= 4(a² + b²) (b² + c²)   //= bcoz expressions has <=

 

after cancelling terms we have

 

2ab²c <= (ac)² + b⁴  

 

Dividing by ab²c

 

2 <= ac/b² + b²/ac

 

thus t + 1/t >= 2

 

(t - 1/t)² >= 0 which is always true but t is not zero

 

t  >= 1/t

 

t² -1 >=  0  (t-1)(1+1) >= 0

t <= -1   and t >= 1

 

for t = 1 we have a GP here 's ur answr!!!!

Rate me if i am correct